Network Working Group M. Groves Internet Draft CESG Intended Status: Informational June 29, 2010 Expires: December 31, 2010 Sakai-Kasahara Key Establishment (SAKKE) draft-groves-sakke-00 Status of this Memo This Internet-Draft is submitted to IETF in full conformance with the provisions of BCP 78 and BCP 79. Internet-Drafts are working documents of the Internet Engineering Task Force (IETF), its areas, and its working groups. Note that other groups may also distribute working documents as Internet- Drafts. Internet-Drafts are draft documents valid for a maximum of six months and may be updated, replaced, or obsoleted by other documents at any time. It is inappropriate to use Internet-Drafts as reference material or to cite them other than as "work in progress." The list of current Internet-Drafts can be accessed at http://www.ietf.org/ietf/1id-abstracts.txt. The list of Internet-Draft Shadow Directories can be accessed at http://www.ietf.org/shadow.html. This Internet-Draft will expire on December 31, 2010. Copyright Notice Copyright (c) 2010 IETF Trust and the persons identified as the document authors. All rights reserved. This document is subject to BCP 78 and the IETF Trust's Legal Provisions Relating to IETF Documents (http://trustee.ietf.org/license-info) in effect on the date of publication of this document. Please review these documents carefully, as they describe your rights and restrictions with respect to this document. Code Components extracted from this document must include Simplified BSD License text as described in Section 4.e of the Trust Legal Provisions and are provided without warranty as described in the BSD License. Groves Informational [Page 1] Internet Draft draft-groves-sakke-00 Jun 29, 2010 Abstract In this document the Sakai-Kasahara Identifier-Based Encryption algorithm (SAKKE) is described. This uses Identifier-Based Encryption to exchange a shared secret from a Sender to a Receiver. Table of Contents 1. Introduction.....................................................2 1.1. Requirements Terminology....................................3 2. Notation and Definitions.........................................3 2.1. Notation....................................................3 2.2. Definitions.................................................5 3. Elliptic Curves and Pairings.....................................6 3.1. E(F_p^2) and the Distortion Map.............................6 3.2. The Tate-Lichtenbaum Pairing................................6 4. Representation of Values.........................................8 5. Supporting Algorithms............................................9 5.1. Hashing to an Integer Range.................................9 6. The SAKKE Cryptosystem...........................................9 6.1. Setup.......................................................9 6.1.1. Secret Key Extraction.................................10 6.1.2. User Provisioning.....................................10 6.2. Key Exchange...............................................10 6.2.1. Sender................................................10 6.2.2. Receiver..............................................11 6.3. Group Communications.......................................11 7. Security Considerations.........................................12 8. References......................................................13 8.1. Normative References.......................................13 8.2. Informative References.....................................13 Appendix A. Test Data..............................................14 1. Introduction This document defines an efficient use of Identifier-Based Encryption (IBE) based on bilinear pairings. The Sakai-Kasahara IBE cryptosystem [S-K] is described for establishment of a shared secret value. This document adds to the IBE options available in [RFC5091], providing an efficient primitive and an additional family of curves. This document is restricted to a particular family of curves (see Section 2.1) which have the benefit of a simple and efficient method of calculating the pairing on which the Sakai-Kasahara IBE cryptosystem is based. Identifier-Based Encryption schemes allow public and private keys to be derived from Identifiers. In fact, the Identifier can itself be viewed as corresponding to a public key or certificate in a traditional public key system. However, in IBE, the Identifier can Groves Informational [Page 2] Internet Draft draft-groves-sakke-00 Jun 29, 2010 be formed by both Sender and Receiver, which obviates the necessity of providing public keys through a third party or of transmitting certified public keys during each session establishment. Furthermore, in an IBE system, calculation of keys can occur as needed, and indeed, messages can be sent to users who are yet to enrol. The Sakai-Kasahara primitive described in this document supports simplex transmission of messages from a Sender to a Receiver. The choice of elliptic curve pairing on which the primitive is based allows simple and efficient implementations. The Sakai-Kasahara Key Establishment scheme described in this document is drawn from the SK-KEM scheme (as modified to support multi-party communications) submitted to the IEEE P1363 Working Group in [SK-KEM]. 1.1. Requirements Terminology The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT", "SHOULD", "SHOULD NOT", "RECOMMENDED", "MAY", and "OPTIONAL" in this document are to be interpreted as described in [RFC2119]. 2. Notation and Definitions 2.1. Notation n A security parameter; the size of symmetric keys in bits to be exchanged by SAKKE. p A prime, which is the order of the finite field F_p. In this document p is always congruent to 3 modulo 4. F_p The finite field of order p. F* The multiplicative group of the non-zero elements in the field F; e.g., (F_p)* is the multiplicative group of the finite field F_p. q An odd prime that divides p + 1. To provide the desired level of security, lg(q) MUST be greater than 2*n. E An elliptic curve defined over F_p, having a subgroup of order q. In this document we use supersingular curves with equation y^2 = x^3 - 3 * x. This curve is chosen because of the efficiency and simplicity advantages it offers. The choice of -3 for the coefficient of x provides advantages for elliptic curve arithmetic which are explained in [P1363]. A further reason for this choice of curve is that Groves Informational [Page 3] Internet Draft draft-groves-sakke-00 Jun 29, 2010 Barreto's trick [Barreto] of eliminating the computation of the denominators when calculating the pairing applies. E(F) The additive group of points of affine coordinates (x,y) with x, y in the field F, that satisfy the curve equation for E. P A point of E(F_p) that generates the cyclic subgroup of order q. The coordinates of P are given by P = (P_x,P_y). These coordinates are in F_p, and they satisfy the curve equation. 0 The null element of any additive group of points on an elliptic curve, also called the point at infinity. F_p^2 The extension field of degree 2 of the field F_p. In this document we use a particular instantiation of this field; F_p^2 = F_p[i] where i^2 + 1 = 0. PF_p The projectivisation of F_p. We define this to be (F_p^2)*/(F_p)*. Note that PF_p is cyclic and has order p + 1, which is divisible by q. G[q] The q-torsion of a group G. This is the subgroup generated by points of order q in G. < , > A version of the Tate-Lichtenbaum pairing. In this document this is a bilinear map from E(F_p)[q] x E(F_p)[q] onto the subgroup of order q in PF_p. A full definition is given in Section 3.2. Hash A cryptographic hash function. The following conventions are assumed for curve operations. Point addition - If A and B are two points on a curve E, their sum is denoted as A + B. Scalar multiplication - If A is a point on a curve, and k an integer, the result of adding A to itself a total of k times is denoted [k]A. We assume that the following concrete representations of mathematical objects are used. Elements of F_p - The p elements of F_p are represented directly using the integers from 0 to p-1. Elements of F_p^2 - The elements of F_p^2 = F_p[i] are represented as x_1 + i * x_2, where x_1 and x_2 are elements of F_p. Elements of PF_p - Elements of PF_p are cosets of (F_p)* in Groves Informational [Page 4] Internet Draft draft-groves-sakke-00 Jun 29, 2010 (F_p^2)*. Every element of F_p^2 can be written unambiguously in the form x_1 + i * x_2, where x_1 and x_2 are elements of F_p. Thus elements of PF_p (except the unique element of order 2) can be represented unambiguously by x_2 / x_1 in F_p. Since q is odd, every element of PF_p[q] can be represented by an element of F_p in this manner. This representation of elements in PF_p[q] allows efficient implementation of PF_p[q] group operations, as these can be defined using arithmetic in F_p. If a and b are elements of F_p representing elements A and B of PF_p[q] respectively, then A * B in PF_p[q] is represented by (a + b)/(1 - a * b) in F_p. 2.2. Definitions Identifier - Each user of an IBE system MUST have a unique, unambiguous identifying string that can be easily derived by all valid communicants. This string is the user's Identifier. An Identifier is an integer in the range 2 to q-1. The method by which Identifiers are formed MUST be defined for each application. Key Management Server (KMS) - The Key Management Server is a trusted 3rd party for the IBE system. It derives system secrets and distributes key material to those authorised to obtain it. Applications MAY support the use mutual communication between the users of multiple KMSs. We denote KMSs by KMS_T, KMS_S etc. Public parameters - The public parameters are a set of parameters that are held by all users of an IBE system. Such a system MAY contain multiple KMSs. Each application of SAKKE MUST define the set of public parameters to be used. The parameters needed are p, q, E, P, g=, Hash and n. Master Secret (z_T) - The Master Secret z_T is the master key generated and privately kept by KMS_T and is used by KMS_T to generate the private keys of the users that it provisions; it is an integer in the range 2 to q-1. KMS Public Key: Z_T = [z_T]P - The KMS Public Key Z_T is used to form Public Key Establishment Keys for all users provisioned by KMS_T; it is a point of order q in E(F_p). It MUST be provisioned by KMS_T to all who are authorised to send messages to users of the IBE system. Receiver Secret Key (RSK) - Each user enrolled in an IBE system is provisioned with a Receiver Secret Key by its KMS. The RSK provided to a user with Identifier a by KMS_T is denoted K_(a,T). In SAKKE, the RSK is a point of order q in E(F_p). Shared Secret Value - The aim of the SAKKE scheme is for the Groves Informational [Page 5] Internet Draft draft-groves-sakke-00 Jun 29, 2010 Sender to securely transmit a Shared Secret Value to the Receiver. The Shared Secret Value is an integer in the range 0 to (2^n) - 1; it is denoted SSV. Encapsulated Data - The Encapsulated Data are used to transmit secret information securely to the Receiver. They can be computed directly from the Receiver's Identifier, the public parameters, the KMS Public Key, and the Shared Secret Value to be transmitted. In SAKKE the Encapsulated Data are a point of order q in E(F_p) and an integer in the range 0 to (2^n) - 1. They are formatted as described in Section 4. 3. Elliptic Curves and Pairings E is a supersingular elliptic curve (of j-invariant 1728). E(F_p) contains a cyclic subgroup of order q, denoted E(F_p)[q], whereas the larger object E(F_p^2) contains the direct product of two cyclic subgroups of order q, denoted E(F_p^2)[q]. P is a generator of E(F_p)[q]. It is specified by the (affine) coordinates (P_x,P_y) in F_p, satisfying the curve equation. Routines for point addition and doubling on E(F_p) can be found in Appendix A.10 of [P1363]. 3.1. E(F_p^2) and the Distortion Map If (Q_x,Q_y) are (affine) coordinates in F_p for some point (denoted Q) on E(F_p)[q], then (-Q_x,iQ_y) are (affine) coordinates in F_p^2 for some point on E(F_p^2)[q]. This latter point is denoted [i]Q, by analogy with the definition for scalar multiplication. The two points P and [i]P together generate E(F_p^2)[q]. The map [i]: E(F_p) -> E(F_p^2) is sometimes termed the distortion map. 3.2. The Tate-Lichtenbaum Pairing We proceed to describe the pairing < , > to be used in SAKKE. We will need to evaluate polynomials f_R that depend on points on E(F_p)[q]. Miller's algorithm [Miller] provides a method for evaluation of f_R(X), where X is some element of E(F_p^2)[q] and R is some element of E(F_p)[q] and f_R is some polynomial over F_p whose divisor is (q)(R) - (q)(0). Note that f_R is defined only up to scalars of F_p. The version of the Tate-Lichtenbaum pairing used in this document is given by = f_R([i]Q)^c / (F_p)*. It satisfies the bilinear relation <[x]R,Q> = = ^x for all Q, R in E(F_p)[q], for all integers x. Note that the domain of definition is restricted to E(F_p)[q] x E(F_p)[q] so that certain optimisations are natural. Groves Informational [Page 6] Internet Draft draft-groves-sakke-00 Jun 29, 2010 We provide pseudocode for computing , with elliptic curve arithmetic expressed in affine coordinates. We make use of Barreto's trick [Barreto] for avoiding the calculation of denominators. Note that this section does not fully describe the most efficient way of computing the pairing; it is possible to compute the pairing without any explicit reference to the extension field F_p^2. This reduces the number and complexity of the operations needed to compute the pairing. Pseudocode begins: Routine for computing the pairing : Input R, Q points on E(F_p)[q]; Initialise variables: v = (F_p)*; // An element of PF_p[q] C = R; // An element of E(F_p)[q] c = (p+1)/q; // An integer for bits of q-1, starting with the second MSB, ending with the LSB, do // gradient of line through C, C, [-2]C. l = 3*( C_x^2 - 1 ) / ( 2*C_y ); //accumulate line evaluated at [i]Q into v v = v^2 * ( l*( Q_x + C_x ) + ( i*Q_y - C_y ) ); C = [2]C; if bit is 1 then // gradient of line through C, R, -C-R. l = ( C_y - R_y )/( C_x - R_x ); //accumulate line evaluated at [i]Q into v v = v * ( l*( Q_x + C_x ) + ( i*Q_y - C_y ) ); C = C+R; end if; end for; t = v^c; return representative in F_p of t; End of routine; Routine for computing representative in F_p of elements of PF_p: Input t, in F_p^2, representing an element of PF_p; Groves Informational [Page 7] Internet Draft draft-groves-sakke-00 Jun 29, 2010 Represent t as a + i*b, with a,b in F_p; return b/a; End of routine; End of pseudocode; 4. Representation of Values This section provides canonical representations of values which MUST be used to ensure interoperability of implementations. The following representations MUST be used for input into hash functions and for transmission. Integers Integers MUST be represented as an octet string, with bit length a multiple of 8. To achieve this, the integer is represented most significant bit first, and padded with zero bits on the left until an octet string of the necessary length is obtained. This is the Octet String representation described in Section 5.5.2 of [P1363]. F_p elements Elements of F_p MUST be represented as integers in the range 0 to p-1 using the octet string representation defined above. Such octet strings MUST have length L = Ceiling(lg(p)/8). PF_p elements Elements of PF_p MUST be represented as an element of F_p using the algorithm in Section 3.2. They are therefore represented as octet strings as defined above and are L octets in length. Representation of the unique element of order 2 in PF_p will not be required. Points on E Elliptic Curve Points MUST be represented in Uncompressed representation as defined in Section 5.5.6 of [P1363a]. For an elliptic curve point ( x , y ) with x and y in F_p, this representation is given by 0x00 || x' || y' , where x' is the octet string representing x, y' is the octet string representing y and 0x00 is a NULL octet. The representation is 2*L+1 octets in length. Encapsulated Data The Encapsulated Data MUST be represented as an Elliptic Curve Point concatenated with an integer in the range 0 to (2 ^ n) - 1. Since the length of the representation of elements of F_p is well defined given p, these data can be unambiguously parsed to retrieve their components. The Encapsulated Data is 2*L + n + 1 octets in length. Groves Informational [Page 8] Internet Draft draft-groves-sakke-00 Jun 29, 2010 5. Supporting Algorithms 5.1. Hashing to an Integer Range We use the function HashToIntegerRange( s, n, hashfn) to hash strings to an integer range. Given a string, s, a hash function, hashfn, and an integer, n, this function returns a value between 0 and n - 1. Input: * an octet string, s * an integer, n <= (2^hashlen)^hashlen * a hash function, hashfn, with output length hashlen bits. Output: * an integer, v in the range 0 to n-1 Method: 1) Let A = hashfn( s ) 2) Let h_0 = 00...00, a string of null bits of length hashlen bits 3) Let l = Ceiling( lg( n ) / hashlen ) 4) For each i in 1 to l do: a) Let h_i = hashfn(h_(i - 1)) b) Let v_i = hashfn(h_i || A), where || denotes concatenation 5) Let v' = v_1 || ... || v_l 6) Let v = v' mod n 6. The SAKKE Cryptosystem This chapter describes the Sakai-Kasahara Key Establishment algorithm. It draws from the cryptosystem first described in [S-K]. 6.1. Setup All users share a set of public parameters with a KMS. In most circumstances it is expected that a system will only use a single KMS. However, it is possible for users provisioned by different KMSs Groves Informational [Page 9] Internet Draft draft-groves-sakke-00 Jun 29, 2010 to interoperate provided that they use a common set of public parameters, and that they each possess the necessary KMS Public Keys. In order to facilitate this interoperation, it is anticipated that parameters will be published in application specific standards. KMS_T chooses its KMS Master Secret, z_T. It MUST randomly select a value in the range 2 to q-1, and assigns this value to z_T. It MUST derive its KMS Public Key, Z_T, by performing the calculation Z_T = [z_T]P. 6.1.1. Secret Key Extraction The KMS derives each Receiver Secret Key from an Identifier and its KMS Master Secret. It MUST derive a Receiver Secret Key for each user that it provisions. For Identifier 'a', the Receiver Secret Key K_(a,T) provided by KMS_T MUST be derived by KMS_T as K_(a,T) = [(a + z_T)^-1]P, where 'a' is interpreted as an integer, and the inversion is performed modulo q. 6.1.2. User Provisioning The KMS MUST provide its KMS Public Key to all users through an authenticated channel. Receiver Secret Keys MUST be supplied to all users through a channel that provides confidentiality and mutual authentication. The mechanisms that provide security for these channels are beyond the scope of this document: they are application specific. Upon receipt of key material, each user MUST verify its Receiver Secret Key. For Identifier 'a', Receiver Secret Keys from KMS_T are verified by checking that the following equation holds: < [a]P + Z , K_(a,T) > = g, where 'a' is interpreted as an integer. 6.2. Key Exchange A Sender forms Encapsulated Data and sends it to the Receiver, who processes it. The result is a shared secret which can be used as keying material for securing further communications. We denote the Sender "A", with Identifier a; we denote the Receiver "B", with Identifier b; Identifiers are to be interpreted as integers in the algorithms below. Let A be provisioned by KMS_T and B be provisioned by KMS_S. 6.2.1. Sender In order to form Encapsulated Data to send to device B who is provisioned by KMS_S, A needs to hold Z_S. It is anticipated that Groves Informational [Page 10] Internet Draft draft-groves-sakke-00 Jun 29, 2010 this will have been provided to A by KMS_T along with its User Private Keys. The Sender MUST carry out the following steps. 1) Select a random ephemeral integer value for the Shared Secret Value SSV in the range 0 to 2^n - 1. 2) Compute r = HashToIntegerRange( SSV || b , q , Hash ). 3) Compute R_(b,S) = [r]([b]P + Z_S) in E(F_p). 4) Compute the Hint, H := SSV xor HashToIntegerRange( g^r, 2^n, Hash ). 5) Form the Encapsulated Data ( R_(b,S) , H ) and transmit it to B. 6) Output SSV for use to derive key material for the application to be keyed. 6.2.2. Receiver Device B receives Encapsulated Data from device A. In order to process this, it requires its Receiver Secret Key, K_(b,S), which will have been provisioned in advance by KMS_S. The method by which keys are provisioned by the KMS is application specific. The Receiver MUST carry out the following steps to derive and verify the Shared Secret Value. 1) Parse the Encapsulated Data ( R_(b,S) , H ) and extract R_(b,S) and H. 2) Compute w := < R_(b,S) , K_(b,S) >. Note that by bilinearity w = g^r. 3) Compute SSV = H xor HashToIntegerRange( w, 2^n, Hash ). 4) Compute r = HashToIntegerRange( SSV || b , q , Hash ). 5) Compute TEST = [r]([b]P + Z_S) in E(F_p). If TEST does not equal R_(b,S) then B MUST NOT use the SSV to derive key material. 6) Output SSV for use to derive key material for the application to be keyed. 6.3. Group Communications The SAKKE scheme can be used to exchange Shared Secret Values for group communications. To provide a Shared Secret to multiple Receivers, a Sender MUST form Encapsulated Data for each of their Groves Informational [Page 11] Internet Draft draft-groves-sakke-00 Jun 29, 2010 Identifiers, and transmit the appropriate data to each Receiver. Any party possessing the group Shared Secret Value MAY extend the group by forming Encapsulated Data for a new group member. Whilst the Sender needs to form multiple Encapsulated Data, the fact that the sending operation avoids pairings means that the extension to multiple Receivers can be carried out more efficiently than for alternative IBE schemes which require the Sender to compute a pairing. 7. Security Considerations This document describes the SAKKE cryptographic algorithm. We assume that the security provided by this algorithm depends entirely on the secrecy of the secret keys it uses, and that for an adversary to defeat this security, he will need to perform computationally intensive cryptanalytic attacks to recover a secret key. Note that a security proof exists for SAKKE in the Random Oracle Model [SK-KEM]. When defining public parameters, guidance on parameter sizes from [SP800-57] SHOULD be followed. Note that the size of the F_p^2 discrete logarithm on which the security rests is 2*lg(p). Table 1 shows bits of security afforded by various sizes of p. If k bits of security are needed, then lg(q) SHOULD be chosen to be at least 2*k. Similarly, if k bits of security are needed, then a Hash with output size at least 2*k SHOULD be chosen. Bits of Security | lg(p) ------------------------- 80 | 512 112 | 1024 128 | 1536 192 | 3840 256 | 7680 Table 1: Comparable strengths, taken from Table 2 of [SP800-57] The KMS Master Secret provides the security for each device provisioned by the KMS. It MUST NOT be revealed to any other entity. Each user's Receiver Secret Key protects the SAKKE communications it receives. This key MUST NOT be revealed to any other entity than the trusted KMS and the authorised user. In order to ensure that the Receiver Secret Key is received only by an authorised device, it MUST be provided through a secure channel. The security offered by this system is no greater than the security provided by this delivery channel. Note that IBE systems have different properties than other asymmetric cryptographic schemes with regards to key recovery. The KMS (and Groves Informational [Page 12] Internet Draft draft-groves-sakke-00 Jun 29, 2010 hence any administrator with appropriate privileges) can create Receiver Secret Keys for arbitrary Identifiers, and procedures to monitor the creation of Receiver Secret Keys such as logging of administrator actions SHOULD be defined by any functioning implementation of SAKKE. Identifiers MUST be defined unambiguously by each application of SAKKE. Note that it is not necessary to hash the data in a format for Identifiers (except in the case where its size would be greater than that of q). In this way any weaknesses that might be caused by collisions in hash functions can be avoided without reliance on the structure of the Identifier format. Applications of SAKKE MAY include a time/date component in their Identifier format to ensure that Identifiers (and hence Receiver Secret Keys) are only valid for a fixed period of time. The randomness of values stipulated to be selected at random in SAKKE described in this document is essential to the security provided by SAKKE. If the ephemeral value r selected by the Sender is not chosen at random then the SSV, which is used to provide key material for further communications, could be predictable. 8. References 8.1. Normative References [MIKEY-SAKKE] Groves, M., "MIKEY-SAKKE: Sakai-Kasahara Key Exchange in Multimedia Internet KEYing (MIKEY)", draft-groves-MIKEY-SAKKE-00 [work in progress], June 2010. [P1363] IEEE P1363-2000, "Standard Specifications for Public-Key Cryptography," 2001. [P1363a] IEEE P1363a, "Standard Specifications for Public-Key Cryptography - Amendment 1: Additional Techniques", 2004. [RFC2119] Bradner, S., "Key words for use in RFCs to Indicate Requirement Levels", BCP 14, RFC 2119, March 1997. 8.2. Informative References [Barreto] Barreto, P., Kim, H., Lynn, B. and M. Scott "Efficient Algorithms for Pairing-Based Cryptosystems", Advances in Cryptology - Crypto 2002, LNCS 2442, Springer-Verlag (2002), pp.354-368. Groves Informational [Page 13] Internet Draft draft-groves-sakke-00 Jun 29, 2010 [Miller] Miller, V., "The Weil pairing, and its efficient calculation", J. Cryptology 17 (2004), 235-261. [RFC5091] Boyen, X. and L. Martin, "Identity Based Cryptography Standard (IBCS) #1: Supersingular Curve Implementations of the BF and BB1 Cryptosystems", RFC 5091, December 2007. [S-K] Sakai, R., Ohgishi, K. and M. Kasahara, "ID based cryptosystem with pairing on elliptic curve", Symposium on Cryptography and Information Security - SCIS, 2003. [SK-KEM] Barbosa, M., Chen, L., Cheng, Z., Chimley, M., Dent, A., Farshim, P., Harrison, K., Malone-Lee, J., Smart, N. and F. Vercauteren, "SK-KEM: An Identity-Based KEM", submission for IEEE P1363.3, June 2006. (http://grouper.ieee.org/groups/1363/IBC/ submissions/Barbosa-SK-KEM-2006-06.pdf) Appendix A. Test Data This appendix provides test data for SAKKE with the public parameters defined in Appendix A of [MIKEY-SAKKE]. "b" represents the Identifier of the Responder. The value "mask" is the value used to mask the SSV, and is defined to be HashToIntegerRange( g^r, 2^n, Hash ). // -------------------------------------------------------- // KMS generates: z = AFF429D3 5F84B110 D094803B 3595A6E2 998BC99F Zx = 5958EF1B 1679BF09 9B3A030D F255AA6A 23C1D8F1 43D4D23F 753E69BD 27A832F3 8CB4AD53 DDEF4260 B0FE8BB4 5C4C1FF5 10EFFE30 0367A37B 61F701D9 14AEF097 24825FA0 707D61A6 DFF4FBD7 273566CD DE352A0B 04B7C16A 78309BE6 40697DE7 47613A5F C195E8B9 F328852A 579DB8F9 9B1D0034 479EA9C5 595F47C4 B2F54FF2 Zy = 1508D375 14DCF7A8 E143A605 8C09A6BF 2C9858CA 37C25806 5AE6BF75 32BC8B5B 63383866 E0753C5A C0E72709 F8445F2E 6178E065 857E0EDA 10F68206 B63505ED 87E534FB 2831FF95 7FB7DC61 9DAE6130 1EEACC2F DA3680EA 4999258A 833CEA8F C67C6D19 487FB449 059F26CC 8AAB655A Groves Informational [Page 14] Internet Draft draft-groves-sakke-00 Jun 29, 2010 B58B7CC7 96E24E9A 39409575 4F5F8BAE // -------------------------------------------------------- // Creating Encapsulated Data b = 3230 31302D30 37007465 6C3A2B34 34313233 34353637 38393000 SSV = 12345678 9ABCDEF0 12345678 9ABCDEF0 r = HashToIntegerRange( 12345678 9ABCDEF0 12345678 9ABCDEF0 32303130 2D303700 74656C3A 2B343431 32333435 36373839 3000, q, SHA-256 ) = 14681280 2E82D50F 25EEA39F 75E4E91A 3E44619A F7AE649F E113CB65 D2B32E84 530A18FA E0FEFC62 757628F6 2F804059 7840FFF4 A517A7C7 F3F7E696 AB38F053 77E4851A D8294152 AAEB6FFC CE211425 6EB96269 757731DB 75868CCE ACF1202C F2263A77 E7F4FA59 986152B4 C7A55506 5A329077 0C86F3BB 8ADE405C 526ED54B Rbx = 157B9F35 6C8A3138 A9532EC2 62B04604 83EA33A8 26247411 D852136C E543020C 52BDF196 E5955121 FF83A183 21E90A5A 7EC1D0E1 B433FEFB FD082C96 8674682A A935EFDA E984F557 2B677D51 31E8C90C CC77519D 7C88B20C 5C829287 B2204A3E E7DBEE5D F7975375 24D7215B 2F3D9698 86720EF4 5CB61745 CB69DA22 C87EE985 Rby = 5D9C7EC1 A67942D3 BF0F82F2 9CC1C1D5 5E0FDBA6 F51B0179 0DA75F06 0BE7E9B0 DCC06CE7 A200E8EB A0F77875 6DF2C587 DE65DE84 67A522EE DA10774C C7043F52 D7B61B65 2109DE22 209C1B80 D0744FCB 2A35C51F 335962FA DBFF52C9 4A60AF82 6795356C 16F0DB7F 995CF68B 7EF7D367 B5F96B76 FC8E4778 09406FC9 7DF810B3 g^r = 298A4F75 823B7B86 AEA87E11 57A4448F DB4B2735 2F364150 47C05A9E 527BE983 1A0625B1 BB59360D EF5E7FA2 52E1A0EF 9E2166E3 B3E0A8BE C4854EED 14BD3B36 D43AA069 9F7D71BF 377149A2 37B95CCD 7C5A812A B69D0F48 0802B79D 620663B3 FF0D5C9B CC991DCD 77587560 D7E48B79 CD1FAB30 18DDDC9F 5342B76D F21D3E61 mask = HashToIntegerRange( Groves Informational [Page 15] Internet Draft draft-groves-sakke-00 Jun 29, 2010 298A4F75 823B7B86 AEA87E11 57A4448F DB4B2735 2F364150 47C05A9E 527BE983 1A0625B1 BB59360D EF5E7FA2 52E1A0EF 9E2166E3 B3E0A8BE C4854EED 14BD3B36 D43AA069 9F7D71BF 377149A2 37B95CCD 7C5A812A B69D0F48 0802B79D 620663B3 FF0D5C9B CC991DCD 77587560 D7E48B79 CD1FAB30 18DDDC9F 5342B76D F21D3E61, 2^128, SHA-256 ) = 6F22CF7F 87627451 4627917B AED7E828 H = 7D169907 1DDEAAA1 5413C703 346B36D8 // -------------------------------------------------------- // Receiver processing // Device receives Kb from KMS Kbx = 0AB631BD 80052A89 5AE44295 753842D5 0B64798C 723D55FC AC4048DC A7BF61C6 F42C26D1 C82A8558 C5C19D50 98F9F706 037793B9 FE2062C3 7BFFDFC6 D5E4308B BD2C3FFB 24767D86 11C08A17 5DE13AE8 E4DB0F77 536877B5 A3262AAD AFD007B7 F07F9A1E 4263A1EC F73E050C CEE68AF6 741EE7BB CF0FB40E C31C2E10 6488EE82 Kby = 8614FA9A F6EAC87A 9FE1C996 13235038 A0473076 B5C0A0EB 1BB321AB AD030905 7A491EB0 4A2A98D1 EA1E57D8 DE924B0B 41642FFE 62642FCA 794F6DFF CCB03B2E E2ACCCFA 469951D2 4778E032 125CE424 628A54FA F11DABCD CD23B72D D76401B5 62A655FB 9B56581A 46CEAC6E 0C0A5E2A F23D30B3 1BC1D6B4 E0068701 646EDDE8 // Device processes Encapsulated Data w = 298A4F75 823B7B86 AEA87E11 57A4448F DB4B2735 2F364150 47C05A9E 527BE983 1A0625B1 BB59360D EF5E7FA2 52E1A0EF 9E2166E3 B3E0A8BE C4854EED 14BD3B36 D43AA069 9F7D71BF 377149A2 37B95CCD 7C5A812A B69D0F48 0802B79D 620663B3 FF0D5C9B CC991DCD 77587560 D7E48B79 CD1FAB30 18DDDC9F 5342B76D F21D3E61 SSV = 12345678 9ABCDEF0 12345678 9ABCDEF0 r = 14681280 2E82D50F 25EEA39F 75E4E91A 3E44619A F7AE649F E113CB65 D2B32E84 530A18FA E0FEFC62 757628F6 2F804059 Groves Informational [Page 16] Internet Draft draft-groves-sakke-00 Jun 29, 2010 7840FFF4 A517A7C7 F3F7E696 AB38F053 77E4851A D8294152 AAEB6FFC CE211425 6EB96269 757731DB 75868CCE ACF1202C F2263A77 E7F4FA59 986152B4 C7A55506 5A329077 0C86F3BB 8ADE405C 526ED54B TESTx = 157B9F35 6C8A3138 A9532EC2 62B04604 83EA33A8 26247411 D852136C E543020C 52BDF196 E5955121 FF83A183 21E90A5A 7EC1D0E1 B433FEFB FD082C96 8674682A A935EFDA E984F557 2B677D51 31E8C90C CC77519D 7C88B20C 5C829287 B2204A3E E7DBEE5D F7975375 24D7215B 2F3D9698 86720EF4 5CB61745 CB69DA22 C87EE985 TESTy = 5D9C7EC1 A67942D3 BF0F82F2 9CC1C1D5 5E0FDBA6 F51B0179 0DA75F06 0BE7E9B0 DCC06CE7 A200E8EB A0F77875 6DF2C587 DE65DE84 67A522EE DA10774C C7043F52 D7B61B65 2109DE22 209C1B80 D0744FCB 2A35C51F 335962FA DBFF52C9 4A60AF82 6795356C 16F0DB7F 995CF68B 7EF7D367 B5F96B76 FC8E4778 09406FC9 7DF810B3 TEST == Rb // -------------------------------------------------------- // HashToIntegerRange( M, q, SHA-256 ) Example M = 12345678 9ABCDEF0 12345678 9ABCDEF0 32303130 2D303700 74656C3A 2B343431 32333435 36373839 3000 A = 8B788A5C BBB8C91D C6B723C4 8E11Ed05 46d670CC D415BF0B 2A910FF3 D71DC0AF h0 = 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 h1 = 66687AAD F862BD77 6C8FC18B 8E9F8E20 08971485 6EE233B3 902A591D 0D5F2925 h2 = 2B32DB6C 2C0A6235 FB1397E8 225EA85E 0F0E6E8C 7B126D00 16CCBDE0 E667151E h3 = 12771355 E46CD47C 71ED1721 FD5319B3 83CCA3A1 F9FCE3AA 1C8CD3BD 37AF20D7 h4 = FE15C0D3 EBE314FA D720A08B 839A004C 2E6386F5 AECC19EC 74807D19 20CB6AEB v1 = 3AC6C147 F1186505 BF602805 AC990278 CE9F64D3 5EDB8538 50BA843B FFAB3510 v2 = 100CC3C4 D9BE0029 3E17F52B 7BE9A785 B2256CDC A309CA4E 41533093 D4D29A07 v3 = B07F9E3C A4C41487 BF362170 277B1B5D DC655F93 81D87C7C 1F7A5BE3 34002297 v4 = 9A0B7023 BD9AC221 979A4CBD AA06B472 Groves Informational [Page 17] Internet Draft draft-groves-sakke-00 Jun 29, 2010 7A64083B 37A5A75D 6479A07B 1218ED46 v mod q = 14681280 2E82D50F 25EEA39F 75E4E91A 3E44619A F7AE649F E113CB65 D2B32E84 530A18FA E0FEFC62 757628F6 2F804059 7840FFF4 A517A7C7 F3F7E696 AB38F053 77E4851A D8294152 AAEB6FFC CE211425 6EB96269 757731DB 75868CCE ACF1202C F2263A77 E7F4FA59 986152B4 C7A55506 5A329077 0C86F3BB 8ADE405C 526ED54B // -------------------------------------------------------- Author's Address Michael Groves CESG Hubble Road Cheltenham GL51 8HJ UK Email: Michael.Groves@cesg.gsi.gov.uk Acknowledgement Funding for the RFC Editor function is provided by the IETF Administrative Support Activity (IASA). Groves Informational [Page 18]