Network Working Group D. McGrew
Internet-Draft Cisco Systems
Intended status: Informational July 6, 2009
Expires: January 7, 2010
Fundamental Elliptic Curve Cryptography Algorithms
draft-mcgrew-fundamental-ecc-00.txt
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Abstract
This note describes the fundamental algorithms of Elliptic Curve
Cryptography (ECC) as they are defined in some early references.
These descriptions may be useful to those who want to implement the
fundamental algorithms without using any of the specialized methods
that were developed in following years. Only elliptic curves based
on fields of character greater than three are in scope.
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Table of Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 4
2. Mathematical Background . . . . . . . . . . . . . . . . . . . 5
2.1. Modular Arithmetic . . . . . . . . . . . . . . . . . . . . 5
2.2. Group Operations . . . . . . . . . . . . . . . . . . . . . 5
2.3. Finite Fields . . . . . . . . . . . . . . . . . . . . . . 6
3. Elliptic Curve Groups . . . . . . . . . . . . . . . . . . . . 7
3.1. Homogenous Coordinates . . . . . . . . . . . . . . . . . . 8
3.2. Group Parameters . . . . . . . . . . . . . . . . . . . . . 8
3.2.1. Security . . . . . . . . . . . . . . . . . . . . . . . 9
4. Elliptic Curve Diffie-Hellman (ECDH) . . . . . . . . . . . . . 10
4.1. Data Types . . . . . . . . . . . . . . . . . . . . . . . . 10
4.2. Compact Representation . . . . . . . . . . . . . . . . . . 10
5. Elliptic Curve ElGamal Signatures (ECES) . . . . . . . . . . . 12
5.1. Keypair Generation . . . . . . . . . . . . . . . . . . . . 12
5.2. Signature Creation . . . . . . . . . . . . . . . . . . . . 12
5.3. Signature Verification . . . . . . . . . . . . . . . . . . 13
5.4. Hash Functions . . . . . . . . . . . . . . . . . . . . . . 13
5.5. Rationale . . . . . . . . . . . . . . . . . . . . . . . . 13
6. Abbreviated ECES Signatures (AECES) . . . . . . . . . . . . . 15
6.1. Keypair Generation . . . . . . . . . . . . . . . . . . . . 15
6.2. Signature Creation . . . . . . . . . . . . . . . . . . . . 15
6.3. Signature Verification . . . . . . . . . . . . . . . . . . 15
7. Interoperability . . . . . . . . . . . . . . . . . . . . . . . 17
7.1. ECDH . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
7.2. ECES, AECES, and ECDSA . . . . . . . . . . . . . . . . . . 17
8. Intellectual Property . . . . . . . . . . . . . . . . . . . . 19
8.1. Disclaimer . . . . . . . . . . . . . . . . . . . . . . . . 19
9. Security Considerations . . . . . . . . . . . . . . . . . . . 20
9.1. Subgroups . . . . . . . . . . . . . . . . . . . . . . . . 20
9.2. Diffie-Hellman . . . . . . . . . . . . . . . . . . . . . . 21
9.3. Group Representation and Security . . . . . . . . . . . . 21
9.4. Signatures . . . . . . . . . . . . . . . . . . . . . . . . 22
10. IANA Considerations . . . . . . . . . . . . . . . . . . . . . 23
11. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . 24
12. References . . . . . . . . . . . . . . . . . . . . . . . . . . 25
12.1. Normative References . . . . . . . . . . . . . . . . . . . 25
12.2. Informative References . . . . . . . . . . . . . . . . . . 26
Appendix A. Random Number Generation . . . . . . . . . . . . . . 29
Appendix B. Example Elliptic Curve Group . . . . . . . . . . . . 30
Author's Address . . . . . . . . . . . . . . . . . . . . . . . . . 31
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1. Introduction
ECC is a public-key technology that offers performance advantages at
higher security levels. It includes an Elliptic Curve version of
Diffie-Hellman key exchange protocol [DH1976] and an Elliptic Curve
version of the ElGamal Signature Algorithm [E1985]. The elliptic
curve versions of these algorithms are referred to as ECDH and ECES,
respectively. The adoption of ECC has been slower than had been
anticipated, perhaps due to the lack of freely available normative
documents and uncertainty over intellectual property rights.
This note contains a description of the fundamental algorithms of ECC
over fields with characteristic greater than three, based directly on
original references. Its intent is to provide the Internet community
with a normative specification of the basic algorithms that predate
any specialized or optimized algorithms.
The rest of the note is organized as follows. Section 2.1,
Section 2.2, and Section 2.3 furnish the necessary terminology and
notation from modular arithmetic, group theory and the theory of
finite fields, respectively. Section 3 defines the groups based on
elliptic curves over finite fields of characteristic greater than
three. Section 4 and Section 5 present the fundamental ECDH and ECES
algorithms, respectively. Section 6 presents an abbreviated form of
ECES. The previous sections contain all of the normative text (the
text that defines the norm for implementations conforming to this
specification), and all of the following sections are purely
informative. Interoperability is discussed in Section 7. Section 8
reviews intellectual property issues. Section 9 summarizes security
considerations. Appendix A describes random number generation and
Appendix B provides an example of an Elliptic Curve group.
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2. Mathematical Background
This section reviews mathematical preliminaries and establishes
terminology and notation that is used below.
2.1. Modular Arithmetic
This section reviews modular arithmetic. Two integers x and y are
said to be congruent modulo n if x - y is an integer multiple of n.
Two integers x and y are coprime when their greatest common divisor
is 1; in this case, there is no third number z such that z divides x
and z divides y.
The set Zp = { 0, 1, 2, ..., p-1 } is closed under the operations of
modular addition, modular subtraction, modular multiplication, and
modular inverse. These operations are as follows.
For each pair of integers a and b in Zp, a + b mod p is equal to
a + b if a + b < p, and is equal to a + b - p otherwise.
For each pair of integers a and b in Zp, a - b mod p is equal to
a - b if a + b > p, and is equal to a + b otherwise.
For each pair of integers a and b in Zp, a * b mod p is equal to
the remainder of a * b divided by p.
For each integer x in Zp that is coprime with p, the inverse of x
modulo p is denoted as 1 / x mod p, and can be computed using the
extended euclidean algorithm (see Section 4.5.2 of [K1981v2], for
example).
Algorithms for these operations are well known; for instance, see
Chapter 4 of [K1981v2].
2.2. Group Operations
This section establishes some terminology and notation for
mathematical groups, which is needed later on. Background references
abound; see [D1966], for example.
A group is a set of elements G and an associated operation that
combines any two elements in G and returns a third element in G. The
operation is denoted as * and its application is denoted as a * b,
for any two elements a and b in G. Repeated application of the group
operation n times to the element a is denoted as a^N, for any element
a in G and any positive integer N. That is, a^2, = a * a,
a^3 = a * a * a, and so on.
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The above definition of a group operation uses multiplicative
notation. Sometimes an alternative called additive notation is used,
in which a * b is denoted as a + b, and a^N is denoted as N * a. In
multiplicative notation, g^N is called exponentiation, while the
equivalent operation in additive notation is called scalar
multiplication. In this document, multiplicative notation is used
throughout for consistency.
Every group has an special element called the identity element, which
we denote as e. For each element a in G, e * a = a * e = a. By
convention, a^0 is equal to the identity element for any a in G.
A cyclic group of order R is a group that contains the R elements
g, g^2, g^3, ..., g^R. The element g is called the generator of the
group. The element g^R is equal to the identity element e. Note
that g^X is equal to g^(X modulo R) for any non-negative integer X.
Given the element a of order N, and an integer i between 1 and N-1,
inclusive, the element a^i can be computed by the "square and
multiply" method outlined in Section 2.1 of [M1983] (see also Knuth,
Vol. 2, Section 4.6.3.), or other methods.
2.3. Finite Fields
This section establishes terminology and notation for finite fields
with prime characteristic.
When p is a prime number, then the set Zp, with the associated
addition, subtraction, multiplication and division operations, is a
finite field with character p. There is a one-to-one correspondence
between the integers between zero and p-1 and the elements of the
field. The field is denoted as Fp.
Equations involving field elements do not include the "mod p"
operation, but it is understood to be implicit. For example, the
statement that x, y, and z are in Fp and
z = x + y
is equivalent to the statement that x, y, and z are in Zp and
z = x + y mod p.
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3. Elliptic Curve Groups
This note only covers elliptic curves over fields with characteristic
greater than three. For other fields, the definition of the elliptic
curve group would be different.
An elliptic curve over a field F is defined by the curve equation
y^2 = x^3 + a*x + b,
where x, y, a, and b are elements of the field Fp [M1985]. A point
on an elliptic curve is a pair (x,y) of values in Fp that satisfy the
curve equation, such that x and y are both in Fp, or it is a special
point (@,@) that represents the identity element (which is called the
"point at infinity"). The order of an elliptic curve group is the
number of distinct points.
Two elliptic curve points (x1,y1) and (x2,y2) are equal whenever
x1=x2 and y1=y2, or when both points are the point at infinity.
The group operation associated with the elliptic curve group is as
follows [BC1989]. To an arbitrary pair of points P and Q specified
by their coordinates (x1,y1) and (x2,y2) respectively, the group
operation assigns a third point P*Q with the coordinates (x3,y3).
These coordinates are computed as follows
(x3,y3) = (@,@) when P is not equal to Q and x1 is equal to x2.
x3 = ((y2-y1)/(x2-x1))^2 - x1 - x2 and
y3 = (x1-x3)*(y2-y1)/(x2-x1) - y1 when P is not equal to Q and
x1 is not equal to x2.
(x3,y3) = (@,@) when P is equal to Q, P is equal to (0,0) and P is
an element of the group.
x3 = ((3*x1^2 + a)/(2*y1))^2 - 2*x1 and
y3 = (x1-x3)*(3*x1^2 + a)/(2*y1) - y1 if P is equal to Q.
In the above equations, a, x1, x2, x3, y1, y2, and y3 are elements of
the field Fp; thus, computation of x3 and y3 in practice use a "mod
p" operation.
The representation of elliptic curve points as a pair of integers in
Zp is known as the affine coordinate representation. This
representation is suitable as an external data representation for
communicating or storing group elements, though the point at infinity
must be treated as a special case.
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Some pairs of integers are not valid elliptic curve points. A valid
pair will satisfy the curve equation, while an invalid pair will not.
3.1. Homogenous Coordinates
An alternative way to implement the group operation is to use
homogeneous coordinates [K1987] (see also [KMOV1991]). This method
is typically more efficient because it does not require a modular
inverse operation.
An elliptic curve point (x,y) is equivalent to a point (X,Y,Z) in
homogeneous coordinates whenever x=X/Z and y=Y/Z.
Let P1=(X1,Y1,Z1) and P2=(X2,Y2,Z2) be points on an elliptic curve
and suppose that the points P1, P2 are not equal to (@,@), P1 is not
equal to P2, and P1 is not equal to -P2. Then the product
P3=(X3,Y3,Z3) = P1 * P2 is given by
X3 = v * (Z2 * (Z1 * u^2 - 2 * X1 * v^2) - v^3),
Y3 = z2 * (3 * X1 * u * v^2 - Y1 * v^3 - Z1 * u^3),
Z3 = 8 * (Y1)^3 * (Z1)^3,
where u = Y2 * Z1 - Y1 * Z2 and v = X2 * Z1 - X1 * Z2.
The product P3=(X3,Y3,Z3) = P1 * P1 is given by
X3 = 2 * Y1 * Z1 * (w^2 - 8 * X1 * Y1^2 * Z1),
Y3 = 4 * Y1^2 * Z1 * (3 * w * X1 - 2 * Y1^2 * Z1) - w^3,
Z3 = 8 * (Y1 * Z1)^3.
In the above equations, a, u, v, w, X1, X2, X3, Y1, Y2, Y3, Z1, Z2,
and Z3 are elements of the field Fp; thus, computation of X3, Y3, and
Z3 in practice use a "mod p" operation.
When converting from affine coordinates to homogeneous coordinates,
it is convenient to set Z to 1. When converting from homogeneous
coordinates to affine coordinates, it is necessary to perform a
modular inverse to find 1/Z mod p.
3.2. Group Parameters
An elliptic curve group over a finite field with characteristic
greater than three is completely specified by the following
parameters:
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The prime number p that indicates the order of the field Fp.
The value a used in the curve equation.
The value b used in the curve equation.
The generator g of the group.
The order n of the group generated by g.
An example of an Elliptic Curve Group is provided in Appendix B.
Each elliptic curve point is associated with with a particular group,
i.e a particular parameter set. Two elliptic curve groups are equal
if and only if each of the parameters in the set are equal. The
elliptic curve group operation is only defined between two points on
the same group. It is an error to apply the group operation to two
elements that are from different groups, or to apply the group
operation to a pair of coordinates that are not a valid point. See
Section 9.3 for further information.
3.2.1. Security
Security is highly dependent on the choice of these parameters. This
section gives normative guidance on acceptable choices. See also
Section 9 for more information.
The order of the group generated by g should be divisible by a large
prime, in order to preclude easy solution of the discrete logarithm
problem [K1987]
With some parameter choices, the discrete log problem is
significantly easier to solve. This includes parameter sets in which
b = 0 and p = 3 (mod 4), and parameter sets in which a = 0 and
p = 2 (mod 3) [MOV1993]. These parameter choices are inferior for
cryptographic purposes and should not be used.
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4. Elliptic Curve Diffie-Hellman (ECDH)
The Diffie-Hellman (DH) key exchange protocol [DH1976] allows two
parties communicating over an insecure channel to agree on a secret
key. It was originally defined in terms of operations in the
multiplicative group of a field with a large prime characteristic.
Massey [M1983] observed that it can be easily generalized so that it
is defined in terms of an arbitrary mathematical group. Miller
[M1985] and Koblitz [K1987] analyzed the DH protocol over an elliptic
curve group. We describe DH following the former reference.
Let G be a group, and g be a generator for that group, and let t
denote the order of G. The DH protocol runs as follows. Party A
chooses an exponent j between 1 and t-1 uniformly at random, computes
g^j and sends that element to B. Party B chooses an exponent k
between 1 and t-1 uniformly at random, computes g^k and sends that
element to A. Each party can compute g^(j*k); party A computes
(g^k)^j, and party B computes (g^j)^k.
See Appendix A regarding generation of random numbers.
4.1. Data Types
An ECDH private key a is an integer in Zt.
The corresponding ECDH public key Y is group element, where Y = g^a.
Each public key is associated with a particular group, i.e. a
particular parameter set as per Section 3.2.
The shared secret computed by both parties is a group element.
Each run of the ECDH protocol is associated with a particular group,
and both of the public keys and the shared secret are elements of
that group.
4.2. Compact Representation
As described in the final paragraph of [M1985], the x-coordinate of
the shared secret value g^(j*k) is a suitable representative for the
entire point whenever exponentiation is used as a one-way function.
In the ECDH key exchange protocol, after the element g^(j*k) has been
computed, the x-coordinate of that value can be used as the shared
secret. We call this compact output.
Following [M1985] again, when compact output is used in ECDH, only
the x-coordinate of an elliptic curve point needs to be transmitted,
instead of both coordinates as in the typical affine coordinate
representation. We call this the compact representation.
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ECDH can be used with or without compact output. Both parties in a
particular run of the ECDH protocol must use the same method. ECDH
can be used with or without compact representation. If compact
representation is used in a particular run of the ECDH protocol, then
compact output must be used as well.
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5. Elliptic Curve ElGamal Signatures (ECES)
The ElGamal signature algorithm was introduced in 1984 [E1984a]
[E1984b] [E1985]. It is based on the discrete logarithm problem in
the multiplicative group of the integers modulo a large prime number.
It is straightforward to extended it to use an elliptic curve group.
In this section we recall a well-specified elliptic curve version of
the ElGamal Signature Algorithm, as described in [A1992] and
[MV1993]. This signature method is called Elliptic Curve ElGamal
Signatures (ECES).
The algorithm uses an elliptic curve group, as described in
Section 3.2. We denote the generator as alpha, and the order of the
generator as n. We follow [MV1993] in describing the algorithms in
terms of mathematical groups.
ECES uses a collision-resistant hash function, so that it can sign
messages of arbitrary length. We denote the hash function as h().
Its input is a bit string of arbitrary length, and its output is an
integer between zero and n-1, inclusive.
ECES uses a function g() from the set of group elements to the set of
integers Zn. This function returns the x-coordinate of the affine
coordinate representation of the elliptic curve point.
5.1. Keypair Generation
The private key a is an integer between 0 and n - 1, inclusive,
generated uniformly at random. The public key is the group element
Q = alpha^a.
5.2. Signature Creation
To sign message m, using the private key a:
1. First, choose an integer k uniformly at random from the set of
all integers k in Zn that are coprime to n. (If n is a prime,
then choose an integer uniformly at random between 1 and n-1.)
(See Appendix A regarding random integers.)
2. Next, compute the group element r = alpha^k.
3. Finally, compute the integer s as
s = (h(m) + a * g(r)) / k (mod n).
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4. If s is equal to zero, then the signature creation must be
repeated, starting at Step 1 and using a newly chosen k value.
The signature for message m is the ordered pair (r, s). Note that
the first component is a group element, and the second is a non-
negative integer.
5.3. Signature Verification
To verify the message m and the signature (r,s) using the public key
Q:
Compute the group element r^s * Q^(-g(r)).
Compute the group element alpha^h(m).
Verify that the two elements previously computed are the same. If
they are identical, then the signature and message pass the
verification; otherwise, they fail.
5.4. Hash Functions
Let H() denote a hash function whose output is a fixed-length bit
string. To use H in ECES, we define the mapping between that output
and the integers between zero and n-1; this realizes the function h()
described above. Given a bit string m, the function h(m) is computed
as follows:
1. H(m) is evaluated; the result is a fixed-length bit string.
2. Convert the resulting bit string to an integer i by treating its
leftmost (initial) bit as the most significant bit of i, and
treating its rightmost (final) bit as the least significant bit
of i.
3. After conversion, reduce i modulo n, where n is the group order.
5.5. Rationale
This subsection is not normative and is provided only as background
information.
The signature verification will pass whenever the signature is
properly generated, because
r^s * Q^(-g(r)) = alpha^(k*s - a*g(r)) = alpha^h(m).
The reason that the random variable k must be coprime with n is so
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that 1/k mod n is defined.
A valid signature with s=0 leaks the secret key, since in that case a
= h(m) / g(r) mod n. We adopt Rivest's suggestion to avoid this
problem [R1992].
As described in the final paragraph of [M1985], for it is suitable to
use the x-coordinate of a particular elliptic curve point as a
representative for that point. This is what the function g() does.
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6. Abbreviated ECES Signatures (AECES)
The ECES system is secure and efficient, but has signatures that are
slightly larger than they need to be. Koyama and Tsuruoka described
a signature system based on Elliptic Curve ElGamal, but with shorter
signatures [KT1994]. Their idea is to include only the x-coordinate
of the EC point in the signature, instead of both coordinates.
Menezes, Qu, and Vanstone independently developed the same idea,
which was the basis for the "Elliptic Curve Signature Scheme with
Appendix (ECSSA)" submission to the IEEE 1363 working group
[MQV1994].
In this section we describe an Elliptic Curve Signature Scheme that
hash a single elliptic curve coordinate in the signature instead of
both coordinates. It is based on ECSSA, but with an inversion
operation moved from the signature operation to the verification
operation, so that the signing operation is more compatible with
ECES. (See [AMV1990] and [A1992] for a discussion of these
alternatives; the security of the methods is equivalent.) We refer
to this scheme as Abbreviated ECES, or AECES.
6.1. Keypair Generation
Keypairs are the same as for ECES and are as described in
Section 5.1.
6.2. Signature Creation
In this section we describe how to compute the signature for a
message m using the private key a.
Signature creation is as for ECES, with the following additional
step:
1. Let the integer s1 be equal to the x-coordinate of r.
The signature is the ordered pair (s1, s). Both signature components
are non-negative integers.
6.3. Signature Verification
Given the message m, the public key Q, and the signature (s1,s)
verification is as follows:
1. Compute the inverse of s modulo q. We denote this value as w.
2. Compute the non-negative integers u and v, where
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u = w * h(m) mod q, and
v = w * s1 mod q.
3. Compute the elliptic curve point R' = alpha^u * Q^v
4. If the x-coordinate of R' is equal to s1, then the signature and
message pass the verification; otherwise, they fail.
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7. Interoperability
The algorithms in this note can be used to interoperate with some
other ECC specifications. This section provides details for each
algorithm.
7.1. ECDH
Section 4 can be used with the Internet Key Exchange (IKE) versions
one [RFC2409] or two [RFC4306]. These algorithms are compatible with
the ECP groups for the defined by [RFC4753], [RFC2409], and
[RFC2412]. The group definition used in this protocol uses an affine
coordinate representation of the public key and uses neither the
compact output nor the compact representation of Section 4.2. Note
that some groups use a negative curve parameter "a" and express this
fact in the curve equation rather than in the parameter. The test
cases in Section 8 of [RFC4753] can be used to test an
implementation; these cases use the multiplicative notation, as does
this note. The KEi and KEr payloads are equal to g^i and g^r,
respectively, with 64 bits of encoding data prepended to them.
The algorithms in Section 4 can be used to interoperate with the IEEE
[P1363] and ANSI [X9.62] standards for ECDH based on fields of
characteristic greater than three.
7.2. ECES, AECES, and ECDSA
The Digital Signature Algorithm (DSA) is based on the discrete
logarithm problem over the multiplicative subgroup of the finite
field large prime order [DSA1991][FIPS186]. The Elliptic Curve
Digital Signature Algorithm (ECDSA) [P1363] [X9.62] is an elliptic
curve version of DSA.
AECES can interoperate with the IEEE [P1363] and ANSI [X9.62]
standards for Elliptic Curve DSA (ECDSA) based on fields of
characteristic greater than three.
An ECES signature can be converted into an ECDSA signature by
discarding the y-coordinate from the elliptic curve point.
There is a strong correspondence between ECES signatures and ECDSA
signatures. In the notation of Section 5, an ECDSA signature
consists of the pair of integers (g(r), s), and signature
verification passes if and only if
A^(h(m)/s) * Q^(g(r)/s) = r,
where the equality of the elliptic curve elements is checked by
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checking for the equality of their x-coordinates. For valid
signatures, (h(m)+a*r)/s mod q = k, and thus the two sides are equal.
An ECDSA signature contains only the x-coordinate g(r), but this is
sufficient to allow the signatures to be checked with the above
method.
Whenever the ECES signature (r, s) is valid for a particular message
m, and public key Q, then there is a valid ECDSA signature (g(r), s)
for the same message and public key.
Whenever the ECDSA signature (c, d) is valid for a particular message
m, and public key Q, then there is a valid ECES signature for the
same message and public key. This signature has the form ((c, f(c)),
d), or ((c, q-f(c)), d) where the function f takes as input an
integer in Zq and is defined as
f(x) = sqrt(x^3 + a*x + b) (mod q).
It is possible to compute the square root modulo q, for instance, by
using Shanks's method [K1987]. However, it is not as efficient to
convert an ECDSA signature (or an AECES signature) to an ECES
signature.
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8. Intellectual Property
Concerns about intellectual property have slowed the adoption of ECC,
because a number of optimizations and specialized algorithms have
been patented in recent years.
All of the normative references in this note were published during or
before October, 1994, and all of the normative text in this note is
based solely on those references.
8.1. Disclaimer
This document is not intended as legal advice. Readers are advised
to consult their own legal advisers if they would like a legal
interpretation of their rights.
The IETF policies and processes regarding intellectual property and
patents are outlined in [RFC3979] and [RFC4879] and at
https://datatracker.ietf.org/ipr/about/.
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9. Security Considerations
The security level of an elliptic curve cryptosystem is determined by
the cryptanalytic algorithm that is the least expensive for an
attacker to implement. There are several algorithms to consider.
The Polhig-Hellman method is a divide-and-conquer technique [PH1978].
If the group order n can be factored as
n = q1 * q2 * ... * qz,
then the discrete log problem over the group can be solved by
independently solving a discrete log problem in groups of order q1,
q2, ..., qz, then combining the results using the Chinese remainder
theorem. The overall computational cost is dominated by that of the
discrete log problem in the subgroup with the largest order.
Shanks algorithm [K1981v3] computes a discrete logarithm in a group
of order n using O(sqrt(n)) operations and O(sqrt(n)) storage. The
Pollard rho algorithm [P1978] computes a discrete logarithm in a
group of order n using O(sqrt(n)) operations, with a negligible
amount of storage, and can be efficiently parallelized.
The Pollard lambda algorithm [P1978] can solve the discrete logarithm
problem using O(sqrt(w)) operations and O(log(w)) storage, when the
exponent belongs to a set of w elements.
The algorithms described above work in any group. There are
specialized algorithms that specifically target elliptic curve
groups. There are no subexponential algorithms against general
elliptic curve groups, though there are methods that target certain
special elliptic curve groups; see [MOV1993] and [FR1994].
9.1. Subgroups
A group consisting of a set of elements S with associated group
operation * is a subgroup of the group with the set of elements G, if
the latter group uses the same group operation and S is a subset of
G. For each elliptic curve equation, there is an elliptic curve group
whose group order is equal to the order of the elliptic curve; that
is, there is a group that contains every point on the curve.
The order m of the elliptic curve is divisible by the order n of the
group associated with the generator; that is, for each elliptic curve
group, m = n * c for some number c. The number c is called the
"cofactor" [P1363]. Each elliptic curve group (e.g. each parameter
set as in Section 3.2) is associated with a particular cofactor.
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It is possible and desirable to use a cofactor equal to 1.
It is common to use a "safe prime group" in the conventional Diffie-
Hellman protocol over the multiplicative group of the prime field Fp
(see Appendix E of [RFC2412] for example). A safe prime group is the
subgroup of prime order q of the multiplicative group of Zp, where
p-1 = 2*q. The use of safe prime groups simplifies protocol design,
implementation and use, because they minimize the effectiveness of
some cryptanalytic attacks. Elliptic curve groups with a cofactor of
1 have similar benefits.
9.2. Diffie-Hellman
Note that the key exchange protocol as defined in Section 4 does not
protect against active attacks; Party A must use some method to
ensure that (g^k) originated with the intended communicant B, rather
than an attacker, and Party B must do the same with (g^j).
It is not sufficient to authenticate the shared secret g^(j*k), since
this leaves the protocol open to attacks that manipulate the public
keys. Instead, the values of the public keys g^x and g^y that are
exchanged should be directly authenticated. This is the strategy
used by protocols that build on Diffie-Hellman and which use end-
entity authentication to protect against active attacks, such as
OAKLEY [RFC2412] and the Internet Key Exchange [RFC2409][RFC4306].
When the cofactor of a group is not equal to 1, there are a number of
attacks that are possible against ECDH. See [VW1996], [AV1996], and
[LL1997].
9.3. Group Representation and Security
The elliptic curve group operation does not explicitly incorporate
the parameter b from the curve equation. This opens the possibility
that a malicious attacker could learn information about an ECDH
private key by submitting a bogus public key [BMM2000]. An attacker
can craft an elliptic curve group G' that has identical parameters to
a group G that is being used in an ECDH protocol, except that b is
different. An attacker can submit a point on G' into a run of the
ECDH protocol that is using group G, and gain information from the
fact that the group operations using the private key of the device
under attack are effectively taking place in G' instead of G.
This attack can gain useful information about an ECDH private key
that is associated with a static public key, that is, a public key
that is used in more than one run of the protocol. However, it does
not gain any useful information against ephemeral keys.
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This sort of attack is thwarted if an ECDH implementation does not
assume that each pair of coordinates in Zp is actually a point on the
appropriate elliptic curve.
9.4. Signatures
Elliptic curve parameters should only be used if they come from a
trusted source; otherwise, some attacks are possible [AV1996],
[V1996].
In principle, any collision-resistant hash function is suitable for
use in ECES. To facilitate interoperability, we recognize the
following hashes as suitable for use as the function H defined in
Section 5.4:
SHA-256, which has a 256-bit output.
SHA-384, which has a 384-bit output.
SHA-512, which has a 512-bit output.
All of these hash functions are defined in [FIPS180-2].
The number of bits in the output of the hash used in ECES should be
equal or close to the number of bits needed to represent the group
order.
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10. IANA Considerations
This note has no actions for IANA. This section should be removed by
the RFC editor before publication as an RFC.
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11. Acknowledgements
The author expresses his thanks to the originators of elliptic curve
cryptography, whose work made this note possible.
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12. References
12.1. Normative References
[A1992] Anderson, J., "Response to the proposed DSS",
Communications of the ACM v.35 n.7 p.50-52, July 1992.
[AMV1990] Agnew, G., Mullin, R., and S. Vanstone, "Improved Digital
Signature Scheme based on Discrete Exponentiation",
Electronics Letters Vol. 26, No. 14, July, 1990.
[BC1989] Bender, A. and G. Castagnoli, "On the Implementation of
Elliptic Curve Cryptosystems", Advances in Cryptology -
CRYPTO '89 Proceedings Spinger Lecture Notes in Computer
Science (LNCS) volume 435, 1989.
[D1966] Deskins, W., "Abstract Algebra", MacMillan Company , 1966.
[DH1976] Diffie, W. and M. Hellman, "New Directions in
Cryptography", IEEE Transactions in Information
Theory IT-22, pp 644-654, 1976.
[E1984a] ElGamal, T., "Cryptography and logarithms over finite
fields", Stanford University UMI Order No. DA 8420519,
1984.
[E1984b] ElGamal, T., "Cryptography and logarithms over finite
fields", Advances in Cryptology - CRYPTO '84
Proceedings Springer Lecture Notes in Computer Science
(LNCS) volume 196, 1984.
[E1985] ElGamal, T., "A public key cryptosystem and a signature
scheme based on discrete logarithms", IEEE Transactions on
Information Theory Vol 30, No. 4, pp. 469-472, 1985.
[FR1994] Frey, G. and H. Ruck, "A remark concerning m-divisibility
and the discrete logarithm in the divisor class group of
curves.", Mathematics of Computation Vol. 62, No. 206, pp.
865-874, 1994.
[K1981v2] Knuth, D., "The Art of Computer Programming, Vol. 2:
Seminumerical Algorithms", Addison Wesley , 1981.
[K1987] Koblitz, N., "Elliptic Curve Cryptosystems", Mathematics
of Computation Vol. 48, 1987, 203-209, 1987.
[M1983] Massey, J., "Logarithms in finite cyclic groups -
cryptographic issues", Processings of the 4th Symposium on
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Information Theory , 1983.
[M1985] Miller, V., "Use of elliptic curves in cryptography",
Advances in Cryptology - CRYPTO '85 Proceedings Springer
Lecture Notes in Computer Science (LNCS) volume 218, 1985.
[MOV1993] Menezes, A., Vanstone, S., and T. Okamoto, "Reducing
Elliptic Curve Logarithms to Logarithms in a Finite
Field", IEEE Transactions on Information Theory Vol 39,
No. 5, pp. 1639-1646, September, 1993.
[MQV1994] Menezes, A., Qu, M., and S. Vanstone, "Submission to the
IEEE P1363 Working Group (Part 6: Elliptic Curve Systems,
Draft 2)", Working Document , October, 1994.
[MV1993] Menezes, A. and S. Vanstone, "Elliptic Curve Cryptosystems
and Their Implementation", Journal of Cryptology Volume 6,
No. 4, pp209-224, 1993.
[R1992] Rivest, R., "Response to the proposed DSS", Communications
of the ACM v.35 n.7 p.41-47., July 1992.
12.2. Informative References
[AV1996] Anderson, R. and S. Vaudenay, "Minding Your P's and Q's",
Advances in Cryptology - ASIACRYPT '96 Proceedings Spinger
Lecture Notes in Computer Science (LNCS) volume 1163,
1996.
[BMM2000] Biehl, I., Meyer, B., and V. Muller, "Differential fault
analysis on elliptic curve cryptosystems", Advances in
Cryptology - CRYPTO 2000 Proceedings Spinger Lecture Notes
in Computer Science (LNCS) volume 1880, 2000.
[DSA1991] "DIGITAL SIGNATURE STANDARD", Federal Register Vol. 56,
August 1991.
[FIPS180-2]
"SECURE HASH STANDARD", Federal Information Processing
Standard (FIPS) 180-2, August 2002.
[FIPS186] "DIGITAL SIGNATURE STANDARD", Federal Information
Processing Standard FIPS-186, 1994.
[K1981v3] Knuth, D., "The Art of Computer Programming, Vol. 3:
Sorting and Searching", Addison Wesley , 1981.
[KMOV1991]
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Koyama, K., Menezes, A., Vanstone, S., and T. Okamoto,
"New Public-Key Schemes Based on Elliptic Curves over the
Ring Zn", Advances in Cryptology - CRYPTO '91
Proceedings Spinger Lecture Notes in Computer Science
(LNCS) volume 576, 1991.
[KT1994] Koyama, K. and Y. Tsuruoka, "Digital signature system
based on elliptic curve and signer device and verifier
device for said system", Japanese Unexamined Patent
Application Publication H6-43809, 1994.
[LL1997] Lim, C. and P. Lee, "A Key Recovery Attack on Discrete
Log-based Schemes Using a Prime Order Subgroup", Advances
in Cryptology - CRYPTO '97 Proceedings Spinger Lecture
Notes in Computer Science (LNCS) volume 1294, 1997.
[P1363] "Standard Specifications for Public Key Cryptography",
Institute of Electric and Electronic Engineers
(IEEE) P1363, 2000.
[P1978] Pollard, J., "Monte Carlo methods for index computation
mod p", Mathematics of Computation Vol. 32, 1978.
[PH1978] Polhig, S. and M. Hellman, "An Improved Algorithm for
Computing Logarithms over GF(p) and its Cryptographic
Significance", IEEE Transactions on Information Theory Vol
24, pp. 106-110, 1978.
[RFC2409] Harkins, D. and D. Carrel, "The Internet Key Exchange
(IKE)", RFC 2409, November 1998.
[RFC2412] Orman, H., "The OAKLEY Key Determination Protocol",
RFC 2412, November 1998.
[RFC3979] Bradner, S., "Intellectual Property Rights in IETF
Technology", BCP 79, RFC 3979, March 2005.
[RFC4306] Kaufman, C., "Internet Key Exchange (IKEv2) Protocol",
RFC 4306, December 2005.
[RFC4753] Fu, D. and J. Solinas, "ECP Groups For IKE and IKEv2",
RFC 4753, January 2007.
[RFC4879] Narten, T., "Clarification of the Third Party Disclosure
Procedure in RFC 3979", BCP 79, RFC 4879, April 2007.
[V1996] Vaudenay, S., "Hidden Collisions on DSS", Advances in
Cryptology - CRYPTO '96 Proceedings Spinger Lecture Notes
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in Computer Science (LNCS) volume 1109, 1996.
[VW1996] van Oorschot, P. and M. Wiener, "On Diffie-Hellman key
agreement with short exponents", Advances in Cryptology -
EUROCRYPT '96 Proceedings Spinger Lecture Notes in
Computer Science (LNCS) volume 1070, 1996.
[X9.62] "Public Key Cryptography for the Financial Services
Industry: The Elliptic Curve Digital Signature Algorithm
(ECDSA)", American National Standards Institute (ANSI)
X9.62.
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Appendix A. Random Number Generation
It is easy to generate an integer uniformly at random between zero
and 2^t, for some positive integer t. Generate a random bit string
that contains exactly t bits, and then convert the bit string to a
non-negative integer by treating the bits as the coefficients in a
base-two expansion of an integer.
It is sometimes necessary to generate an integer r uniformly at
random so that r satisfies a certain property P, for example, lying
within a certain interval. A simple way to do this is with the
rejection method:
1. Generate a candidate number c uniformly at random from a set that
includes many numbers that satisfy property P.
2. If c satisfies property P, then return c. Otherwise, return to
Step 1.
For example, to generate a number between 1 and n-1, repeatedly
generate integers between zero and 2^t, stopping at the first integer
that falls within that interval.
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Appendix B. Example Elliptic Curve Group
For concreteness, we recall an elliptic curve defined by Solinas and
Yu in [RFC4753] and referred to as P-256, which is believed to
provide a 128-bit security level. We use the notation of
Section 3.2, and express the generator in the affine coordinate
representation g=(gx,gy), where the values gx and gy are in Zp.
p: FFFFFFFF00000001000000000000000000000000FFFFFFFFFFFFFFFFFFFFFFFF
a: - 3
b: 5AC635D8AA3A93E7B3EBBD55769886BC651D06B0CC53B0F63BCE3C3E27D2604B
n: FFFFFFFF00000000FFFFFFFFFFFFFFFFBCE6FAADA7179E84F3B9CAC2FC632551
gx: 6B17D1F2E12C4247F8BCE6E563A440F277037D812DEB33A0F4A13945D898C296
gy: 4FE342E2FE1A7F9B8EE7EB4A7C0F9E162BCE33576B315ECECBB6406837BF51F5
Note that p can also be expressed as
p = 2^(256)-2^(224)+2^(192)+2^(96)-1.
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Author's Address
David A. McGrew
Cisco Systems
510 McCarthy Blvd.
Milpitas, CA 95035
US
Phone: (408) 525 8651
Email: mcgrew@cisco.com
URI: http://www.mindspring.com/~dmcgrew/dam.htm
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