Internet DRAFT - draft-jonsson-pkcs1-v2dot1

draft-jonsson-pkcs1-v2dot1







Network Working Group                                         J. Jonsson
Internet Draft                                                B. Kaliski
expires in six months                                   RSA Laboratories
<draft-jonsson-pkcs1-v2dot1-00.txt>                          August 2002

                PKCS #1: RSA Cryptography Specifications
                              Version 2.1



Status of this Memo

   This document is an Internet-Draft and is subject to all provisions
   of Section 10 of RFC2026 except that the right to produce derivative
   works is not granted.

   Internet-Drafts are working documents of the Internet Engineering
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Copyright Notice

   Copyright (C) The Internet Society (2002).  All Rights Reserved.

Abstract

   This memo represents a republication of PKCS #1 v2.1 from RSA
   Laboratories' Public-Key Cryptography Standards (PKCS) series, and
   change control is retained within the PKCS process.  The body of
   this document is taken directly from the PKCS #1 v2.1 document.

   This document provides recommendations for the implementation of
   public-key cryptography based on the RSA algorithm.






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Table of Contents

   1.       Introduction.....................................3
   2.       Notation.........................................4
   3.       Key types........................................6
   3.1      RSA public key...................................7
   3.2      RSA private key..................................7
   4.       Data conversion primitives.......................9
   4.1      I2OSP............................................9
   4.2      OS2IP...........................................10
   5.       Cryptographic primitives........................11
   5.1      Encryption and decryption primitives............11
   5.2      Signature and verification primitives...........13
   6.       Overview of schemes.............................15
   7.       Encryption schemes..............................16
   7.1      RSAES-OAEP......................................17
   7.2      RSAES-PKCS1-v1_5................................24
   8.       Signature schemes with appendix.................28
   8.1      RSASSA-PSS......................................29
   8.2      RSASSA-PKCS1-v1_5...............................32
   9        Encoding methods for signatures with appendix...36
   9.1      EMSA-PSS........................................37
   9.2      EMSA-PKCS1-v1_5.................................42
   A.       ASN.1 syntax....................................44
   A.1      RSA key representation..........................44
   A.2      Scheme identification...........................46
   B.       Supporting techniques...........................52
   B.1      Hash functions..................................52
   B.2      Mask Generation Functions.......................54
   C.       ASN.1 module....................................55
   D.       Intellectual property considerations............63
   E.       Revision history................................63
   F.       References......................................64
   G.       About PKCS......................................69
            Security Considerations.........................69
            Acknowledgements................................69
            Full Copyright Statement........................70














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1. Introduction

   This document provides recommendations for the implementation of
   public-key cryptography based on the RSA algorithm [42], covering
   the following aspects:

    * Cryptographic primitives

    * Encryption schemes

    * Signature schemes with appendix

    * ASN.1 syntax for representing keys and for identifying the
      schemes 

   The recommendations are intended for general application within
   computer and communications systems, and as such include a fair
   amount of flexibility. It is expected that application standards
   based on these specifications may include additional constraints.
   The recommendations are intended to be compatible with the standard
   IEEE-1363-2000 [26] and draft standards currently being developed
   by the ANSI X9F1 [1] and IEEE P1363 [27] working groups.

   This document supersedes PKCS #1 version 2.0 [35][44] but includes
   compatible techniques. 

   The organization of this document is as follows:

    * Section 1 is an introduction.

    * Section 2 defines some notation used in this document.

    * Section 3 defines the RSA public and private key types.

    * Sections 4 and 5 define several primitives, or basic
      mathematical operations. Data conversion primitives are in 
      Section 4, and cryptographic primitives (encryption-decryption,
      signature-verification) are in Section 5.

    * Sections 6, 7, and 8 deal with the encryption and signature
      schemes in this document. Section 6 gives an overview. Along
      with the methods found in PKCS #1 v1.5, Section 7 defines an
      OAEP-based [3] encryption scheme and Section 8 defines a
      PSS-based [4][5] signature scheme with appendix. 

    * Section 9 defines the encoding methods for the signature
      schemes in Section 8. 




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    * Appendix A defines the ASN.1 syntax for the keys defined in
      Section 3 and the schemes in Sections 7 and 8. 

    * Appendix B defines the hash functions and the mask generation
      function used in this document, including ASN.1 syntax for
      the techniques. 

    * Appendix C gives an ASN.1 module.

    * Appendices D, E, F and G cover intellectual property issues,
      outline the revision history of PKCS #1, give references to
      other publications and standards, and provide general
      information about the Public-Key Cryptography Standards. 

2. Notation

   c              ciphertext representative, an integer between 0
                  and n-1 

   C              ciphertext, an octet string

   d              RSA private exponent 

   d_i            additional factor r_i's CRT exponent, a positive
                  integer such that 

                     e * d_i == 1 (mod (r_i-1)), i = 3, ..., u

   dP             p's CRT exponent, a positive integer such that

                     e * dP == 1 (mod (p-1))

   dQ             q's CRT exponent, a positive integer such that 

                     e * dQ == 1 (mod (q-1))

   e              RSA public exponent

   EM             encoded message, an octet string

   emBits         (intended) length in bits of an encoded message EM

   emLen          (intended) length in octets of an encoded message EM

   GCD(. , .)     greatest common divisor of two nonnegative integers 

   Hash           hash function




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   hLen           output length in octets of hash function Hash

   k              length in octets of the RSA modulus n

   K              RSA private key

   L              optional RSAES-OAEP label, an octet string

   LCM(., ..., .) least common multiple of a list of nonnegative
                  integers 

   m              message representative, an integer between 0 and n-1

   M              message, an octet string

   mask           MGF output, an octet string

   maskLen       (intended) length of the octet string mask

   MGF           mask generation function

   mgfSeed       seed from which mask is generated, an octet string

   mLen          length in octets of a message M

   n             RSA modulus, n = r_1 * r_2 * ... * r_u , u >= 2

   (n, e)        RSA public key

   p, q          first two prime factors of the RSA modulus n

   qInv          CRT coefficient, a positive integer less than p such
                 that 

                    q * qInv == 1 (mod p)

   r_i           prime factors of the RSA modulus n, including 
                 r_1 = p, r_2 = q, and additional factors if any 

   s             signature representative, an integer between 0 and 
                 n-1

   S             signature, an octet string

   sLen          length in octets of the EMSA-PSS salt






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   t_i           additional prime factor r_i's CRT coefficient, a
                 positive integer less than r_i such that 

                    r_1 * r_2 * ... * r_(i-1) * t_i == 1 (mod r_i) , 

                 i = 3, ... , u 

   u             number of prime factors of the RSA modulus, u >= 2

   x             a nonnegative integer

   X             an octet string corresponding to x

   xLen          (intended) length of the octet string X

   0x            indicator of hexadecimal representation of an octet
                 or an octet string; "0x48" denotes the octet with
                 hexadecimal value 48; "(0x)48 09 0e" denotes the
                 string of three consecutive octets with hexadecimal
                 value 48, 09, and 0e, respectively

   \lambda(n)    LCM(r_1-1, r_2-1, ... , r_u-1)

   \xor          bit-wise exclusive-or of two octet strings

   \ceil(.)      ceiling function; \ceil(x) is the smallest integer
                 larger than or equal to the real number x 

   ||            concatenation operator

   ==            congruence symbol; a == b (mod n) means that the
                 integer n divides the integer a - b 

   Note. The CRT can be applied in a non-recursive as well as a
   recursive way. In this document a recursive approach following
   Garner's algorithm [22] is used. See also Note 1 in Section 3.2. 

3. Key types

   Two key types are employed in the primitives and schemes defined in
   this document: RSA public key and RSA private key. Together, an RSA
   public key and an RSA private key form an RSA key pair.

   This specification supports so-called "multi-prime" RSA where the
   modulus may have more than two prime factors. The benefit of
   multi-prime RSA is lower computational cost for the decryption and
   signature primitives, provided that the CRT (Chinese Remainder
   Theorem) is used. Better performance can be achieved on single



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   processor platforms, but to a greater extent on multiprocessor
   platforms, where the modular exponentiations involved can be done
   in parallel. 

   For a discussion on how multi-prime affects the security of the RSA
   cryptosystem, the reader is referred to [49]. 

3.1 RSA public key

   For the purposes of this document, an RSA public key consists of
   two components: 
   
      n        the RSA modulus, a positive integer
      e        the RSA public exponent, a positive integer

   In a valid RSA public key, the RSA modulus n is a product of u
   distinct odd primes r_i, i = 1, 2, ..., u, where u >= 2, and the
   RSA public exponent e is an integer between 3 and n - 1 satisfying
   GCD(e, \lambda(n)) = 1, where \lambda(n) = LCM(r_1 - 1, ...,
   r_u - 1). By convention, the first two primes r_1 and r_2 may also
   be denoted p and q respectively. 

   A recommended syntax for interchanging RSA public keys between
   implementations is given in Appendix A.1.1; an implementation's
   internal representation may differ. 

3.2 RSA private key

   For the purposes of this document, an RSA private key may have
   either of two representations. 

   1. The first representation consists of the pair (n, d), where the
   components have the following meanings:

      n        the RSA modulus, a positive integer
      d        the RSA private exponent, a positive integer

   2. The second representation consists of a quintuple (p, q, dP, dQ,
   qInv) and a (possibly empty) sequence of triplets (r_i, d_i, t_i),
   i = 3, ..., u, one for each prime not in the quintuple, where the
   components have the following meanings: 

      p        the first factor, a positive integer
      q        the second factor, a positive integer
      dP       the first factor's CRT exponent, a positive integer
      dQ       the second factor's CRT exponent, a positive integer
      qInv     the (first) CRT coefficient, a positive integer




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      r_i   the i-th factor, a positive integer
      d_i   the i-th factor's CRT exponent, a positive integer
      t_i   the i-th factor's CRT coefficient, a positive integer

   In a valid RSA private key with the first representation, the RSA
   modulus n is the same as in the corresponding RSA public key and is
   the product of u distinct odd primes r_i, i = 1, 2, ..., u, where u
   >= 2. The RSA private exponent d is a positive integer less than n
   satisfying 

      e * d == 1 (mod \lambda(n)), 

   where e is the corresponding RSA public exponent and \lambda(n) is
   defined as in Section 3.1.

   In a valid RSA private key with the second representation, the two
   factors p and q are the first two prime factors of the RSA modulus
   n (i.e., r_1 and r_2), the CRT exponents dP and dQ are positive
   integers less than p and q respectively satisfying 

      e * dP == 1 (mod (p-1))
      e * dQ == 1 (mod (q-1)) , 

   and the CRT coefficient qInv is a positive integer less than p
   satisfying 

      q * qInv == 1 (mod p). 

   If u > 2, the representation will include one or more triplets
   (r_i, d_i, t_i), i = 3, ..., u. The factors r_i are the additional
   prime factors of the RSA modulus n. Each CRT exponent d_i (i = 3,
   ..., u) satisfies 

      e * d_i == 1 (mod (r_i - 1)).

   Each CRT coefficient t_i (i = 3, ..., u) is a positive integer less
   than r_i satisfying 

      R_i * t_i == 1 (mod r_i) ,

   where R_i = r_1 * r_2 * ... * r_(i-1).

   A recommended syntax for interchanging RSA private keys between
   implementations, which includes components from both
   representations, is given in Appendix A.1.2; an implementation's
   internal representation may differ. 





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   Notes.

   1. The definition of the CRT coefficients here and the formulas
      that use them in the primitives in Section 5 generally follow
      Garner's algorithm [22] (see also Algorithm 14.71 in [37]). 
      However, for compatibility with the representations of RSA
      private keys in PKCS #1 v2.0 and previous versions, the roles of
      p and q are reversed compared to the rest of the primes. Thus,
      the first CRT coefficient, qInv, is defined as the inverse of 
      q mod p, rather than as the inverse of R_1 mod r_2, i.e., of 
      p mod q.    

   2. Quisquater and Couvreur [40] observed the benefit of applying
      the Chinese Remainder Theorem to RSA operations. 

4. Data conversion primitives

   Two data conversion primitives are employed in the schemes defined
   in this document:

    * I2OSP - Integer-to-Octet-String primitive

    * OS2IP - Octet-String-to-Integer primitive

   For the purposes of this document, and consistent with ASN.1
   syntax, an octet string is an ordered sequence of octets (eight-bit
   bytes). The sequence is indexed from first (conventionally,
   leftmost) to last (rightmost). For purposes of conversion to and
   from integers, the first octet is considered the most significant
   in the following conversion primitives. 

4.1 I2OSP

   I2OSP converts a nonnegative integer to an octet string of a
   specified length.

   I2OSP (x, xLen)

   Input:
   x        nonnegative integer to be converted
   xLen     intended length of the resulting octet string

   Output:
   X        corresponding octet string of length xLen

   Error: "integer too large"





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   Steps:

   1. If x >= 256^xLen, output "integer too large" and stop.

   2. Write the integer x in its unique xLen-digit representation in
      base 256: 

         x = x_(xLen-1) 256^(xLen-1) + x_(xLen-2) 256^(xLen-2) + ... 
         + x_1 256 + x_0,

      where 0 <= x_i < 256 (note that one or more leading digits will
      be zero if x is less than 256^(xLen-1)). 

   3. Let the octet X_i have the integer value x_(xLen-i) for 1 <= i
      <= xLen. Output the octet string

         X = X_1 X_2 ... X_xLen.

4.2 OS2IP

   OS2IP converts an octet string to a nonnegative integer.

   OS2IP (X)

   Input:
   X        octet string to be converted

   Output:
   x        corresponding nonnegative integer

   Steps:

   1. Let X_1 X_2 ... X_xLen be the octets of X from first to last,
      and let x_(xLen-i) be the integer value of the octet X_i for 
      1 <= i <= xLen. 

   2. Let x = x_(xLen-1) 256^(xLen-1) + x_(xLen-2) 256^(xLen-2) + ...
      + x_1 256 + x_0. 

   3. Output x.











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5. Cryptographic primitives

   Cryptographic primitives are basic mathematical operations on which
   cryptographic schemes can be built. They are intended for
   implementation in hardware or as software modules, and are not
   intended to provide security apart from a scheme. 

   Four types of primitive are specified in this document, organized
   in pairs: encryption and decryption; and signature and
   verification. 

   The specifications of the primitives assume that certain conditions
   are met by the inputs, in particular that RSA public and private
   keys are valid.

5.1 Encryption and decryption primitives

   An encryption primitive produces a ciphertext representative from a
   message representative under the control of a public key, and a
   decryption primitive recovers the message representative from the
   ciphertext representative under the control of the corresponding
   private key. 

   One pair of encryption and decryption primitives is employed in the
   encryption schemes defined in this document and is specified here:
   RSAEP/RSADP. RSAEP and RSADP involve the same mathematical
   operation, with different keys as input. 

   The primitives defined here are the same as IFEP-RSA/IFDP-RSA in
   IEEE Std 1363-2000 [26] (except that support for multi-prime RSA
   has been added) and are compatible with PKCS #1 v1.5. 

   The main mathematical operation in each primitive is
   exponentiation. 

5.1.1 RSAEP

   RSAEP ((n, e), m)

   Input:
   (n, e)   RSA public key
   m        message representative, an integer between 0 and n - 1

   Output:
   c        ciphertext representative, an integer between 0 and n - 1

   Error: "message representative out of range"




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   Assumption: RSA public key (n, e) is valid

   Steps:

   1. If the message representative m is not between 0 and n - 1,
      output "message representative out of range" and stop. 

   2. Let c = m^e mod n.

   3. Output c.

5.1.2   RSADP

   RSADP (K, c)

   Input:
   K        RSA private key, where K has one of the following forms:
             - a pair (n, d)
             - a quintuple (p, q, dP, dQ, qInv) and a possibly empty
               sequence of triplets (r_i, d_i, t_i), i = 3, ..., u 
   c        ciphertext representative, an integer between 0 and n - 1

   Output:
   m        message representative, an integer between 0 and n - 1

   Error: "ciphertext representative out of range"

   Assumption: RSA private key K is valid

   Steps:

   1. If the ciphertext representative c is not between 0 and n - 1,
      output "ciphertext representative out of range" and stop. 

   2. The message representative m is computed as follows.

      a. If the first form (n, d) of K is used, let m = c^d mod n.

      b. If the second form (p, q, dP, dQ, qInv) and (r_i, d_i, t_i)
         of K is used, proceed as follows: 

         i.    Let m_1 = c^dP mod p and m_2 = c^dQ mod q.

         ii.   If u > 2, let m_i = c^(d_i) mod r_i, i = 3, ..., u.

         iii.  Let h = (m_1 - m_2) * qInv mod p.

         iv.   Let m = m_2 + q * h.



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         v.    If u > 2, let R = r_1 and for i = 3 to u do

               1. Let R = R * r_(i-1).

               2. Let h = (m_i - m) * t_i mod r_i.

               3. Let m = m + R * h.

   3.   Output m.

   Note. Step 2.b can be rewritten as a single loop, provided that one
   reverses the order of p and q. For consistency with PKCS #1 v2.0,
   however, the first two primes p and q are treated separately from
   the additional primes. 

5.2 Signature and verification primitives

   A signature primitive produces a signature representative from a
   message representative under the control of a private key, and a
   verification primitive recovers the message representative from the
   signature representative under the control of the corresponding
   public key. One pair of signature and verification primitives is
   employed in the signature schemes defined in this document and is
   specified here: RSASP1/RSAVP1. 

   The primitives defined here are the same as IFSP-RSA1/IFVP-RSA1 in
   IEEE 1363-2000 [26] (except that support for multi-prime RSA has
   been added) and are compatible with PKCS #1 v1.5.

   The main mathematical operation in each primitive is
   exponentiation, as in the encryption and decryption primitives of
   Section 5.1. RSASP1 and RSAVP1 are the same as RSADP and RSAEP
   except for the names of their input and output arguments; they are
   distinguished as they are intended for different purposes. 

5.2.1 RSASP1

   RSASP1 (K, m)

   Input:
   K        RSA private key, where K has one of the following forms:
             - a pair (n, d)
             - a quintuple (p, q, dP, dQ, qInv) and a (possibly empty)
               sequence of triplets (r_i, d_i, t_i), i = 3, ..., u 
   m        message representative, an integer between 0 and n - 1

   Output:
   s        signature representative, an integer between 0 and n - 1



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   Error: "message representative out of range"

   Assumption: RSA private key K is valid

   Steps:

   1. If the message representative m is not between 0 and n - 1,
      output "message representative out of range" and stop. 

   2. The signature representative s is computed as follows.

      a. If the first form (n, d) of K is used, let s = m^d mod n.

      b. If the second form (p, q, dP, dQ, qInv) and (r_i, d_i, t_i)
         of K is used, proceed as follows: 

         i.    Let s_1 = m^dP mod p and s_2 = m^dQ mod q.

         ii.   If u > 2, let s_i = m^(d_i) mod r_i, i = 3, ..., u.

         iii.  Let h = (s_1 - s_2) * qInv mod p.

         iv.   Let s = s_2 + q * h.

         v.    If u > 2, let R = r_1 and for i = 3 to u do

               1. Let R = R * r_(i-1).

               2. Let h = (s_i - s) * t_i mod r_i.

               3. Let s = s + R * h.

   3. Output s.

   Note. Step 2.b can be rewritten as a single loop, provided that one
   reverses the order of p and q. For consistency with PKCS #1 v2.0,
   however, the first two primes p and q are treated separately from
   the additional primes. 

5.2.2 RSAVP1

   RSAVP1 ((n, e), s)

   Input:
   (n, e)   RSA public key
   s        signature representative, an integer between 0 and n - 1





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   Output:
   m        message representative, an integer between 0 and n - 1

   Error: "signature representative out of range"

   Assumption: RSA public key (n, e) is valid

   Steps:

   1. If the signature representative s is not between 0 and n - 1,
      output "signature representative out of range" and stop. 

   2. Let m = s^e mod n.

   3. Output m.

6. Overview of schemes

   A scheme combines cryptographic primitives and other techniques to
   achieve a particular security goal. Two types of scheme are
   specified in this document: encryption schemes and signature
   schemes with appendix. 

   The schemes specified in this document are limited in scope in that
   their operations consist only of steps to process data with an RSA
   public or private key, and do not include steps for obtaining or
   validating the key. Thus, in addition to the scheme operations, an
   application will typically include key management operations by
   which parties may select RSA public and private keys for a scheme
   operation. The specific additional operations and other details are
   outside the scope of this document. 

   As was the case for the cryptographic primitives (Section 5), the
   specifications of scheme operations assume that certain conditions
   are met by the inputs, in particular that RSA public and private
   keys are valid. The behavior of an implementation is thus
   unspecified when a key is invalid. The impact of such unspecified
   behavior depends on the application. Possible means of addressing
   key validation include explicit key validation by the application;
   key validation within the public-key infrastructure; and assignment
   of liability for operations performed with an invalid key to the
   party who generated the key. 

   A generally good cryptographic practice is to employ a given RSA
   key pair in only one scheme. This avoids the risk that
   vulnerability in one scheme may compromise the security of the
   other, and may be essential to maintain provable security. While
   RSAES-PKCS1-v1_5 (Section 7.2) and RSASSA-PKCS1-v1_5 (Section 8.2)



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   have traditionally been employed together without any known bad
   interactions (indeed, this is the model introduced by PKCS #1
   v1.5), such a combined use of an RSA key pair is not recommended
   for new applications.

   To illustrate the risks related to the employment of an RSA key
   pair in more than one scheme, suppose an RSA key pair is employed
   in both RSAES-OAEP (Section 7.1) and RSAES-PKCS1-v1_5. Although
   RSAES-OAEP by itself would resist attack, an opponent might be able
   to exploit a weakness in the implementation of RSAES-PKCS1-v1_5 to
   recover messages encrypted with either scheme. As another example,
   suppose an RSA key pair is employed in both RSASSA-PSS (Section
   8.1) and RSASSA-PKCS1-v1_5. Then the security proof for RSASSA-PSS
   would no longer be sufficient since the proof does not account for
   the possibility that signatures might be generated with a second
   scheme. Similar considerations may apply if an RSA key pair is
   employed in one of the schemes defined here and in a variant
   defined elsewhere. 

7. Encryption schemes

   For the purposes of this document, an encryption scheme consists of
   an encryption operation and a decryption operation, where the
   encryption operation produces a ciphertext from a message with a
   recipient's RSA public key, and the decryption operation recovers
   the message from the ciphertext with the recipient's corresponding
   RSA private key. 

   An encryption scheme can be employed in a variety of applications. 
   A typical application is a key establishment protocol, where the
   message contains key material to be delivered confidentially from
   one party to another. For instance, PKCS #7 [45] employs such a
   protocol to deliver a content-encryption key from a sender to a
   recipient; the encryption schemes defined here would be suitable
   key-encryption algorithms in that context. 

   Two encryption schemes are specified in this document: RSAES-OAEP
   and RSAES-PKCS1-v1_5. RSAES-OAEP is recommended for new
   applications; RSAES-PKCS1-v1_5 is included only for compatibility
   with existing applications, and is not recommended for new
   applications.

   The encryption schemes given here follow a general model similar to
   that employed in IEEE Std 1363-2000 [26], combining encryption and
   decryption primitives with an encoding method for encryption. The
   encryption operations apply a message encoding operation to a
   message to produce an encoded message, which is then converted to
   an integer message representative. An encryption primitive is



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   applied to the message representative to produce the ciphertext.
   Reversing this, the decryption operations apply a decryption
   primitive to the ciphertext to recover a message representative,
   which is then converted to an octet string encoded message. A
   message decoding operation is applied to the encoded message to
   recover the message and verify the correctness of the decryption. 

   To avoid implementation weaknesses related to the way errors are
   handled within the decoding operation (see [6] and [36]), the
   encoding and decoding operations for RSAES-OAEP and
   RSAES-PKCS1-v1_5 are embedded in the specifications of the
   respective encryption schemes rather than defined in separate
   specifications. Both encryption schemes are compatible with the
   corresponding schemes in PKCS #1 v2.0. 

7.1 RSAES-OAEP

   RSAES-OAEP combines the RSAEP and RSADP primitives (Sections 5.1.1
   and 5.1.2) with the EME-OAEP encoding method (step 1.b in Section
   7.1.1 and step 3 in Section 7.1.2). EME-OAEP is based on Bellare
   and Rogaway's Optimal Asymmetric Encryption scheme [3]. (OAEP
   stands for "Optimal Asymmetric Encryption Padding."). It is
   compatible with the IFES scheme defined in IEEE Std 1363-2000 [26],
   where the encryption and decryption primitives are IFEP-RSA and
   IFDP-RSA and the message encoding method is EME-OAEP. RSAES-OAEP
   can operate on messages of length up to k - 2hLen - 2 octets, where
   hLen is the length of the output from the underlying hash function
   and k is the length in octets of the recipient's RSA modulus. 

   Assuming that computing e-th roots modulo n is infeasible and the
   mask generation function in RSAES-OAEP has appropriate properties,
   RSAES-OAEP is semantically secure against adaptive chosen-
   ciphertext attacks. This assurance is provable in the sense that
   the difficulty of breaking RSAES-OAEP can be directly related to
   the difficulty of inverting the RSA function, provided that the
   mask generation function is viewed as a black box or random oracle;
   see [21] and the note below for further discussion.

   Both the encryption and the decryption operations of RSAES-OAEP
   take the value of a label L as input. In this version of PKCS #1, L
   is the empty string; other uses of the label are outside the scope
   of this document. See Appendix A.2.1 for the relevant ASN.1
   syntax.

   RSAES-OAEP is parameterized by the choice of hash function and mask
   generation function. This choice should be fixed for a given RSA
   key. Suggested hash and mask generation functions are given in
   Appendix B.



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   Note. Recent results have helpfully clarified the security
   properties of the OAEP encoding method [3] (roughly the procedure
   described in step 1.b in Section 7.1.1). The background is as
   follows. In 1994, Bellare and Rogaway [3] introduced a security
   concept that they denoted plaintext awareness (PA94). They proved
   that if a deterministic public-key encryption primitive (e.g.,
   RSAEP) is hard to invert without the private key, then the
   corresponding OAEP-based encryption scheme is plaintext-aware (in
   the random oracle model), meaning roughly that an adversary cannot
   produce a valid ciphertext without actually "knowing" the
   underlying plaintext. Plaintext awareness of an encryption scheme
   is closely related to the resistance of the scheme against
   chosen-ciphertext attacks. In such attacks, an adversary is given
   the opportunity to send queries to an oracle simulating the
   decryption primitive. Using the results of these queries, the
   adversary attempts to decrypt a challenge ciphertext. 

   However, there are two flavors of chosen-ciphertext attacks, and
   PA94 implies security against only one of them. The difference
   relies on what the adversary is allowed to do after she is given
   the challenge ciphertext. The indifferent attack scenario (denoted
   CCA1) does not admit any queries to the decryption oracle after the
   adversary is given the challenge ciphertext, whereas the adaptive
   scenario (denoted CCA2) does (except that the decryption oracle
   refuses to decrypt the challenge ciphertext once it is published).
   In 1998, Bellare and Rogaway, together with Desai and Pointcheval
   [2], came up with a new, stronger notion of plaintext awareness
   (PA98) that does imply security against CCA2.

   To summarize, there have been two potential sources for
   misconception: that PA94 and PA98 are equivalent concepts; or that
   CCA1 and CCA2 are equivalent concepts. Either assumption leads to
   the conclusion that the Bellare-Rogaway paper implies security of
   OAEP against CCA2, which it does not.

   (Footnote: It might be fair to mention that PKCS #1 v2.0 cites [3]
   and claims that "a chosen ciphertext attack is ineffective against
   a plaintext-aware encryption scheme such as RSAES-OAEP" without
   specifying the kind of plaintext awareness or chosen ciphertext
   attack considered.)

   OAEP has never been proven secure against CCA2; in fact, Victor
   Shoup [48] has demonstrated that such a proof does not exist in the
   general case. Put briefly, Shoup showed that an adversary in the
   CCA2 scenario who knows how to partially invert the encryption
   primitive but does not know how to invert it completely may well be
   able to break the scheme. For example, one may imagine an attacker
   who is able to break RSAES-OAEP if she knows how to recover all but



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   the first 20 bytes of a random integer encrypted with RSAEP. Such
   an attacker does not need to be able to fully invert RSAEP, because
   she does not use the first 20 octets in her attack. 

   Still, RSAES-OAEP is secure against CCA2, which was proved by
   Fujisaki, Okamoto, Pointcheval, and Stern [21] shortly after the
   announcement of Shoup's result. Using clever lattice reduction
   techniques, they managed to show how to invert RSAEP completely
   given a sufficiently large part of the pre-image. This observation,
   combined with a proof that OAEP is secure against CCA2 if the
   underlying encryption primitive is hard to partially invert, fills
   the gap between what Bellare and Rogaway proved about RSAES-OAEP
   and what some may have believed that they proved. Somewhat
   paradoxically, we are hence saved by an ostensible weakness in
   RSAEP (i.e., the whole inverse can be deduced from parts of it).

   Unfortunately however, the security reduction is not efficient for
   concrete parameters. While the proof successfully relates an
   adversary Adv against the CCA2 security of RSAES-OAEP to an
   algorithm Inv inverting RSA, the probability of success for Inv is
   only approximately \epsilon^2 / 2^18, where \epsilon is the
   probability of success for Adv. 

   (Footnote: In [21] the probability of success for the inverter was
   \epsilon^2 / 4. The additional factor 1 / 2^16 is due to the eight
   fixed zero bits at the beginning of the encoded message EM, which
   are not present in the variant of OAEP considered in [21] (Inv must
   apply Adv twice to invert RSA, and each application corresponds to
   a factor 1 / 2^8).) 

   In addition, the running time for Inv is approximately t^2, where t
   is the running time of the adversary. The consequence is that we
   cannot exclude the possibility that attacking RSAES-OAEP is
   considerably easier than inverting RSA for concrete parameters.
   Still, the existence of a security proof provides some assurance
   that the RSAES-OAEP construction is sounder than ad hoc
   constructions such as RSAES-PKCS1-v1_5.

   Hybrid encryption schemes based on the RSA-KEM key encapsulation
   paradigm offer tight proofs of security directly applicable to
   concrete parameters; see [30] for discussion. Future versions of
   PKCS #1 may specify schemes based on this paradigm. 









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7.1.1 Encryption operation

   RSAES-OAEP-ENCRYPT ((n, e), M, L)

   Options:
   Hash    hash function (hLen denotes the length in octets of the
           hash function output) 
   MGF     mask generation function

   Input:
   (n, e)  recipient's RSA public key (k denotes the length in octets
           of the RSA modulus n) 
   M       message to be encrypted, an octet string of length mLen,
           where mLen <= k - 2hLen - 2 
   L       optional label to be associated with the message; the default
           value for L, if L is not provided, is the empty string

   Output:
   C       ciphertext, an octet string of length k

   Errors: "message too long"; "label too long"

   Assumption: RSA public key (n, e) is valid

   Steps:

   1. Length checking:

      a. If the length of L is greater than the input limitation for
         the hash function (2^61 - 1 octets for SHA-1), output "label
         too long" and stop. 

      b. If mLen > k - 2hLen - 2, output "message too long" and stop.

   2. EME-OAEP encoding (see Figure 1 below):

      a. If the label L is not provided, let L be the empty string. 
         Let lHash = Hash(L), an octet string of length hLen (see the
         note below).

      b. Generate an octet string PS consisting of k - mLen - 2hLen -
         2 zero octets. The length of PS may be zero.

      c. Concatenate lHash, PS, a single octet with hexadecimal value
         0x01, and the message M to form a data block DB of length k -
         hLen - 1 octets as 

            DB = lHash || PS || 0x01 || M.



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      d. Generate a random octet string seed of length hLen. 

      e. Let dbMask = MGF(seed, k - hLen - 1).

      f. Let maskedDB = DB \xor dbMask.

      g. Let seedMask = MGF(maskedDB, hLen).

      h. Let maskedSeed = seed \xor seedMask.

      i. Concatenate a single octet with hexadecimal value 0x00,
         maskedSeed, and maskedDB to form an encoded message EM of
         length k octets as 

            EM = 0x00 || maskedSeed || maskedDB.

   3. RSA encryption:

      a. Convert the encoded message EM to an integer message
         representative m (see Section 4.2): 

            m = OS2IP (EM).

      b. Apply the RSAEP encryption primitive (Section 5.1.1) to the
         RSA public key (n, e) and the message representative m to
         produce an integer ciphertext representative c: 

            c = RSAEP ((n, e), m). 

      c. Convert the ciphertext representative c to a ciphertext C of
         length k octets (see Section 4.1):

            C = I2OSP (c, k).

   4. Output the ciphertext C.

   Note. If L is the empty string, the corresponding hash value lHash
   has the following hexadecimal representation for different choices
   of Hash: 

   SHA-1:   (0x)da39a3ee 5e6b4b0d 3255bfef 95601890 afd80709
   SHA-256: (0x)e3b0c442 98fc1c14 9afbf4c8 996fb924 27ae41e4 649b934c
                a495991b 7852b855
   SHA-384: (0x)38b060a7 51ac9638 4cd9327e b1b1e36a 21fdb711 14be0743
                4c0cc7bf 63f6e1da 274edebf e76f65fb d51ad2f1 4898b95b
   SHA-512: (0x)cf83e135 7eefb8bd f1542850 d66d8007 d620e405 0b5715dc
                83f4a921 d36ce9ce 47d0d13c 5d85f2b0 ff8318d2 877eec2f
                63b931bd 47417a81 a538327a f927da3e



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   __________________________________________________________________

                             +----------+---------+-------+
                        DB = |  lHash   |    PS   |   M   |
                             +----------+---------+-------+
                                            |
                  +----------+              V
                  |   seed   |--> MGF ---> xor
                  +----------+              |
                        |                   |
               +--+     V                   |
               |00|    xor <----- MGF <-----|
               +--+     |                   |
                 |      |                   |
                 V      V                   V
               +--+----------+----------------------------+
         EM =  |00|maskedSeed|          maskedDB          |
               +--+----------+----------------------------+
   __________________________________________________________________

   Figure 1: EME-OAEP encoding operation. lHash is the hash of the
   optional label L. Decoding operation follows reverse steps to
   recover M and verify lHash and PS. 

7.1.2 Decryption operation

   RSAES-OAEP-DECRYPT (K, C, L)

   Options:
   Hash     hash function (hLen denotes the length in octets of the
            hash function output) 
   MGF      mask generation function

   Input:   
   K        recipient's RSA private key (k denotes the length in
            octets of the RSA modulus n) 
   C        ciphertext to be decrypted, an octet string of length k,
            where k = 2hLen + 2 
   L        optional label whose association with the message is to be
            verified; the default value for L, if L is not provided,
            is the empty string 

   Output:
   M        message, an octet string of length mLen, where mLen ˙ k -
            2hLen - 2

   Error: "decryption error"




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   Steps:

   1. Length checking: 

      a. If the length of L is greater than the input limitation for
         the hash function (2^61 - 1 octets for SHA-1), output
         "decryption error" and stop. 

      b. If the length of the ciphertext C is not k octets, output
         "decryption error" and stop. 

      c. If k < 2hLen + 2, output "decryption error" and stop.

   2.    RSA decryption:

      a. Convert the ciphertext C to an integer ciphertext
         representative c (see Section 4.2): 

            c = OS2IP (C).

      b. Apply the RSADP decryption primitive (Section 5.1.2) to the
         RSA private key K and the ciphertext representative c to
         produce an integer message representative m: 

            m = RSADP (K, c).

         If RSADP outputs "ciphertext representative out of range"
         (meaning that c >= n), output "decryption error" and stop. 

      c. Convert the message representative m to an encoded message EM
         of length k octets (see Section 4.1):

            EM = I2OSP (m, k).

   3. EME-OAEP decoding:

      a. If the label L is not provided, let L be the empty string. 
         Let lHash = Hash(L), an octet string of length hLen (see the
         note in Section 7.1.1). 

      b. Separate the encoded message EM into a single octet Y, an
         octet string maskedSeed of length hLen, and an octet string
         maskedDB of length k - hLen - 1 as 

            EM = Y || maskedSeed || maskedDB.

      c. Let seedMask = MGF(maskedDB, hLen).




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      d. Let seed = maskedSeed \xor seedMask.

      e. Let dbMask = MGF(seed, k - hLen - 1).

      f. Let DB = maskedDB \xor dbMask.

      g. Separate DB into an octet string lHash' of length hLen, a
         (possibly empty) padding string PS consisting of octets with 
         hexadecimal value 0x00, and a message M as

            DB = lHash' || PS || 0x01 || M.

         If there is no octet with hexadecimal value 0x01 to separate
         PS from M, if lHash does not equal lHash', or if Y is
         nonzero, output "decryption error" and stop. (See the note
         below.)

   4. Output the message M.

   Note. Care must be taken to ensure that an opponent cannot
   distinguish the different error conditions in Step 3.f, whether by
   error message or timing, or, more generally, learn partial
   information about the encoded message EM. Otherwise an opponent may
   be able to obtain useful information about the decryption of the
   ciphertext C, leading to a chosen-ciphertext attack such as the one
   observed by Manger [36].

7.2 RSAES-PKCS1-v1_5

   RSAES-PKCS1-v1_5 combines the RSAEP and RSADP primitives (Sections
   5.1.1 and 5.1.2) with the EME-PKCS1-v1_5 encoding method (step 1 in
   Section 7.2.1 and step 3 in Section 7.2.2). It is mathematically
   equivalent to the encryption scheme in PKCS #1 v1.5.
   RSAES-PKCS1-v1_5 can operate on messages of length up to k - 11
   octets (k is the octet length of the RSA modulus), although care
   should be taken to avoid certain attacks on low-exponent RSA due to
   Coppersmith, Franklin, Patarin, and Reiter when long messages are
   encrypted (see the third bullet in the notes below and [10]; [14]
   contains an improved attack). As a general rule, the use of this
   scheme for encrypting an arbitrary message, as opposed to a
   randomly generated key, is not recommended.

   It is possible to generate valid RSAES-PKCS1-v1_5 ciphertexts
   without knowing the corresponding plaintexts, with a reasonable
   probability of success. This ability can be exploited in a chosen-
   ciphertext attack as shown in [6]. Therefore, if RSAES-PKCS1-v1_5
   is to be used, certain easily implemented countermeasures should be
   taken to thwart the attack found in [6]. Typical examples include



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   the addition of structure to the data to be encoded, rigorous
   checking of PKCS #1 v1.5 conformance (and other redundancy) in
   decrypted messages, and the consolidation of error messages in a
   client-server protocol based on PKCS #1 v1.5. These can all be
   effective countermeasures and do not involve changes to a PKCS #1
   v1.5-based protocol. See [7] for a further discussion of these and
   other countermeasures. It has recently been shown that the security
   of the SSL/TLS handshake protocol [17], which uses RSAES-PKCS1-v1_5
   and certain countermeasures, can be related to a variant of the RSA
   problem; see [32] for discussion. 

   Note. The following passages describe some security recommendations
   pertaining to the use of RSAES-PKCS1-v1_5. Recommendations from
   version 1.5 of this document are included as well as new
   recommendations motivated by cryptanalytic advances made in the
   intervening years.

    * It is recommended that the pseudorandom octets in step 2 in
      Section 7.2.1 be generated independently for each encryption
      process, especially if the same data is input to more than one
      encryption process. Hůstad's results [24] are one motivation for
      this recommendation.

    * The padding string PS in step 2 in Section 7.2.1 is at least
      eight octets long, which is a security condition for public-key
      operations that makes it difficult for an attacker to recover
      data by trying all possible encryption blocks.

    * The pseudorandom octets can also help thwart an attack due to
      Coppersmith et al. [10] (see [14] for an improvement of the
      attack) when the size of the message to be encrypted is kept
      small. The attack works on low-exponent RSA when similar
      messages are encrypted with the same RSA public key. More 
      specifically, in one flavor of the attack, when two inputs to
      RSAEP agree on a large fraction of bits (8/9) and low-exponent
      RSA (e = 3) is used to encrypt both of them, it may be possible
      to recover both inputs with the attack. Another flavor of the
      attack is successful in decrypting a single ciphertext when a
      large fraction (2/3) of the input to RSAEP is already known. For
      typical applications, the message to be encrypted is short
      (e.g., a 128-bit symmetric key) so not enough information will
      be known or common between two messages to enable the attack. 
      However, if a long message is encrypted, or if part of a message
      is known, then the attack may be a concern. In any case, the
      RSAES-OAEP scheme overcomes the attack.






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7.2.1 Encryption operation

   RSAES-PKCS1-V1_5-ENCRYPT ((n, e), M)

   Input: 
   (n, e)   recipient's RSA public key (k denotes the length in octets
            of the modulus n) 
   M        message to be encrypted, an octet string of length mLen,
            where mLen <= k - 11 

   Output:    
   C        ciphertext, an octet string of length k

   Error: "message too long"

   Steps:

   1. Length checking: If mLen > k - 11, output "message too long" and
      stop.

   2. EME-PKCS1-v1_5 encoding:

      a. Generate an octet string PS of length k - mLen - 3 consisting
         of pseudo-randomly generated nonzero octets. The length of PS
         will be at least eight octets. 

      b. Concatenate PS, the message M, and other padding to form an
         encoded message EM of length k octets as 

            EM = 0x00 || 0x02 || PS || 0x00 || M.

   3. RSA encryption:

      a. Convert the encoded message EM to an integer message
         representative m (see Section 4.2): 

            m = OS2IP (EM).

      b. Apply the RSAEP encryption primitive (Section 5.1.1) to the
         RSA public key (n, e) and the message representative m to
         produce an integer ciphertext representative c: 

            c = RSAEP ((n, e), m).

      c. Convert the ciphertext representative c to a ciphertext C of
         length k octets (see Section 4.1):

            C = I2OSP (c, k).



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   4. Output the ciphertext C.

7.2.2 Decryption operation

   RSAES-PKCS1-V1_5-DECRYPT (K, C)

   Input:
   K        recipient's RSA private key
   C        ciphertext to be decrypted, an octet string of length k,
            where k is the length in octets of the RSA modulus n 

   Output:
   M        message, an octet string of length at most k - 11

   Error: "decryption error"

   Steps:

   1. Length checking: If the length of the ciphertext C is not k
      octets (or if k < 11), output "decryption error" and stop. 

   2. RSA decryption: 

      a. Convert the ciphertext C to an integer ciphertext
         representative c (see Section 4.2): 

            c = OS2IP (C).

      b. Apply the RSADP decryption primitive (Section 5.1.2) to the
         RSA private key (n, d) and the ciphertext representative c to
         produce an integer message representative m:

            m = RSADP ((n, d), c).

         If RSADP outputs "ciphertext representative out of range"
         (meaning that c >= n), output "decryption error" and stop. 

      c. Convert the message representative m to an encoded message EM
         of length k octets (see Section 4.1): 

            EM = I2OSP (m, k).

   3. EME-PKCS1-v1_5 decoding: Separate the encoded message EM into an
      octet string PS consisting of nonzero octets and a message M as 

         EM = 0x00 || 0x02 || PS || 0x00 || M.





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      If the first octet of EM does not have hexadecimal value 0x00,
      if the second octet of EM does not have hexadecimal value 0x02,
      if there is no octet with hexadecimal value 0x00 to separate PS
      from M, or if the length of PS is less than 8 octets, output
      "decryption error" and stop. (See the note below.) 

   4. Output M.

   Note. Care shall be taken to ensure that an opponent cannot
   distinguish the different error conditions in Step 3, whether by
   error message or timing. Otherwise an opponent may be able to
   obtain useful information about the decryption of the ciphertext C,
   leading to a strengthened version of Bleichenbacher's attack [6];
   compare to Manger's attack [36]. 

8. Signature schemes with appendix

   For the purposes of this document, a signature scheme with appendix
   consists of a signature generation operation and a signature
   verification operation, where the signature generation operation
   produces a signature from a message with a signer's RSA private
   key, and the signature verification operation verifies the
   signature on the message with the signer's corresponding RSA public
   key. To verify a signature constructed with this type of scheme it
   is necessary to have the message itself. In this way, signature
   schemes with appendix are distinguished from signature schemes with
   message recovery, which are not supported in this document. 

   A signature scheme with appendix can be employed in a variety of
   applications. For instance, the signature schemes with appendix
   defined here would be suitable signature algorithms for X.509
   certificates [28]. Related signature schemes could be employed in
   PKCS #7 [45], although for technical reasons the current version of
   PKCS #7 separates a hash function from a signature scheme, which is
   different than what is done here; see the note in Appendix A.2.3
   for more discussion. 

   Two signature schemes with appendix are specified in this document:
   RSASSA-PSS and RSASSA-PKCS1-v1_5. Although no attacks are known
   against RSASSA-PKCS1-v1_5, in the interest of increased robustness,
   RSASSA-PSS is recommended for eventual adoption in new
   applications. RSASSA-PKCS1-v1_5 is included for compatibility with
   existing applications, and while still appropriate for new
   applications, a gradual transition to RSASSA-PSS is encouraged. 

   The signature schemes with appendix given here follow a general
   model similar to that employed in IEEE Std 1363-2000 [26],
   combining signature and verification primitives with an encoding



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   method for signatures. The signature generation operations apply a
   message encoding operation to a message to produce an encoded
   message, which is then converted to an integer message
   representative. A signature primitive is applied to the message
   representative to produce the signature. Reversing this, the
   signature verification operations apply a signature verification
   primitive to the signature to recover a message representative,
   which is then converted to an octet string encoded message. A
   verification operation is applied to the message and the encoded
   message to determine whether they are consistent. 

   If the encoding method is deterministic (e.g., EMSA-PKCS1-v1_5),
   the verification operation may apply the message encoding operation
   to the message and compare the resulting encoded message to the
   previously derived encoded message. If there is a match, the
   signature is considered valid. If the method is randomized (e.g.,
   EMSA-PSS), the verification operation is typically more
   complicated. For example, the verification operation in EMSA-PSS
   extracts the random salt and a hash output from the encoded message
   and checks whether the hash output, the salt, and the message are
   consistent; the hash output is a deterministic function in terms of
   the message and the salt. 

   For both signature schemes with appendix defined in this document,
   the signature generation and signature verification operations are
   readily implemented as "single-pass" operations if the signature is
   placed after the message. See PKCS #7 [45] for an example format in
   the case of RSASSA-PKCS1-v1_5. 

8.1 RSASSA-PSS

   RSASSA-PSS combines the RSASP1 and RSAVP1 primitives with the
   EMSA-PSS encoding method. It is compatible with the IFSSA scheme as
   amended in the IEEE P1363a draft [27], where the signature and
   verification primitives are IFSP-RSA1 and IFVP-RSA1 as defined in
   IEEE Std 1363-2000 [26] and the message encoding method is EMSA4.
   EMSA4 is slightly more general than EMSA-PSS as it acts on bit
   strings rather than on octet strings. EMSA-PSS is equivalent to
   EMSA4 restricted to the case that the operands as well as the hash
   and salt values are octet strings. 

   The length of messages on which RSASSA-PSS can operate is either
   unrestricted or constrained by a very large number, depending on
   the hash function underlying the EMSA-PSS encoding method. 

   Assuming that computing e-th roots modulo n is infeasible and the
   hash and mask generation functions in EMSA-PSS have appropriate
   properties, RSASSA-PSS provides secure signatures. This assurance



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   is provable in the sense that the difficulty of forging signatures
   can be directly related to the difficulty of inverting the RSA
   function, provided that the hash and mask generation functions are
   viewed as black boxes or random oracles. The bounds in the security
   proof are essentially "tight", meaning that the success probability
   and running time for the best forger against RSASSA-PSS are very
   close to the corresponding parameters for the best RSA inversion
   algorithm; see [4][13][31] for further discussion.

   In contrast to the RSASSA-PKCS1-v1_5 signature scheme, a hash
   function identifier is not embedded in the EMSA-PSS encoded
   message, so in theory it is possible for an adversary to substitute
   a different (and potentially weaker) hash function than the one
   selected by the signer. Therefore, it is recommended that the
   EMSA-PSS mask generation function be based on the same hash
   function. In this manner the entire encoded message will be
   dependent on the hash function and it will be difficult for an
   opponent to substitute a different hash function than the one
   intended by the signer. This matching of hash functions is only for
   the purpose of preventing hash function substitution, and is not
   necessary if hash function substitution is addressed by other means
   (e.g., the verifier accepts only a designated hash function). See
   [34] for further discussion of these points. The provable security
   of RSASSA-PSS does not rely on the hash function in the mask
   generation function being the same as the hash function applied to
   the message. 

   RSASSA-PSS is different from other RSA-based signature schemes in
   that it is probabilistic rather than deterministic, incorporating a
   randomly generated salt value. The salt value enhances the security
   of the scheme by affording a "tighter" security proof than
   deterministic alternatives such as Full Domain Hashing (FDH); see
   [4] for discussion. However, the randomness is not critical to
   security. In situations where random generation is not possible, a
   fixed value or a sequence number could be employed instead, with
   the resulting provable security similar to that of FDH [12]. 

8.1.1 Signature generation operation

   RSASSA-PSS-SIGN (K, M)

   Input:
   K        signer's RSA private key
   M        message to be signed, an octet string

   Output:    
   S        signature, an octet string of length k, where k is the
            length in octets of the RSA modulus n 



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   Errors: "message too long;" "encoding error"

   Steps:
   1. EMSA-PSS encoding: Apply the EMSA-PSS encoding operation
      (Section 9.1.1) to the message M to produce an encoded message
      EM of length \ceil ((modBits - 1)/8) octets such that the bit
      length of the integer OS2IP (EM) (see Section 4.2) is at most
      modBits - 1, where modBits is the length in bits of the RSA
      modulus n:

         EM = EMSA-PSS-ENCODE (M, modBits - 1).

      Note that the octet length of EM will be one less than k if
      modBits - 1 is divisible by 8 and equal to k otherwise. If the
      encoding operation outputs "message too long," output "message
      too long" and stop. If the encoding operation outputs "encoding
      error," output "encoding error" and stop. 

   2. RSA signature:

      a. Convert the encoded message EM to an integer message
         representative m (see Section 4.2): 

            m = OS2IP (EM).

      b. Apply the RSASP1 signature primitive (Section 5.2.1) to the
         RSA private key K and the message representative m to produce
         an integer signature representative s: 

            s = RSASP1 (K, m).

      c. Convert the signature representative s to a signature S of
         length k octets (see Section 4.1):

            S = I2OSP (s, k).

   3. Output the signature S.

8.1.2 Signature verification operation

   RSASSA-PSS-VERIFY ((n, e), M, S)

   Input:
   (n, e)   signer's RSA public key
   M        message whose signature is to be verified, an octet string
   S        signature to be verified, an octet string of length k,
            where k is the length in octets of the RSA modulus n 




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   Output: 
   "valid signature" or "invalid signature"

   Steps:

   1. Length checking: If the length of the signature S is not k
      octets, output "invalid signature" and stop.

   2. RSA verification: 

      a. Convert the signature S to an integer signature
         representative s (see Section 4.2):

            s = OS2IP (S).

      b. Apply the RSAVP1 verification primitive (Section 5.2.2) to
         the RSA public key (n, e) and the signature representative s
         to produce an integer message representative m:

            m = RSAVP1 ((n, e), s).

         If RSAVP1 output "signature representative out of range,"
         output "invalid signature" and stop.

      c. Convert the message representative m to an encoded message EM
         of length emLen = \ceil ((modBits - 1)/8) octets, where
         modBits is the length in bits of the RSA modulus n (see
         Section 4.1):

            EM = I2OSP (m, emLen).

         Note that emLen will be one less than k if modBits - 1 is
         divisible by 8 and equal to k otherwise. If I2OSP outputs
         "integer too large," output "invalid signature" and stop. 

   3. EMSA-PSS verification: Apply the EMSA-PSS verification operation
      (Section 9.1.2) to the message M and the encoded message EM to
      determine whether they are consistent: 

         Result = EMSA-PSS-VERIFY (M, EM, modBits - 1).

   4. If Result = "consistent," output "valid signature." Otherwise,
      output "invalid signature."

8.2 RSASSA-PKCS1-v1_5

   RSASSA-PKCS1-v1_5 combines the RSASP1 and RSAVP1 primitives with
   the EMSA-PKCS1-v1_5 encoding method. It is compatible with the



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   IFSSA scheme defined in IEEE Std 1363-2000 [26], where the
   signature and verification primitives are IFSP-RSA1 and IFVP-RSA1
   and the message encoding method is EMSA-PKCS1-v1_5 (which is not
   defined in IEEE Std 1363-2000, but is in the IEEE P1363a draft
   [27]).

   The length of messages on which RSASSA-PKCS1-v1_5 can operate is
   either unrestricted or constrained by a very large number,
   depending on the hash function underlying the EMSA-PKCS1-v1_5
   method. 

   Assuming that computing e-th roots modulo n is infeasible and the
   hash function in EMSA-PKCS1-v1_5 has appropriate properties,
   RSASSA-PKCS1-v1_5 is conjectured to provide secure signatures. More
   precisely, forging signatures without knowing the RSA private key
   is conjectured to be computationally infeasible. Also, in the
   encoding method EMSA-PKCS1-v1_5, a hash function identifier is
   embedded in the encoding. Because of this feature, an adversary
   trying to find a message with the same signature as a previously
   signed message must find collisions of the particular hash function
   being used; attacking a different hash function than the one
   selected by the signer is not useful to the adversary. See [34] for
   further discussion.

   Note. As noted in PKCS #1 v1.5, the EMSA-PKCS1-v1_5 encoding method
   has the property that the encoded message, converted to an integer
   message representative, is guaranteed to be large and at least
   somewhat "random". This prevents attacks of the kind proposed by
   Desmedt and Odlyzko [16] where multiplicative relationships between
   message representatives are developed by factoring the message
   representatives into a set of small values (e.g., a set of small
   primes). Coron, Naccache, and Stern [15] showed that a stronger
   form of this type of attack could be quite effective against some
   instances of the ISO/IEC 9796-2 signature scheme. They also
   analyzed the complexity of this type of attack against the
   EMSA-PKCS1-v1_5 encoding method and concluded that an attack would
   be impractical, requiring more operations than a collision search
   on the underlying hash function (i.e., more than 2^80 operations).
   Coppersmith, Halevi, and Jutla [11] subsequently extended Coron et
   al.'s attack to break the ISO/IEC 9796-1 signature scheme with
   message recovery. The various attacks illustrate the importance of
   carefully constructing the input to the RSA signature primitive,
   particularly in a signature scheme with message recovery. 
   Accordingly, the EMSA-PKCS-v1_5 encoding method explicitly
   includes a hash operation and is not intended for signature schemes
   with message recovery. Moreover, while no attack is known against
   the EMSA-PKCS-v1_5 encoding method, a gradual transition to




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   EMSA-PSS is recommended as a precaution against future
   developments.

8.2.1 Signature generation operation

   RSASSA-PKCS1-V1_5-SIGN (K, M)

   Input:
   K        signer's RSA private key
   M        message to be signed, an octet string

   Output:
   S        signature, an octet string of length k, where k is the
            length in octets of the RSA modulus n

   Errors: "message too long"; "RSA modulus too short"

   Steps:

   1. EMSA-PKCS1-v1_5 encoding: Apply the EMSA-PKCS1-v1_5 encoding
      operation (Section 9.2) to the message M to produce an encoded
      message EM of length k octets: 

         EM = EMSA-PKCS1-V1_5-ENCODE (M, k).

      If the encoding operation outputs "message too long," output
      "message too long" and stop. If the encoding operation outputs
      "intended encoded message length too short," output "RSA modulus
      too short" and stop. 

   2. RSA signature:

      a. Convert the encoded message EM to an integer message
         representative m (see Section 4.2): 

            m = OS2IP (EM).

      b. Apply the RSASP1 signature primitive (Section 5.2.1) to the
         RSA private key K and the message representative m to produce
         an integer signature representative s: 

            s = RSASP1 (K, m).

      c. Convert the signature representative s to a signature S of
         length k octets (see Section 4.1): 

            S = I2OSP (s, k).




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   3. Output the signature S.

8.2.2 Signature verification operation

   RSASSA-PKCS1-V1_5-VERIFY ((n, e), M, S)

   Input:
   (n, e)   signer's RSA public key
   M        message whose signature is to be verified, an octet string
   S        signature to be verified, an octet string of length k,
            where k is the length in octets of the RSA modulus n 

   Output: 
   "valid signature" or "invalid signature"

   Errors: "message too long"; "RSA modulus too short"

   Steps:

   1. Length checking: If the length of the signature S is not k
      octets, output "invalid signature" and stop. 

   2. RSA verification: 

      a. Convert the signature S to an integer signature
         representative s (see Section 4.2): 

            s = OS2IP (S).

      b. Apply the RSAVP1 verification primitive (Section 5.2.2) to
         the RSA public key (n, e) and the signature representative s
         to produce an integer message representative m:

            m = RSAVP1 ((n, e), s).

         If RSAVP1 outputs "signature representative out of range,"
         output "invalid signature" and stop. 

      c. Convert the message representative m to an encoded message EM
         of length k octets (see Section 4.1): 

            EM' = I2OSP (m, k).

         If I2OSP outputs "integer too large," output "invalid
         signature" and stop.






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   3. EMSA-PKCS1-v1_5 encoding: Apply the EMSA-PKCS1-v1_5 encoding
      operation (Section 9.2) to the message M to produce a second
      encoded message EM' of length k octets: 

         EM' = EMSA-PKCS1-V1_5-ENCODE (M, k).

      If the encoding operation outputs "message too long," output
      "message too long" and stop. If the encoding operation outputs
      "intended encoded message length too short," output "RSA modulus
      too short" and stop. 

   4. Compare the encoded message EM and the second encoded message
      EM'. If they are the same, output "valid signature"; otherwise,
      output "invalid signature." 

   Note. Another way to implement the signature verification operation
   is to apply a "decoding" operation (not specified in this document)
   to the encoded message to recover the underlying hash value, and
   then to compare it to a newly computed hash value. This has the
   advantage that it requires less intermediate storage (two hash
   values rather than two encoded messages), but the disadvantage that
   it requires additional code.

9. Encoding methods for signatures with appendix

   Encoding methods consist of operations that map between octet
   string messages and octet string encoded messages, which are
   converted to and from integer message representatives in the
   schemes. The integer message representatives are processed via the
   primitives. The encoding methods thus provide the connection
   between the schemes, which process messages, and the primitives. 

   An encoding method for signatures with appendix, for the purposes
   of this document, consists of an encoding operation and optionally
   a verification operation. An encoding operation maps a message M to
   an encoded message EM of a specified length. A verification
   operation determines whether a message M and an encoded message EM
   are consistent, i.e., whether the encoded message EM is a valid
   encoding of the message M. 

   The encoding operation may introduce some randomness, so that
   different applications of the encoding operation to the same
   message will produce different encoded messages, which has benefits
   for provable security. For such an encoding method, both an
   encoding and a verification operation are needed unless the
   verifier can reproduce the randomness (e.g., by obtaining the salt
   value from the signer). For a deterministic encoding method only an
   encoding operation is needed. 



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   Two encoding methods for signatures with appendix are employed in
   the signature schemes and are specified here: EMSA-PSS and
   EMSA-PKCS1-v1_5. 

9.1 EMSA-PSS

   This encoding method is parameterized by the choice of hash
   function, mask generation function, and salt length. These options
   should be fixed for a given RSA key, except that the salt length
   can be variable (see [31] for discussion). Suggested hash and mask
   generation functions are given in Appendix B. The encoding method
   is based on Bellare and Rogaway's Probabilistic Signature Scheme
   (PSS) [4][5]. It is randomized and has an encoding operation and a
   verification operation. 

   Figure 2 illustrates the encoding operation.

   __________________________________________________________________

                                  +-----------+
                                  |     M     |
                                  +-----------+
                                        |
                                        V
                                      Hash
                                        |
                                        V
                          +--------+----------+----------+
                     M' = |Padding1|  mHash   |   salt   |
                          +--------+----------+----------+
                                         |
               +--------+----------+     V
         DB =  |Padding2|maskedseed|   Hash
               +--------+----------+     |
                         |               |
                         V               |    +--+
                        xor <--- MGF <---|    |bc|
                         |               |    +--+
                         |               |      |
                         V               V      V
               +-------------------+----------+--+
         EM =  |    maskedDB       |maskedseed|bc|
               +-------------------+----------+--+
   __________________________________________________________________

   Figure 2: EMSA-PSS encoding operation. Verification operation
   follows reverse steps to recover salt, then forward steps to
   recompute and compare H. 



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   Notes.

   1. The encoding method defined here differs from the one in Bellare
      and Rogaway's submission to IEEE P1363a [5] in three respects: 

       * It applies a hash function rather than a mask generation
         function to the message. Even though the mask generation
         function is based on a hash function, it seems more natural
         to apply a hash function directly. 

       * The value that is hashed together with the salt value is the
         string (0x)00 00 00 00 00 00 00 00 || mHash rather than the
         message M itself. Here, mHash is the hash of M. Note that the
         hash function is the same in both steps. See Note 3 below for
         further discussion. (Also, the name "salt" is used instead of
         "seed", as it is more reflective of the value's role.) 

       * The encoded message in EMSA-PSS has nine fixed bits; the
         first bit is 0 and the last eight bits form a "trailer
         field", the octet 0xbc. In the original scheme, only the
         first bit is fixed. The rationale for the trailer field is
         for compatibility with the Rabin-Williams IFSP-RW signature
         primitive in IEEE Std 1363-2000 [26] and the corresponding
         primitive in the draft ISO/IEC 9796-2 [29]. 

   2. Assuming that the mask generation function is based on a hash
      function, it is recommended that the hash function be the same
      as the one that is applied to the message; see Section 8.1 for
      further discussion.

    3. Without compromising the security proof for RSASSA-PSS, one may
      perform steps 1 and 2 of EMSA-PSS-ENCODE and EMSA-PSS-VERIFY
      (the application of the hash function to the message) outside
      the module that computes the rest of the signature operation, so
      that mHash rather than the message M itself is input to the
      module. In other words, the security proof for RSASSA-PSS still
      holds even if an opponent can control the value of mHash. This
      is convenient if the module has limited I/O bandwidth, e.g., a
      smart card. Note that previous versions of PSS [4][5] did not
      have this property. Of course, it may be desirable for other
      security reasons to have the module process the full
      message. For instance, the module may need to "see" what it is
      signing if it does not trust the component that computes the
      hash value. 

   4. Typical salt lengths in octets are hLen (the length of the
      output of the hash function Hash) and 0. In both cases the
      security of RSASSA-PSS can be closely related to the hardness of



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      inverting RSAVP1. Bellare and Rogaway [4] give a tight lower
      bound for the security of the original RSA-PSS scheme, which
      corresponds roughly to the former case, while Coron [12] gives a
      lower bound for the related Full Domain Hashing scheme, which
      corresponds roughly to the latter case. In [13] Coron provides a
      general treatment with various salt lengths ranging from 0 to
      hLen; see [27] for discussion. See also [31], which adapts the
      security proofs in [4][13] to address the differences between
      the original and the present version of RSA-PSS as listed in
      Note 1 above.

   5. As noted in IEEE P1363a [27], the use of randomization in
      signature schemes - such as the salt value in EMSA-PSS - may
      provide a "covert channel" for transmitting information other
      than the message being signed. For more on covert channels, see
      [50].

9.1.1 Encoding operation

   EMSA-PSS-ENCODE (M, emBits)

   Options:

   Hash     hash function (hLen denotes the length in octets of the
            hash function output) 
   MGF      mask generation function
   sLen     intended length in octets of the salt

   Input:
   M        message to be encoded, an octet string 
   emBits   maximal bit length of the integer OS2IP (EM) (see Section
            4.2), at least 8hLen + 8sLen + 9 

   Output:
   EM       encoded message, an octet string of length emLen = 
            \ceil (emBits/8) 

   Errors:  "encoding error"; "message too long"

   Steps:

   1. If the length of M is greater than the input limitation for the
      hash function (2^61 - 1 octets for SHA-1), output "message too
      long" and stop. 

   2. Let mHash = Hash(M), an octet string of length hLen.

   3. If emLen < hLen + sLen + 2, output "encoding error" and stop.



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   4. Generate a random octet string salt of length sLen; if sLen = 0,
      then salt is the empty string. 

   5. Let 

         M' = (0x)00 00 00 00 00 00 00 00 || mHash || salt;

      M' is an octet string of length 8 + hLen + sLen with eight
      initial zero octets. 

   6. Let H = Hash(M'), an octet string of length hLen.

   7. Generate an octet string PS consisting of emLen - sLen - hLen -
      2 zero octets. The length of PS may be 0. 

   8. Let DB = PS || 0x01 || salt; DB is an octet string of length
      emLen - hLen - 1. 

   9. Let dbMask = MGF(H, emLen - hLen - 1).

  10. Let maskedDB = DB \xor dbMask.

  11. Set the leftmost 8emLen - emBits bits of the leftmost octet in
      maskedDB to zero. 

  12. Let EM = maskedDB || H || 0xbc.

  13. Output EM.

9.1.2 Verification operation

   EMSA-PSS-VERIFY (M, EM, emBits)

   Options:
   Hash     hash function (hLen denotes the length in octets of the
            hash function output)
   MGF      mask generation function
   sLen     intended length in octets of the salt

   Input:
   M        message to be verified, an octet string
   EM       encoded message, an octet string of length emLen = 
            \ceil (emBits/8) 
   emBits   maximal bit length of the integer OS2IP (EM) (see Section
            4.2), at least 8hLen + 8sLen + 9 

   Output: 
   "consistent" or "inconsistent"



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   Steps:

   1. If the length of M is greater than the input limitation for the
      hash function (2^61 - 1 octets for SHA-1), output "inconsistent"
      and stop. 

   2. Let mHash = Hash(M), an octet string of length hLen.

   3. If emLen < hLen + sLen + 2, output "inconsistent" and stop.

   4. If the rightmost octet of EM does not have hexadecimal value
      0xbc, output "inconsistent" and stop. 

   5. Let maskedDB be the leftmost emLen - hLen - 1 octets of EM, and
      let H be the next hLen octets.

   6. If the leftmost 8emLen - emBits bits of the leftmost octet in
      maskedDB are not all equal to zero, output "inconsistent" and
      stop.

   7. Let dbMask = MGF(H, emLen - hLen - 1).

   8. Let DB = maskedDB \xor dbMask.

   9. Set the leftmost 8emLen - emBits bits of the leftmost octet in
      DB to zero.

  10. If the emLen - hLen - sLen - 2 leftmost octets of DB are not
      zero or if the octet at position emLen - hLen - sLen - 1 (the
      leftmost position is "position 1") does not have hexadecimal
      value 0x01, output "inconsistent" and stop. 

  11. Let salt be the last sLen octets of DB.

  12. Let

         M' = (0x)00 00 00 00 00 00 00 00 || mHash || salt ;

      M' is an octet string of length 8 + hLen + sLen with eight
      initial zero octets. 

  13. Let H' = Hash(M'), an octet string of length hLen.

  14. If H = H', output "consistent." Otherwise, output
      "inconsistent." 






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9.2 EMSA-PKCS1-v1_5

   This encoding method is deterministic and only has an encoding
   operation.

   EMSA-PKCS1-v1_5-ENCODE (M, emLen)

   Option:
   Hash     hash function (hLen denotes the length in octets of the
            hash function output)

   Input:
   M        message to be encoded
   emLen    intended length in octets of the encoded message, at least
            tLen + 11, where tLen is the octet length of the DER
            encoding T of a certain value computed during the encoding
            operation

   Output:
   EM       encoded message, an octet string of length emLen

   Errors: 
   "message too long"; "intended encoded message length too short" 

   Steps:

   1. Apply the hash function to the message M to produce a hash value
      H: 

         H = Hash(M).

      If the hash function outputs "message too long," output "message
      too long" and stop. 

   2. Encode the algorithm ID for the hash function and the hash value
      into an ASN.1 value of type DigestInfo (see Appendix A.2.4) with
      the Distinguished Encoding Rules (DER), where the type
      DigestInfo has the syntax 

      DigestInfo ::= SEQUENCE {
          digestAlgorithm AlgorithmIdentifier, 
          digest OCTET STRING 
      }

      The first field identifies the hash function and the second
      contains the hash value. Let T be the DER encoding of the
      DigestInfo value (see the notes below) and let tLen be the
      length in octets of T. 



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   3. If emLen < tLen + 11, output "intended encoded message length
      too short" and stop. 

   4. Generate an octet string PS consisting of emLen - tLen - 3
      octets with hexadecimal value 0xff. The length of PS will be at
      least 8 octets. 

   5. Concatenate PS, the DER encoding T, and other padding to form
      the encoded message EM as 

         EM = 0x00 || 0x01 || PS || 0x00 || T.

   6. Output EM.

   Notes.

   1. For the six hash functions mentioned in Appendix B.1, the DER
      encoding T of the DigestInfo value is equal to the following: 

      MD2:     (0x)30 20 30 0c 06 08 2a 86 48 86 f7 0d 02 02 05 00 04
                   10 || H.   
      MD5:     (0x)30 20 30 0c 06 08 2a 86 48 86 f7 0d 02 05 05 00 04
                   10 || H.
      SHA-1:   (0x)30 21 30 09 06 05 2b 0e 03 02 1a 05 00 04 14 || H. 
      SHA-256: (0x)30 31 30 0d 06 09 60 86 48 01 65 03 04 02 01 05 00
                   04 20 || H.
      SHA-384: (0x)30 41 30 0d 06 09 60 86 48 01 65 03 04 02 02 05 00
                   04 30 || H.
      SHA-512: (0x)30 51 30 0d 06 09 60 86 48 01 65 03 04 02 03 05 00
                   04 40 || H.

   2. In version 1.5 of this document, T was defined as the BER
      encoding, rather than the DER encoding, of the DigestInfo value.
      In particular, it is possible - at least in theory - that the
      verification operation defined in this document (as well as in
      version 2.0) rejects a signature that is valid with respect to
      the specification given in PKCS #1 v1.5. This occurs if other
      rules than DER are applied to DigestInfo (e.g., an indefinite
      length encoding of the underlying SEQUENCE type). While this is
      unlikely to be a concern in practice, a cautious implementer may
      choose to employ a verification operation based on a BER
      decoding operation as specified in PKCS #1 v1.5. In this manner,
      compatibility with any valid implementation based on PKCS #1
      v1.5 is obtained. Such a verification operation should indicate
      whether the underlying BER encoding is a DER encoding and hence
      whether the signature is valid with respect to the specification
      given in this document. 




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A. ASN.1 syntax

A.1 RSA key representation

   This section defines ASN.1 object identifiers for RSA public and
   private keys, and defines the types RSAPublicKey and
   RSAPrivateKey. The intended application of these definitions
   includes X.509 certificates, PKCS #8 [46], and PKCS #12 [47].

   The object identifier rsaEncryption identifies RSA public and
   private keys as defined in Appendices A.1.1 and A.1.2. The
   parameters field associated with this OID in a value of type
   AlgorithmIdentifier shall have a value of type NULL. 

   rsaEncryption    OBJECT IDENTIFIER ::= { pkcs-1 1 }

   The definitions in this section have been extended to support
   multi-prime RSA, but are backward compatible with previous
   versions. 

A.1.1 RSA public key syntax

   An RSA public key should be represented with the ASN.1 type
   RSAPublicKey:

      RSAPublicKey ::= SEQUENCE {
          modulus           INTEGER,  -- n
          publicExponent    INTEGER   -- e 
      }

   The fields of type RSAPublicKey have the following meanings:

    * modulus is the RSA modulus n.

    * publicExponent is the RSA public exponent e.

A.1.2 RSA private key syntax

   An RSA private key should be represented with the ASN.1 type
   RSAPrivateKey:

      RSAPrivateKey ::= SEQUENCE {
          version           Version, 
          modulus           INTEGER,  -- n
          publicExponent    INTEGER,  -- e
          privateExponent   INTEGER,  -- d
          prime1            INTEGER,  -- p
          prime2            INTEGER,  -- q



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          exponent1         INTEGER,  -- d mod (p-1)
          exponent2         INTEGER,  -- d mod (q-1) 
          coefficient       INTEGER,  -- (inverse of q) mod p
          otherPrimeInfos   OtherPrimeInfos OPTIONAL 
      }

   The fields of type RSAPrivateKey have the following meanings:

    * version is the version number, for compatibility with future
      revisions of this document. It shall be 0 for this version of
      the document, unless multi-prime is used, in which case it shall
      be 1. 

         Version ::= INTEGER { two-prime(0), multi(1) }
            (CONSTRAINED BY
            {-- version must be multi if otherPrimeInfos present --}) 

    * modulus is the RSA modulus n.

    * publicExponent is the RSA public exponent e.

    * privateExponent is the RSA private exponent d.

    * prime1 is the prime factor p of n.

    * prime2 is the prime factor q of n.

    * exponent1 is d mod (p - 1).

    * exponent2 is d mod (q - 1).

    * coefficient is the CRT coefficient q^(-1) mod p.

    * otherPrimeInfos contains the information for the additional
      primes r_3, ..., r_u, in order. It shall be omitted if version
      is 0 and shall contain at least one instance of OtherPrimeInfo
      if version is 1. 

         OtherPrimeInfos ::= SEQUENCE SIZE(1..MAX) OF OtherPrimeInfo

         OtherPrimeInfo ::= SEQUENCE {
             prime             INTEGER,  -- ri
             exponent          INTEGER,  -- di
             coefficient       INTEGER   -- ti
         }

   The fields of type OtherPrimeInfo have the following meanings:




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    * prime is a prime factor r_i of n, where i ■ 3.

    * exponent is d_i = d mod (r_i - 1).

    * coefficient is the CRT coefficient t_i = (r_1 * r_2 * ... *
      r_(i-1))^(-1) mod r_i. 

   Note. It is important to protect the RSA private key against both
   disclosure and modification. Techniques for such protection are
   outside the scope of this document. Methods for storing and
   distributing private keys and other cryptographic data are
   described in PKCS #12 and #15.

A.2 Scheme identification

   This section defines object identifiers for the encryption and
   signature schemes. The schemes compatible with PKCS #1 v1.5 have
   the same definitions as in PKCS #1 v1.5. The intended application
   of these definitions includes X.509 certificates and PKCS #7. 

   Here are type identifier definitions for the PKCS #1 OIDs:

      PKCS1Algorithms    ALGORITHM-IDENTIFIER ::= {
          { OID rsaEncryption              PARAMETERS NULL } |
          { OID md2WithRSAEncryption       PARAMETERS NULL } |
          { OID md5WithRSAEncryption       PARAMETERS NULL } |
          { OID sha1WithRSAEncryption      PARAMETERS NULL } |
          { OID sha256WithRSAEncryption    PARAMETERS NULL } |
          { OID sha384WithRSAEncryption    PARAMETERS NULL } |
          { OID sha512WithRSAEncryption    PARAMETERS NULL } |
          { OID id-RSAES-OAEP PARAMETERS RSAES-OAEP-params } | 
          PKCS1PSourceAlgorithms                             |
          { OID id-RSASSA-PSS PARAMETERS RSASSA-PSS-params } , 
          ...  -- Allows for future expansion --
      }

A.2.1 RSAES-OAEP

   The object identifier id-RSAES-OAEP identifies the RSAES-OAEP
   encryption scheme.

      id-RSAES-OAEP    OBJECT IDENTIFIER ::= { pkcs-1 7 }

   The parameters field associated with this OID in a value of type
   AlgorithmIdentifier shall have a value of type RSAES-OAEP-params: 






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      RSAES-OAEP-params ::= SEQUENCE {
          hashAlgorithm     [0] HashAlgorithm    DEFAULT sha1, 
          maskGenAlgorithm  [1] MaskGenAlgorithm DEFAULT mgf1SHA1, 
          pSourceAlgorithm  [2] PSourceAlgorithm DEFAULT pSpecifiedEmpty
      }

   The fields of type RSAES-OAEP-params have the following meanings:

    * hashAlgorithm identifies the hash function. It shall be an
      algorithm ID with an OID in the set OAEP-PSSDigestAlgorithms. 
      For a discussion of supported hash functions, see Appendix B.1. 

         HashAlgorithm ::= AlgorithmIdentifier { 
            {OAEP-PSSDigestAlgorithms} 
         }

         OAEP-PSSDigestAlgorithms    ALGORITHM-IDENTIFIER ::= {
             { OID id-sha1 PARAMETERS NULL   }|
             { OID id-sha256 PARAMETERS NULL }|
             { OID id-sha384 PARAMETERS NULL }|
             { OID id-sha512 PARAMETERS NULL }, 
             ...  -- Allows for future expansion --
         }

      The default hash function is SHA-1:
   
         sha1    HashAlgorithm ::= {
             algorithm   id-sha1, 
             parameters  SHA1Parameters : NULL
         }

         SHA1Parameters ::= NULL

    * maskGenAlgorithm identifies the mask generation function. It
      shall be an algorithm ID with an OID in the set
      PKCS1MGFAlgorithms, which for this version shall consist of
      id-mgf1, identifying the MGF1 mask generation function (see
      Appendix B.2.1). The parameters field associated with id-mgf1
      shall be an algorithm ID with an OID in the set
      OAEP-PSSDigestAlgorithms, identifying the hash function on which
      MGF1 is based. 

         MaskGenAlgorithm ::= AlgorithmIdentifier { 
            {PKCS1MGFAlgorithms}
         }






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         PKCS1MGFAlgorithms    ALGORITHM-IDENTIFIER ::= {
             { OID id-mgf1 PARAMETERS HashAlgorithm }, 
             ...  -- Allows for future expansion --
         }

      The default mask generation function is MGF1 with SHA-1: 

         mgf1SHA1    MaskGenAlgorithm ::= {
             algorithm   id-mgf1, 
             parameters  HashAlgorithm : sha1
         }

    * pSourceAlgorithm identifies the source (and possibly the value)
      of the label L. It shall be an algorithm ID with an OID in the
      set PKCS1PSourceAlgorithms, which for this version shall consist
      of id-pSpecified, indicating that the label is specified
      explicitly. The parameters field associated with id-pSpecified
      shall have a value of type OCTET STRING, containing the
      label. In previous versions of this specification, the term
      "encoding parameters" was used rather than "label", hence the
      name of the type below. 

         PSourceAlgorithm ::= AlgorithmIdentifier { 
            {PKCS1PSourceAlgorithms} 
         }

         PKCS1PSourceAlgorithms    ALGORITHM-IDENTIFIER ::= {
             { OID id-pSpecified PARAMETERS EncodingParameters }, 
             ...  -- Allows for future expansion --
         }

         id-pSpecified    OBJECT IDENTIFIER ::= { pkcs-1 9 }

         EncodingParameters ::= OCTET STRING(SIZE(0..MAX))
 
      The default label is an empty string (so that lHash will contain
      the hash of the empty string):

         pSpecifiedEmpty    PSourceAlgorithm ::= {
             algorithm   id-pSpecified, 
             parameters  EncodingParameters : emptyString
         }

         emptyString    EncodingParameters ::= ''H 

      If all of the default values of the fields in RSAES-OAEP-params
      are used, then the algorithm identifier will have the following
      value:



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         rSAES-OAEP-Default-Identifier  RSAES-AlgorithmIdentifier ::= {
             algorithm   id-RSAES-OAEP, 
             parameters  RSAES-OAEP-params : {
                 hashAlgorithm       sha1, 
                 maskGenAlgorithm    mgf1SHA1, 
                 pSourceAlgorithm    pSpecifiedEmpty
             }
         }

         RSAES-AlgorithmIdentifier ::= AlgorithmIdentifier {
            {PKCS1Algorithms} 
         } 

A.2.2 RSAES-PKCS1-v1_5

   The object identifier rsaEncryption (see Appendix A.1) identifies
   the RSAES-PKCS1-v1_5 encryption scheme. The parameters field
   associated with this OID in a value of type AlgorithmIdentifier
   shall have a value of type NULL. This is the same as in PKCS #1
   v1.5. 

      rsaEncryption    OBJECT IDENTIFIER ::= { pkcs-1 1 }

A.2.3 RSASSA-PSS

   The object identifier id-RSASSA-PSS identifies the RSASSA-PSS
   encryption scheme. 

      id-RSASSA-PSS    OBJECT IDENTIFIER ::= { pkcs-1 10 }

   The parameters field associated with this OID in a value of type
   AlgorithmIdentifier shall have a value of type RSASSA-PSS-params: 

      RSASSA-PSS-params ::= SEQUENCE {
          hashAlgorithm      [0] HashAlgorithm    DEFAULT sha1, 
          maskGenAlgorithm   [1] MaskGenAlgorithm DEFAULT mgf1SHA1, 
          saltLength         [2] INTEGER          DEFAULT 20,
          trailerField       [3] TrailerField     DEFAULT trailerFieldBC
      }

   The fields of type RSASSA-PSS-params have the following meanings:

    * hashAlgorithm identifies the hash function. It shall be an
      algorithm ID with an OID in the set OAEP-PSSDigestAlgorithms
      (see Appendix A.2.1). The default hash function is SHA-1. 

    * maskGenAlgorithm identifies the mask generation function. It
      shall be an algorithm ID with an OID in the set



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      PKCS1MGFAlgorithms (see Appendix A.2.1). The default mask 
      generation function is MGF1 with SHA-1. For MGF1 (and more
      generally, for other mask generation functions based on a hash
      function), it is recommended that the underlying hash function
      be the same as the one identified by hashAlgorithm; see Note 2
      in Section 9.1 for further comments. 

    * saltLength is the octet length of the salt. It shall be an
      integer. For a given hashAlgorithm, the default value of
      saltLength is the octet length of the hash value. Unlike the
      other fields of type RSASSA-PSS-params, saltLength does not need
      to be fixed for a given RSA key pair.

    * trailerField is the trailer field number, for compatibility with
      the draft IEEE P1363a [27]. It shall be 1 for this version of
      the document, which represents the trailer field with
      hexadecimal value 0xbc. Other trailer fields (including the
      trailer field HashID || 0xcc in IEEE P1363a) are not supported
      in this document. 

         TrailerField ::= INTEGER { trailerFieldBC(1) }

      If the default values of the hashAlgorithm, maskGenAlgorithm,
      and trailerField fields of RSASSA-PSS-params are used, then the 
      algorithm identifier will have the following value: 

         rSASSA-PSS-Default-Identifier  RSASSA-AlgorithmIdentifier ::= {
             algorithm   id-RSASSA-PSS, 
             parameters  RSASSA-PSS-params : {
                 hashAlgorithm       sha1, 
                 maskGenAlgorithm    mgf1SHA1, 
                 saltLength          20,
                 trailerField        trailerFieldBC
             }
         }

         RSASSA-AlgorithmIdentifier ::= 
             AlgorithmIdentifier { {PKCS1Algorithms} } 

   Note. In some applications, the hash function underlying a
   signature scheme is identified separately from the rest of the
   operations in the signature scheme. For instance, in PKCS #7 [45],
   a hash function identifier is placed before the message and a
   "digest encryption" algorithm identifier (indicating the rest of
   the operations) is carried with the signature. In order for PKCS #7
   to support the RSASSA-PSS signature scheme, an object identifier
   would need to be defined for the operations in RSASSA-PSS after the
   hash function (analogous to the RSAEncryption OID for the



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   RSASSA-PKCS1-v1_5 scheme). S/MIME CMS [25] takes a different
   approach. Although a hash function identifier is placed before the
   message, an algorithm identifier for the full signature scheme may
   be carried with a CMS signature (this is done for DSA signatures).
   Following this convention, the id-RSASSA-PSS OID can be used to
   identify RSASSA-PSS signatures in CMS. Since CMS is considered the
   successor to PKCS #7 and new developments such as the addition of
   support for RSASSA-PSS will be pursued with respect to CMS rather
   than PKCS #7, an OID for the "rest of" RSASSA-PSS is not defined in
   this version of PKCS #1.

A.2.4 RSASSA-PKCS1-v1_5

   The object identifier for RSASSA-PKCS1-v1_5 shall be one of the
   following. The choice of OID depends on the choice of hash
   algorithm: MD2, MD5, SHA-1, SHA-256, SHA-384, or SHA-512. Note that
   if either MD2 or MD5 is used, then the OID is just as in PKCS #1
   v1.5. For each OID, the parameters field associated with this OID
   in a value of type AlgorithmIdentifier shall have a value of type
   NULL. The OID should be chosen in accordance with the following
   table: 

      Hash algorithm   OID
      --------------------------------------------------------
      MD2              md2WithRSAEncryption    ::= {pkcs-1 2}
      MD5              md5WithRSAEncryption    ::= {pkcs-1 4}
      SHA-1            sha1WithRSAEncryption   ::= {pkcs-1 5} 
      SHA-256          sha256WithRSAEncryption ::= {pkcs-1 11}
      SHA-384          sha384WithRSAEncryption ::= {pkcs-1 12}
      SHA-512          sha512WithRSAEncryption ::= {pkcs-1 13}

   The EMSA-PKCS1-v1_5 encoding method includes an ASN.1 value of type
   DigestInfo, where the type DigestInfo has the syntax 

      DigestInfo ::= SEQUENCE { 
          digestAlgorithm DigestAlgorithm,
          digest OCTET STRING 
      }

   digestAlgorithm identifies the hash function and shall be an
   algorithm ID with an OID in the set PKCS1-v1-5DigestAlgorithms. For
   a discussion of supported hash functions, see Appendix B.1. 

      DigestAlgorithm ::= 
          AlgorithmIdentifier { {PKCS1-v1-5DigestAlgorithms} }






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      PKCS1-v1-5DigestAlgorithms    ALGORITHM-IDENTIFIER ::= {
          { OID id-md2 PARAMETERS NULL    }|
          { OID id-md5 PARAMETERS NULL    }|
          { OID id-sha1 PARAMETERS NULL   }|
          { OID id-sha256 PARAMETERS NULL }|
          { OID id-sha384 PARAMETERS NULL }|
          { OID id-sha512 PARAMETERS NULL } 
      }
 
B. Supporting techniques

   This section gives several examples of underlying functions
   supporting the encryption schemes in Section 7 and the encoding
   methods in Section 9. A range of techniques is given here to allow
   compatibility with existing applications as well as migration to
   new techniques. While these supporting techniques are appropriate
   for applications to implement, none of them is required to be
   implemented. It is expected that profiles for PKCS #1 v2.1 will be
   developed that specify particular supporting techniques.

   This section also gives object identifiers for the supporting
   techniques. 

B.1 Hash functions

   Hash functions are used in the operations contained in Sections 7
   and 9. Hash functions are deterministic, meaning that the output is
   completely determined by the input. Hash functions take octet
   strings of variable length, and generate fixed length octet
   strings. The hash functions used in the operations contained in
   Sections 7 and 9 should generally be collision-resistant. This
   means that it is infeasible to find two distinct inputs to the hash
   function that produce the same output. A collision-resistant hash
   function also has the desirable property of being one-way; this
   means that given an output, it is infeasible to find an input whose
   hash is the specified output. In addition to the requirements, the
   hash function should yield a mask generation function (Appendix
   B.2) with pseudorandom output. 

   Six hash functions are given as examples for the encoding methods
   in this document: MD2 [33], MD5 [41], SHA-1 [38], and the proposed
   algorithms SHA-256, SHA-384, and SHA-512 [39]. For the RSAES-OAEP
   encryption scheme and EMSA-PSS encoding method, only SHA-1 and
   SHA-256/384/512 are recommended. For the EMSA-PKCS1-v1_5 encoding
   method, SHA-1 or SHA-256/384/512 are recommended for new
   applications. MD2 and MD5 are recommended only for compatibility
   with existing applications based on PKCS #1 v1.5. 




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   The object identifiers id-md2, id-md5, id-sha1, id-sha256,
   id-sha384, and id-sha512, identify the respective hash functions: 

      id-md2      OBJECT IDENTIFIER ::= { 
          iso(1) member-body(2) us(840) rsadsi(113549) 
          digestAlgorithm(2) 2
      }

      id-md5      OBJECT IDENTIFIER ::= { 
          iso(1) member-body(2) us(840) rsadsi(113549) 
          digestAlgorithm(2) 5
      }

      id-sha1    OBJECT IDENTIFIER ::= {
          iso(1) identified-organization(3) oiw(14) secsig(3)
          algorithms(2) 26 
      }

      id-sha256    OBJECT IDENTIFIER ::= {
          joint-iso-itu-t(2) country(16) us(840) organization(1)
          gov(101) csor(3) nistalgorithm(4) hashalgs(2) 1 
      }

      id-sha384    OBJECT IDENTIFIER ::= {
          joint-iso-itu-t(2) country(16) us(840) organization(1)
          gov(101) csor(3) nistalgorithm(4) hashalgs(2) 2 
      }

      id-sha512    OBJECT IDENTIFIER ::= {
          joint-iso-itu-t(2) country(16) us(840) organization(1)
          gov(101) csor(3) nistalgorithm(4) hashalgs(2) 3 
      }

   The parameters field associated with these OIDs in a value of type
   AlgorithmIdentifier shall have a value of type NULL. 

   Note. Version 1.5 of PKCS #1 also allowed for the use of MD4 in
   signature schemes. The cryptanalysis of MD4 has progressed
   significantly in the intervening years. For example, Dobbertin [18]
   demonstrated how to find collisions for MD4 and that the first two
   rounds of MD4 are not one-way [20]. Because of these results and
   others (e.g. [8]), MD4 is no longer recommended. There have also
   been advances in the cryptanalysis of MD2 and MD5, although not
   enough to warrant removal from existing applications. Rogier and
   Chauvaud [43] demonstrated how to find collisions in a modified
   version of MD2. No one has demonstrated how to find collisions for
   the full MD5 algorithm, although partial results have been found
   (e.g. [9][19]).



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   To address these concerns, SHA-1, SHA-256, SHA-384, or SHA-512 are
   recommended for new applications. As of today, the best (known)
   collision attacks against these hash functions are generic attacks
   with complexity 2^(L/2), where L is the bit length of the hash
   output. For the signature schemes in this document, a collision
   attack is easily translated into a signature forgery. Therefore,
   the value L / 2 should be at least equal to the desired security
   level in bits of the signature scheme (a security level of B bits
   means that the best attack has complexity 2^B). The same rule of
   thumb can be applied to RSAES-OAEP; it is recommended that the bit
   length of the seed (which is equal to the bit length of the hash
   output) be twice the desired security level in bits.

B.2 Mask generation functions

   A mask generation function takes an octet string of variable length
   and a desired output length as input, and outputs an octet string
   of the desired length. There may be restrictions on the length of
   the input and output octet strings, but such bounds are generally
   very large. Mask generation functions are deterministic; the octet
   string output is completely determined by the input octet string.
   The output of a mask generation function should be pseudorandom:
   Given one part of the output but not the input, it should be
   infeasible to predict another part of the output. The provable
   security of RSAES-OAEP and RSASSA-PSS relies on the random nature
   of the output of the mask generation function, which in turn relies
   on the random nature of the underlying hash.

   One mask generation function is given here: MGF1, which is based on
   a hash function. MGF1 coincides with the mask generation functions
   defined in IEEE Std 1363-2000 [26] and the draft ANSI X9.44
   [1]. Future versions of this document may define other mask
   generation functions. 

B.2.1 MGF1

   MGF1 is a Mask Generation Function based on a hash function.

   MGF1 (mgfSeed, maskLen)

   Options:
   Hash     hash function (hLen denotes the length in octets of the
            hash function output) 

   Input:   
   mgfSeed  seed from which mask is generated, an octet string
   maskLen  intended length in octets of the mask, at most 2^32 hLen




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   Output:   
   mask     mask, an octet string of length maskLen

   Error:   "mask too long"

   Steps:

   1. If maskLen > 2^32 hLen, output "mask too long" and stop.

   2. Let T be the empty octet string.

   3. For counter from 0 to \ceil (maskLen / hLen) - 1, do the
      following: 

      a. Convert counter to an octet string C of length 4 octets (see
         Section 4.1): 

            C = I2OSP (counter, 4) .

      b. Concatenate the hash of the seed mgfSeed and C to the octet
         string T: 

            T = T || Hash(mgfSeed || C) .

   4. Output the leading maskLen octets of T as the octet string mask.
 
   The object identifier id-mgf1 identifies the MGF1 mask generation
   function:

   id-mgf1    OBJECT IDENTIFIER ::= { pkcs-1 8 }

   The parameters field associated with this OID in a value of type
   AlgorithmIdentifier shall have a value of type hashAlgorithm,
   identifying the hash function on which MGF1 is based.

C. ASN.1 module

   PKCS-1 {
       iso(1) member-body(2) us(840) rsadsi(113549) pkcs(1) pkcs-1(1) 
       modules(0) pkcs-1(1) 
   }
   
   -- $ Revision: 2.1 $
   
   -- This module has been checked for conformance with the ASN.1 
   -- standard by the OSS ASN.1 Tools

   DEFINITIONS EXPLICIT TAGS ::=
   


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   BEGIN

   -- EXPORTS ALL
   -- All types and values defined in this module are exported for use 
   -- in other ASN.1 modules.
   
   IMPORTS
   
   id-sha256, id-sha384, id-sha512 
       FROM NIST-SHA2 {
           joint-iso-itu-t(2) country(16) us(840) organization(1) 
           gov(101) csor(3) nistalgorithm(4) modules(0) sha2(1)
       };
   
   -- ============================
   --   Basic object identifiers 
   -- ============================
   
   -- The DER encoding of this in hexadecimal is:
   -- (0x)06 08 
   --        2A 86 48 86 F7 0D 01 01
   --
   pkcs-1    OBJECT IDENTIFIER ::= { 
       iso(1) member-body(2) us(840) rsadsi(113549) pkcs(1) 1
   }
   
   --
   -- When rsaEncryption is used in an AlgorithmIdentifier the 
   -- parameters MUST be present and MUST be NULL.
   --
   rsaEncryption    OBJECT IDENTIFIER ::= { pkcs-1 1 }
   
   --
   -- When id-RSAES-OAEP is used in an AlgorithmIdentifier the 
   -- parameters MUST be present and MUST be RSAES-OAEP-params.
   --
   id-RSAES-OAEP    OBJECT IDENTIFIER ::= { pkcs-1 7 }
   
   --
   -- When id-pSpecified is used in an AlgorithmIdentifier the 
   -- parameters MUST be an OCTET STRING.
   --
   id-pSpecified    OBJECT IDENTIFIER ::= { pkcs-1 9 }
   
   -- When id-RSASSA-PSS is used in an AlgorithmIdentifier the 
   -- parameters MUST be present and MUST be RSASSA-PSS-params.
   --
   id-RSASSA-PSS    OBJECT IDENTIFIER ::= { pkcs-1 10 }



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   --
   -- When the following OIDs are used in an AlgorithmIdentifier the
   -- parameters MUST be present and MUST be NULL.
   --
   md2WithRSAEncryption       OBJECT IDENTIFIER ::= { pkcs-1 2 }
   md5WithRSAEncryption       OBJECT IDENTIFIER ::= { pkcs-1 4 }
   sha1WithRSAEncryption      OBJECT IDENTIFIER ::= { pkcs-1 5 }
   sha256WithRSAEncryption    OBJECT IDENTIFIER ::= { pkcs-1 11 }
   sha384WithRSAEncryption    OBJECT IDENTIFIER ::= { pkcs-1 12 }
   sha512WithRSAEncryption    OBJECT IDENTIFIER ::= { pkcs-1 13 }
   
   --
   -- This OID really belongs in a module with the secsig OIDs.
   --
   id-sha1    OBJECT IDENTIFIER ::= {
       iso(1) identified-organization(3) oiw(14) secsig(3) 
       algorithms(2) 26 
   }
   
   --
   -- OIDs for MD2 and MD5, allowed only in EMSA-PKCS1-v1_5.
   --
   id-md2 OBJECT IDENTIFIER ::= {
       iso(1) member-body(2) us(840) rsadsi(113549) digestAlgorithm(2) 2
   }
   
   id-md5 OBJECT IDENTIFIER ::= {
       iso(1) member-body(2) us(840) rsadsi(113549) digestAlgorithm(2) 5
   }
   
   --
   -- When id-mgf1 is used in an AlgorithmIdentifier the parameters MUST
   -- be present and MUST be a HashAlgorithm, for example sha1.
   --
   id-mgf1    OBJECT IDENTIFIER ::= { pkcs-1 8 }
   
   -- ================
   --   Useful types
   -- ================

   ALGORITHM-IDENTIFIER ::= CLASS {
       &id    OBJECT IDENTIFIER  UNIQUE, 
       &Type  OPTIONAL
   }
       WITH SYNTAX { OID &id [PARAMETERS &Type] }






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   --
   -- Note: the parameter InfoObjectSet in the following definitions 
   -- allows a distinct information object set to be specified for sets
   -- of algorithms such as:  
   -- DigestAlgorithms    ALGORITHM-IDENTIFIER ::= {
   --     { OID id-md2  PARAMETERS NULL }|
   --     { OID id-md5  PARAMETERS NULL }|
   --     { OID id-sha1 PARAMETERS NULL }
   -- }
   --
   
   AlgorithmIdentifier { ALGORITHM-IDENTIFIER:InfoObjectSet } ::= 
   SEQUENCE {
       algorithm  ALGORITHM-IDENTIFIER.&id({InfoObjectSet}), 
       parameters 
           ALGORITHM-IDENTIFIER.&Type({InfoObjectSet}{@.algorithm})  
               OPTIONAL
   }
   
   -- ==============
   --   Algorithms
   -- ==============
   
   --
   -- Allowed EME-OAEP and EMSA-PSS digest algorithms.
   --
   OAEP-PSSDigestAlgorithms    ALGORITHM-IDENTIFIER ::= {
       { OID id-sha1 PARAMETERS NULL   }|
       { OID id-sha256 PARAMETERS NULL }|
       { OID id-sha384 PARAMETERS NULL }|
       { OID id-sha512 PARAMETERS NULL }, 
       ...  -- Allows for future expansion --
   }
   
   --
   -- Allowed EMSA-PKCS1-v1_5 digest algorithms.
   --
   PKCS1-v1-5DigestAlgorithms    ALGORITHM-IDENTIFIER ::= {
       { OID id-md2 PARAMETERS NULL    }|
       { OID id-md5 PARAMETERS NULL    }|
       { OID id-sha1 PARAMETERS NULL   }|
       { OID id-sha256 PARAMETERS NULL }|
       { OID id-sha384 PARAMETERS NULL }|
       { OID id-sha512 PARAMETERS NULL } 
   }






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   sha1    HashAlgorithm ::= {
       algorithm   id-sha1, 
       parameters  SHA1Parameters : NULL
   }
   
   HashAlgorithm ::= AlgorithmIdentifier { {OAEP-PSSDigestAlgorithms} }
   
   SHA1Parameters ::= NULL
   
   --
   -- Allowed mask generation function algorithms. 
   -- If the identifier is id-mgf1, the parameters are a HashAlgorithm. 
   --
   PKCS1MGFAlgorithms    ALGORITHM-IDENTIFIER ::= {
       { OID id-mgf1 PARAMETERS HashAlgorithm }, 
       ...  -- Allows for future expansion --
   }
   
   --
   -- Default AlgorithmIdentifier for id-RSAES-OAEP.maskGenAlgorithm and
   -- id-RSASSA-PSS.maskGenAlgorithm.
   --
   mgf1SHA1    MaskGenAlgorithm ::= {
       algorithm   id-mgf1, 
       parameters  HashAlgorithm : sha1
   }
   
   MaskGenAlgorithm ::= AlgorithmIdentifier { {PKCS1MGFAlgorithms} }
   
   --
   -- Allowed algorithms for pSourceAlgorithm.
   --
   PKCS1PSourceAlgorithms    ALGORITHM-IDENTIFIER ::= {
       { OID id-pSpecified PARAMETERS EncodingParameters }, 
       ...  -- Allows for future expansion --
   }
   
   EncodingParameters ::= OCTET STRING(SIZE(0..MAX))
   
   -- 
   -- This identifier means that the label L is an empty string, so the
   -- digest of the empty string appears in the RSA block before 
   -- masking.
   --
   pSpecifiedEmpty    PSourceAlgorithm ::= {
       algorithm   id-pSpecified, 
       parameters  EncodingParameters : emptyString
   }



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   PSourceAlgorithm ::= AlgorithmIdentifier { {PKCS1PSourceAlgorithms} }
   
   emptyString    EncodingParameters ::= ''H 
   
   --
   -- Type identifier definitions for the PKCS #1 OIDs.
   --
   PKCS1Algorithms    ALGORITHM-IDENTIFIER ::= {
       { OID rsaEncryption              PARAMETERS NULL } |
       { OID md2WithRSAEncryption       PARAMETERS NULL } |
       { OID md5WithRSAEncryption       PARAMETERS NULL } |
       { OID sha1WithRSAEncryption      PARAMETERS NULL } |
       { OID sha256WithRSAEncryption    PARAMETERS NULL } |
       { OID sha384WithRSAEncryption    PARAMETERS NULL } |
       { OID sha512WithRSAEncryption    PARAMETERS NULL } |
       { OID id-RSAES-OAEP PARAMETERS RSAES-OAEP-params } | 
       PKCS1PSourceAlgorithms                             |
       { OID id-RSASSA-PSS PARAMETERS RSASSA-PSS-params } , 
       ...  -- Allows for future expansion --
   }
   
   -- ===================
   --   Main structures
   -- ===================
   
   RSAPublicKey ::= SEQUENCE {
       modulus           INTEGER,  -- n
       publicExponent    INTEGER   -- e 
   }
   
   -- 
   -- Representation of RSA private key with information for the CRT 
   -- algorithm.
   --
   RSAPrivateKey ::= SEQUENCE {
       version           Version, 
       modulus           INTEGER,  -- n
       publicExponent    INTEGER,  -- e
       privateExponent   INTEGER,  -- d
       prime1            INTEGER,  -- p
       prime2            INTEGER,  -- q
       exponent1         INTEGER,  -- d mod (p-1)
       exponent2         INTEGER,  -- d mod (q-1) 
       coefficient       INTEGER,  -- (inverse of q) mod p
       otherPrimeInfos   OtherPrimeInfos OPTIONAL 
   }





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   Version ::= INTEGER { two-prime(0), multi(1) }
       (CONSTRAINED BY {
           -- version must be multi if otherPrimeInfos present --
       })
   
   OtherPrimeInfos ::= SEQUENCE SIZE(1..MAX) OF OtherPrimeInfo

   OtherPrimeInfo ::= SEQUENCE {
       prime             INTEGER,  -- ri
       exponent          INTEGER,  -- di
       coefficient       INTEGER   -- ti
   }
   
   -- 
   -- AlgorithmIdentifier.parameters for id-RSAES-OAEP.
   -- Note that the tags in this Sequence are explicit.
   --
   RSAES-OAEP-params ::= SEQUENCE {
       hashAlgorithm      [0] HashAlgorithm     DEFAULT sha1, 
       maskGenAlgorithm   [1] MaskGenAlgorithm  DEFAULT mgf1SHA1, 
       pSourceAlgorithm   [2] PSourceAlgorithm  DEFAULT pSpecifiedEmpty 
   }
   
   -- 
   -- Identifier for default RSAES-OAEP algorithm identifier.
   -- The DER Encoding of this is in hexadecimal:
   -- (0x)30 0D
   --        06 09 
   --           2A 86 48 86 F7 0D 01 01 07
   --        30 00   
   -- Notice that the DER encoding of default values is "empty". 
   --
   
   rSAES-OAEP-Default-Identifier    RSAES-AlgorithmIdentifier ::= {
       algorithm   id-RSAES-OAEP, 
       parameters  RSAES-OAEP-params : {
           hashAlgorithm       sha1, 
           maskGenAlgorithm    mgf1SHA1, 
           pSourceAlgorithm    pSpecifiedEmpty
       }
   }
   
   RSAES-AlgorithmIdentifier ::= 
       AlgorithmIdentifier { {PKCS1Algorithms} }
   






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   -- 
   -- AlgorithmIdentifier.parameters for id-RSASSA-PSS.
   -- Note that the tags in this Sequence are explicit.
   --
   RSASSA-PSS-params ::= SEQUENCE {
       hashAlgorithm      [0] HashAlgorithm      DEFAULT sha1, 
       maskGenAlgorithm   [1] MaskGenAlgorithm   DEFAULT mgf1SHA1, 
       saltLength         [2] INTEGER            DEFAULT 20,
       trailerField       [3] TrailerField       DEFAULT trailerFieldBC
   }
   
   TrailerField ::= INTEGER { trailerFieldBC(1) }
   
   -- 
   -- Identifier for default RSASSA-PSS algorithm identifier
   -- The DER Encoding of this is in hexadecimal:
   -- (0x)30 0D
   --        06 09 
   --           2A 86 48 86 F7 0D 01 01 0A
   --        30 00   
   -- Notice that the DER encoding of default values is "empty". 
   --
   rSASSA-PSS-Default-Identifier    RSASSA-AlgorithmIdentifier ::= {
       algorithm   id-RSASSA-PSS, 
       parameters  RSASSA-PSS-params : {
           hashAlgorithm       sha1, 
           maskGenAlgorithm    mgf1SHA1, 
           saltLength          20,
           trailerField        trailerFieldBC
       }
   }
   
   RSASSA-AlgorithmIdentifier ::= 
       AlgorithmIdentifier { {PKCS1Algorithms} } 
   
   --
   -- Syntax for the EMSA-PKCS1-v1_5 hash identifier.
   --
   DigestInfo ::= SEQUENCE {
       digestAlgorithm DigestAlgorithm,
       digest OCTET STRING 
   }
   
   DigestAlgorithm ::= 
       AlgorithmIdentifier { {PKCS1-v1-5DigestAlgorithms} }
   
   END  -- PKCS1Definitions




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D. Intellectual property considerations

   The RSA public-key cryptosystem is described in U.S. Patent
   4,405,829, which expired on September 20, 2000. RSA Security
   Inc. makes no other patent claims on the constructions described 
   in this document, although specific underlying techniques may be
   covered.   

   Multi-prime RSA is described in U.S. Patent 5,848,159.

   The University of California has indicated that it has a patent
   pending on the PSS signature scheme [5]. It has also provided a
   letter to the IEEE P1363 working group stating that if the PSS
   signature scheme is included in an IEEE standard, "the University of
   California will, when that standard is adopted, FREELY license any
   conforming implementation of PSS as a technique for achieving a
   digital signature with appendix" [23]. The PSS signature scheme is
   specified in the IEEE P1363a draft [27], which was in ballot
   resolution when this document was published.

   License to copy this document is granted provided that it is
   identified as "RSA Security Inc. Public-Key Cryptography Standards
   (PKCS)" in all material mentioning or referencing this document. 

   RSA Security Inc. makes no other representations regarding
   intellectual property claims by other parties. Such determination
   is the responsibility of the user. 
 
E. Revision history

   Versions 1.0 - 1.3

      Versions 1.0 - 1.3 were distributed to participants in RSA Data
      Security, Inc.'s Public-Key Cryptography Standards meetings in
      February and March 1991. 

   Version 1.4

      Version 1.4 was part of the June 3, 1991 initial public release
      of PKCS. Version 1.4 was published as NIST/OSI Implementors'
      Workshop document SEC-SIG-91-18.

   Version 1.5

      Version 1.5 incorporated several editorial changes, including
      updates to the references and the addition of a revision history.
      The following substantive changes were made: 




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       - Section 10: "MD4 with RSA" signature and verification processes
         were added. 
       - Section 11: md4WithRSAEncryption object identifier was added.

      Version 1.5 was republished as IETF RFC 2313.
   
   Version 2.0
   
      Version 2.0 incorporated major editorial changes in terms of
      the document structure and introduced the RSAES-OAEP encryption
      scheme. This version continued to support the encryption and
      signature processes in version 1.5, although the hash algorithm
      MD4 was no longer allowed due to cryptanalytic advances in the
      intervening years. Version 2.0 was republished as IETF RFC 2437
      [35]. 

   Version 2.1

      Version 2.1 introduces multi-prime RSA and the RSASSA-PSS
      signature scheme with appendix along with several editorial
      improvements. This version continues to support the schemes in
      version 2.0. 
 
F. References

   [1]   ANSI X9F1 Working Group. ANSI X9.44 Draft D2: Key
         Establishment Using Integer Factorization Cryptography.
         Working Draft, March 2002.

   [2]   M. Bellare, A. Desai, D. Pointcheval and P. Rogaway.
         Relations Among Notions of Security for Public-Key Encryption
         Schemes. In H. Krawczyk, editor, Advances in Cryptology -
         Crypto '98, volume 1462 of Lecture Notes in Computer Science,
         pp. 26 - 45. Springer Verlag, 1998.

   [3]   M. Bellare and P. Rogaway. Optimal Asymmetric Encryption -
         How to Encrypt with RSA. In A. De Santis, editor, Advances in
         Cryptology - Eurocrypt '94, volume 950 of Lecture Notes in
         Computer Science, pp. 92 - 111. Springer Verlag, 1995.

   [4]   M. Bellare and P. Rogaway. The Exact Security of Digital
         Signatures - How to Sign with RSA and Rabin. In U. Maurer,
         editor, Advances in Cryptology - Eurocrypt '96, volume 1070
         of Lecture Notes in Computer Science, pp. 399 - 416. Springer
         Verlag, 1996.






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   [5]   M. Bellare and P. Rogaway. PSS: Provably Secure Encoding
         Method for Digital Signatures. Submission to IEEE P1363
         working group, August 1998. Available from
         http://grouper.ieee.org/groups/1363/.

   [6]   D. Bleichenbacher. Chosen Ciphertext Attacks Against
         Protocols Based on the RSA Encryption Standard PKCS #1. In
         H. Krawczyk, editor, Advances in Cryptology - Crypto '98,
         volume 1462 of Lecture Notes in Computer Science, pp. 1 -
         12. Springer Verlag, 1998.

   [7]   D. Bleichenbacher, B. Kaliski and J. Staddon. Recent Results
         on PKCS #1: RSA Encryption Standard. RSA Laboratories'
         Bulletin No. 7, June 1998. 

   [8]   B. den Boer and A. Bosselaers. An Attack on the Last Two
         Rounds of MD4. In J. Feigenbaum, editor, Advances in
         Cryptology - Crypto '91, volume 576 of Lecture Notes in
         Computer Science, pp. 194 - 203. Springer Verlag, 1992. 

   [9]   B. den Boer and A. Bosselaers. Collisions for the Compression
         Function of MD5. In T. Helleseth, editor, Advances in
         Cryptology - Eurocrypt '93, volume 765 of Lecture Notes in
         Computer Science, pp. 293 - 304. Springer Verlag, 1994. 

   [10]  D. Coppersmith, M. Franklin, J. Patarin and
         M. Reiter. Low-Exponent RSA with Related Messages. In
         U. Maurer, editor, Advances in Cryptology - Eurocrypt '96,
         volume 1070 of Lecture Notes in Computer Science, pp. 1 -
         9. Springer Verlag, 1996. 

   [11]  D. Coppersmith, S. Halevi and C. Jutla. ISO 9796-1 and the
         New Forgery Strategy. Presented at the rump session of Crypto
         '99, August 1999. 

   [12]  J.-S. Coron. On the Exact Security of Full Domain Hashing. In
         M. Bellare, editor, Advances in Cryptology - Crypto 2000,
         volume 1880 of Lecture Notes in Computer Science, pp. 229 -
         235. Springer Verlag, 2000. 

   [13]  J.-S. Coron. Optimal Security Proofs for PSS and Other
         Signature Schemes. In L. Knudsen, editor, Advances in
         Cryptology - Eurocrypt 2002, volume 2332 of Lecture Notes in
         Computer Science, pp. 272 - 287. Springer Verlag, 2002. 







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   [14]  J.-S. Coron, M. Joye, D. Naccache and P. Paillier. New
         Attacks on PKCS #1 v1.5 Encryption. In B. Preneel, editor,
         Advances in Cryptology - Eurocrypt 2000, volume 1807 of
         Lecture Notes in Computer Science, pp. 369 - 379. Springer
         Verlag, 2000. 

   [15]  J.-S. Coron, D. Naccache and J. P. Stern. On the Security of
         RSA Padding. In M. Wiener, editor, Advances in Cryptology -
         Crypto '99, volume 1666 of Lecture Notes in Computer Science,
         pp. 1 - 18. Springer Verlag, 1999.

   [16]  Y. Desmedt and A.M. Odlyzko. A Chosen Text Attack on the RSA
         Cryptosystem and Some Discrete Logarithm Schemes. In
         H.C. Williams, editor, Advances in Cryptology - Crypto '85,
         volume 218 of Lecture Notes in Computer Science, pp. 516 -
         522. Springer Verlag, 1986. 

   [17]  T. Dierks and C. Allen. IETF RFC 2246: The TLS Protocol
         Version 1.0. January 1999. 

   [18]  H. Dobbertin. Cryptanalysis of MD4. In D. Gollmann, editor,
         Fast Software Encryption '96, volume 1039 of Lecture Notes in
         Computer Science, pp. 55 - 72. Springer Verlag, 1996. 

   [19]  H. Dobbertin. Cryptanalysis of MD5 Compress. Presented at the
         rump session of Eurocrypt '96, May 1996. 

   [20]  H. Dobbertin. The First Two Rounds of MD4 are Not One-Way. In
         S. Vaudenay, editor, Fast Software Encryption '98, volume
         1372 in Lecture Notes in Computer Science, pp. 284 -
         292. Springer Verlag, 1998. 

   [21]  E. Fujisaki, T. Okamoto, D. Pointcheval and J. Stern.
         RSA-OAEP is Secure under the RSA Assumption. In J. Kilian,
         editor, Advances in Cryptology - Crypto 2001, volume 2139 of
         Lecture Notes in Computer Science, pp. 260 - 274. Springer
         Verlag, 2001.

   [22]  H. Garner. The Residue Number System. IRE Transactions on
         Electronic Computers, EC-8 (6), pp. 140 - 147, June 1959.

   [23]  M.L. Grell. Re: Encoding Methods PSS/PSS-R. Letter to IEEE
         P1363 working group, University of California, June 15,
         1999. Available from
         http://grouper.ieee.org/groups/1363/P1363/patents.html. 






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   [24]  J. Hůstad. Solving Simultaneous Modular Equations of Low
         Degree. SIAM Journal of Computing, volume 17, pp. 336 - 341,
         1988. 

   [25]  R. Housley. IETF RFC 2630: Cryptographic Message Syntax. June
         1999. 

   [26]  IEEE Std 1363-2000: Standard Specifications for Public Key
         Cryptography. IEEE, August 2000.

   [27]  IEEE P1363 working group. IEEE P1363a D10: Standard
         Specifications for Public Key Cryptography: Additional
         Techniques. November 1, 2001. Available from
         http://grouper.ieee.org/groups/1363/. 

   [28]  ISO/IEC 9594-8:1997: Information technology - Open Systems
         Interconnection - The Directory: Authentication
         Framework. 1997. 

   [29]  ISO/IEC FDIS 9796-2: Information Technology - Security
         Techniques - Digital Signature Schemes Giving Message
         Recovery - Part 2: Integer Factorization Based Mechanisms.
         Final Draft International Standard, December 
         2001. 

   [30]  ISO/IEC 18033-2: Information Technology - Security Techniques
         - Encryption Algorithms - Part 2: Asymmetric Ciphers.
         V. Shoup, editor, Text for 2nd Working Draft, January 2002. 

   [31]  J. Jonsson. Security Proof for the RSA-PSS Signature Scheme
         (extended abstract). Second Open NESSIE Workshop. September
         2001. Full version available from
         http://eprint.iacr.org/2001/053/. 

   [32]  J. Jonsson and B. Kaliski. On the Security of RSA Encryption
         in TLS. In Advances in Cryptology - Crypto 2002, to appear. 

   [33]  B. Kaliski. IETF RFC 1319: The MD2 Message-Digest
         Algorithm. April 1992. 

   [34]  B. Kaliski. On Hash Function Identification in Signature
         Schemes. In B. Preneel, editor, RSA Conference 2002,
         Cryptographers' Track, volume 2271 of Lecture Notes in
         Computer Science╣ pp. 1 - 16. Springer Verlag, 2002. 

   [35]  B. Kaliski and J. Staddon. IETF RFC 2437: PKCS #1: RSA
         Cryptography Specifications Version 2.0. October 1998. 




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   [36]  J. Manger. A Chosen Ciphertext Attack on RSA Optimal
         Asymmetric Encryption Padding (OAEP) as Standardized in PKCS
         #1 v2.0. In J. Kilian, editor, Advances in Cryptology -
         Crypto 2001, volume 2139 of Lecture Notes in Computer
         Science, pp. 260 - 274. Springer Verlag, 2001. 

   [37]  A. Menezes, P. van Oorschot and S. Vanstone. Handbook of
         Applied Cryptography. CRC Press, 1996.

   [38]  National Institute of Standards and Technology (NIST). FIPS
         Publication 180-1: Secure Hash Standard. April 1994.

   [39]  National Institute of Standards and Technology (NIST). Draft
         FIPS 180-2: Secure Hash Standard. Draft, May 2001. Available
         from http://www.nist.gov/sha/. 

   [40]  J.-J. Quisquater and C. Couvreur. Fast Decipherment Algorithm
         for RSA Public-Key Cryptosystem. Electronics Letters, 18
         (21), pp. 905 - 907, October 1982.

   [41]  R. Rivest. IETF RFC 1321: The MD5 Message-Digest Algorithm.
         April 1992.

   [42]  R. Rivest, A. Shamir and L. Adleman. A Method for Obtaining
         Digital Signatures and Public-Key Cryptosystems.
         Communications of the ACM, 21 (2), pp. 120-126, February
         1978.

   [43]  N. Rogier and P. Chauvaud. The Compression Function of MD2 is
         not Collision Free. Presented at Selected Areas of
         Cryptography '95. Carleton University, Ottawa, Canada. May
         1995.

   [44]  RSA Laboratories. PKCS #1 v2.0: RSA Encryption Standard.
         October 1998.

   [45]  RSA Laboratories. PKCS #7 v1.5: Cryptographic Message Syntax
         Standard. November 1993. (Republished as IETF RFC 2315.)

   [46]  RSA Laboratories. PKCS #8 v1.2: Private-Key Information
         Syntax Standard. November 1993.

   [47]  RSA Laboratories. PKCS #12 v1.0: Personal Information
         Exchange Syntax Standard. June 1999.

   [48]  V. Shoup. OAEP Reconsidered. In J. Kilian, editor, Advances
         in Cryptology - Crypto 2001, volume 2139 of Lecture Notes in
         Computer Science, pp. 239 - 259. Springer Verlag, 2001.



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   [49]  R. D. Silverman. A Cost-Based Security Analysis of Symmetric
         and Asymmetric Key Lengths. RSA Laboratories Bulletin No. 13,
         April 2000. Available from
         http://www.rsasecurity.com.rsalabs/bulletins/.

   [50]  G. J. Simmons. Subliminal communication is easy using the
         DSA. In T. Helleseth, editor, Advances in Cryptology -
         Eurocrypt '93, volume 765 of Lecture Notes in Computer
         Science, pp. 218-232. Springer-Verlag, 1993.
 
G. About PKCS

   The Public-Key Cryptography Standards are specifications produced
   by RSA Laboratories in cooperation with secure systems developers
   worldwide for the purpose of accelerating the deployment of
   public-key cryptography. First published in 1991 as a result of
   meetings with a small group of early adopters of public-key
   technology, the PKCS documents have become widely referenced and
   implemented. Contributions from the PKCS series have become part of
   many formal and de facto standards, including ANSI X9 and IEEE
   P1363 documents, PKIX, SET, S/MIME, SSL/TLS, and WAP/WTLS. 

   Further development of PKCS occurs through mailing list discussions
   and occasional workshops, and suggestions for improvement are
   welcome. For more information, contact: 

      PKCS Editor
      RSA Laboratories
      174 Middlesex Turnpike
      Bedford, MA  01730 USA
      pkcs-editor@rsasecurity.com
      http://www.rsasecurity.com/rsalabs/pkcs

Security Considerations

   Security issues are discussed throughout this memo.

Acknowledgements

   This document is based on a contribution of RSA Laboratories, a
   the research center of RSA Security Inc.  Any substantial use of the
   text from this document must acknowledge RSA Security Inc.  RSA
   Security Inc. requests that all material mentioning or referencing
   this document identify this as "RSA Security Inc. PKCS #1 v2.1".







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Full Copyright Statement

   Copyright (C) The Internet Society (2002).  All Rights Reserved.

   This document and translations of it may be copied and furnished to
   others, and derivative works that comment on or otherwise explain it
   or assist in its implementation may be prepared, copied, published
   and distributed, in whole or in part, without restriction of any
   kind, provided that the above copyright notice and this paragraph are
   included on all such copies and derivative works.  In addition, the
   ASN.1 module presented in Appendix A may be used in whole or in part
   without inclusion of the copyright notice.  However, this document
   itself may not be modified in any way, such as by removing the
   copyright notice or references to the Internet Society or other
   Internet organizations, except as needed for the purpose of
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   The limited permissions granted above are perpetual and will not be
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