Network Working Group D. McGrew
Internet-Draft Cisco Systems
Intended status: Informational October 26, 2009
Expires: April 29, 2010
Fundamental Elliptic Curve Cryptography Algorithms
draft-mcgrew-fundamental-ecc-01.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 defined
over fields of characteristic greater than three are in scope; these
curves are those used in Suite B.
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Table of Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.1. Conventions Used In This Document . . . . . . . . . . . . 4
2. Mathematical Background . . . . . . . . . . . . . . . . . . . 5
2.1. Modular Arithmetic . . . . . . . . . . . . . . . . . . . . 5
2.2. Group Operations . . . . . . . . . . . . . . . . . . . . . 5
2.3. Finite Fields . . . . . . . . . . . . . . . . . . . . . . 6
3. Elliptic Curve Groups . . . . . . . . . . . . . . . . . . . . 8
3.1. Homogeneous Coordinates . . . . . . . . . . . . . . . . . 9
3.2. Group Parameters . . . . . . . . . . . . . . . . . . . . . 10
3.2.1. Security . . . . . . . . . . . . . . . . . . . . . . . 10
4. Elliptic Curve Diffie-Hellman (ECDH) . . . . . . . . . . . . . 11
4.1. Data Types . . . . . . . . . . . . . . . . . . . . . . . . 11
4.2. Compact Representation . . . . . . . . . . . . . . . . . . 11
5. Elliptic Curve ElGamal Signatures (ECES) . . . . . . . . . . . 13
5.1. Keypair Generation . . . . . . . . . . . . . . . . . . . . 13
5.2. Signature Creation . . . . . . . . . . . . . . . . . . . . 13
5.3. Signature Verification . . . . . . . . . . . . . . . . . . 14
5.4. Hash Functions . . . . . . . . . . . . . . . . . . . . . . 14
5.5. Rationale . . . . . . . . . . . . . . . . . . . . . . . . 14
6. Abbreviated ECES Signatures (AECES) . . . . . . . . . . . . . 16
6.1. Keypair Generation . . . . . . . . . . . . . . . . . . . . 16
6.2. Signature Creation . . . . . . . . . . . . . . . . . . . . 16
6.3. Signature Verification . . . . . . . . . . . . . . . . . . 16
7. Interoperability . . . . . . . . . . . . . . . . . . . . . . . 18
7.1. ECDH . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
7.2. ECES, AECES, and ECDSA . . . . . . . . . . . . . . . . . . 18
8. Intellectual Property . . . . . . . . . . . . . . . . . . . . 20
8.1. Disclaimer . . . . . . . . . . . . . . . . . . . . . . . . 20
9. Security Considerations . . . . . . . . . . . . . . . . . . . 21
9.1. Subgroups . . . . . . . . . . . . . . . . . . . . . . . . 21
9.2. Diffie-Hellman . . . . . . . . . . . . . . . . . . . . . . 22
9.3. Group Representation and Security . . . . . . . . . . . . 22
9.4. Signatures . . . . . . . . . . . . . . . . . . . . . . . . 22
10. IANA Considerations . . . . . . . . . . . . . . . . . . . . . 24
11. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . 25
12. References . . . . . . . . . . . . . . . . . . . . . . . . . . 26
12.1. Normative References . . . . . . . . . . . . . . . . . . . 26
12.2. Informative References . . . . . . . . . . . . . . . . . . 27
Appendix A. Key Words . . . . . . . . . . . . . . . . . . . . . . 30
Appendix B. Random Number Generation . . . . . . . . . . . . . . 31
Appendix C. Example Elliptic Curve Group . . . . . . . . . . . . 32
Author's Address . . . . . . . . . . . . . . . . . . . . . . . . . 33
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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 B describes random number generation and
Appendix C provides an example of an Elliptic Curve group.
1.1. Conventions Used In This Document
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 Appendix A.
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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 > 1 such that z
divides x and z divides y.
The set Zq = { 0, 1, 2, ..., q-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 Zq, a + b mod q is equal to
a + b if a + b < q, and is equal to a + b - q otherwise.
For each pair of integers a and b in Zq, a - b mod q is equal to
a - b if a - b >= 0, and is equal to a - b + q otherwise.
For each pair of integers a and b in Zq, a * b mod q is equal to
the remainder of a * b divided by q.
For each integer x in Zq that is coprime with q, the inverse of x
modulo q is denoted as 1 / x mod q, 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 together with an 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. The operation is associative, that
is, for all a, b and c in G, a * (b * c) is identical to (a * b) * c.
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.
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That is, a^2, = a * a, a^3 = a * a * a, and so on. The associativity
of the group operation ensures that the computation of a^n is
unambiguous; any grouping of the terms gives the same result.
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.
Every group element a has a unique inverse element b such that a * b
= b * a = e. The inverse of a is denoted as a^-1 in multiplicative
notation. (In additive notation, the inverse of a is denoted as -a.)
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 addition,
subtraction, multiplication and division operations, is a finite
field with characteristic p. Each nonzero element x in Zp has an
inverse 1/x. There is a one-to-one correspondence between the
integers between zero and p-1, inclusive, and the elements of the
field. The field is denoted as Fp.
Equations involving field elements do not explicitly denote 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
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is equivalent to the statement that x, y, and z are in the set { 0,
1, ..., p-1 } 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; these are the curves used in Suite B [SuiteB].
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, and the
discriminant 16*(4*a^3 - 27*b^2) is nonzero [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
inverse of the point (x1,y1) is the point (x1,-y1).
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 and y1 is equal to 0,
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 and y1 is
not equal to 0.
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 must reduce
the right-hand-side modulo p.
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
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communicating or storing group elements, though the point at infinity
must be treated as a special case.
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. Homogeneous 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
inversion operation.
An elliptic curve point (x,y) (other than the point at infinity
(@,@)) is equivalent to a point (X,Y,Z) in homogeneous coordinates
whenever x=X/Z mod p and y=Y/Z mod p.
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^-1. Then the product
P3=(X3,Y3,Z3) = P1 * P2 is given by
X3 = v * (Z2 * (Z1 * u^2 - 2 * X1 * v^2) - v^3) mod p,
Y3 = z2 * (3 * X1 * u * v^2 - Y1 * v^3 - Z1 * u^3) mod p,
Z3 = 8 * (Y1)^3 * (Z1)^3 mod p,
where u = Y2 * Z1 - Y1 * Z2 mod p and v = X2 * Z1 - X1 * Z2 mod p.
When the points P1 and P2 are equal, then (X1/Z1, Y1/Z1) is equal to
(X2/Z2, Y2/Z2), which is true if and only if u and v are both equal
to zero.
The product P3=(X3,Y3,Z3) = P1 * P1 is given by
X3 = 2 * Y1 * Z1 * (w^2 - 8 * X1 * Y1^2 * Z1) mod p,
Y3 = 4 * Y1^2 * Z1 * (3 * w * X1 - 2 * Y1^2 * Z1) - w^3 mod p,
Z3 = 8 * (Y1 * Z1)^3 mod p,
where w = 3 * X1^2 + a * Z1^2 mod p. In the above equations, a, u,
v, w, X1, X2, X3, Y1, Y2, Y3, Z1, Z2, and Z3 are integers in the set
Fp.
When converting from affine coordinates to homogeneous coordinates,
it is convenient to set Z to 1. When converting from homogeneous
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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:
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 C.
Each elliptic curve point is associated 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 informative guidance.
The order of the group generated by g MUST 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 B regarding generation of random numbers.
4.1. Data Types
An ECDH private key z is an integer in Zt.
The corresponding ECDH public key Y is the group element, where Y =
g^z. 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 extend 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, with prime field order p, curve equation parameters a
and b. We follow [MV1993] in describing the algorithms in terms of
mathematical groups, and denoting the generator as alpha, and its
order as n.
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 z is an integer between 0 and n - 1, inclusive,
generated uniformly at random. The public key is the group element
Q = alpha^z.
5.2. Signature Creation
To sign message m, using the private key z:
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 B regarding random integers.)
2. Next, compute the group element r = alpha^k.
3. Finally, compute the integer s as
s = (h(m) + z * 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 - z*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], 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
uses a single elliptic curve coordinate in the signature instead of
both coordinates. It is based on [KT1994] and [MQV1994], but with
the finite field 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 z.
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 n. We denote this value as w.
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2. Compute the non-negative integers u and v, where
u = w * h(m) mod n, and
v = w * s1 mod n.
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. To use IEEE P1363 ECDH in a
manner that will interoperate with this specification, the following
options and parameter choices should be used: prime curves with a
cofactor of 1, the ECSVDP-DH primitive, and the Key Derivation
Function must be the "identity" function (equivalently, omit the KDF
step and output the shared secret value directly).
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 or AECES signature
by discarding the y-coordinate from the elliptic curve point.
There is a strong correspondence between ECES signatures and ECDSA or
AECES signatures. In the notation of Section 5, an ECDSA (or AECES)
signature consists of the pair of integers (g(r), s), and signature
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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
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 (or AECES) 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 AECES or ECDSA signature
(g(r), s) for the same message and public key.
Whenever an AECES or 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 for ECDH (as defined in Section 4)
were published during or before 1989, those for ECES were published
during or before 1993, and those for AECES were published during or
before October, 1994. All of the normative text for these algorithms
is based solely on their respective 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 Pohlig-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 [VW1994].
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 nonempty 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.
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.
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].
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In principle, any collision-resistant hash function is suitable for
use in ECES or AECES. 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 or AECES
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, and all of the
reviewers, who provided valuable constructive feedback.
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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.
[KT1994] Koyama, K. and Y. Tsuruoka, "Digital signature system
based on elliptic curve and signer device and verifier
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device for said system", Japanese Unexamined Patent
Application Publication H6-43809, February 18, 1994.
[M1983] Massey, J., "Logarithms in finite cyclic groups -
cryptographic issues", Proceedings of the 4th Symposium on
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.
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[K1981v3] Knuth, D., "The Art of Computer Programming, Vol. 3:
Sorting and Searching", Addison Wesley , 1981.
[KMOV1991]
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.
[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] Pohlig, 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.
[RFC2119] Bradner, S., "Key words for use in RFCs to Indicate
Requirement Levels", BCP 14, RFC 2119, March 1997.
[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.
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[SuiteB] "NSA Suite B Cryptography", Web Page http://www.nsa.gov/
ia/programs/suiteb_cryptography/index.shtml.
[V1996] Vaudenay, S., "Hidden Collisions on DSS", Advances in
Cryptology - CRYPTO '96 Proceedings Spinger Lecture Notes
in Computer Science (LNCS) volume 1109, 1996.
[VW1994] van Oorschot, P. and M. Wiener, "Parallel Collision Search
with Application to Hash Functions and Discrete
Logarithms", Proceedings of the 2nd ACM Conference on
Computer and communications security pp. 210-218, 1994.
[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. Key Words
The definitions of these key words are quoted from [RFC2119] and are
commonly used in Internet standards. They are reproduced in this
note in order to avoid a normative reference from after 1994.
1. MUST - This word, or the terms "REQUIRED" or "SHALL", mean that
the definition is an absolute requirement of the specification.
2. MUST NOT - This phrase, or the phrase "SHALL NOT", mean that the
definition is an absolute prohibition of the specification.
3. SHOULD - This word, or the adjective "RECOMMENDED", mean that
there may exist valid reasons in particular circumstances to
ignore a particular item, but the full implications must be
understood and carefully weighed before choosing a different
course.
4. SHOULD NOT - This phrase, or the phrase "NOT RECOMMENDED" mean
that there may exist valid reasons in particular circumstances
when the particular behavior is acceptable or even useful, but
the full implications should be understood and the case carefully
weighed before implementing any behavior described with this
label.
5. MAY - This word, or the adjective "OPTIONAL", mean that an item
is truly optional. One vendor may choose to include the item
because a particular marketplace requires it or because the
vendor feels that it enhances the product while another vendor
may omit the same item. An implementation which does not include
a particular option MUST be prepared to interoperate with another
implementation which does include the option, though perhaps with
reduced functionality. In the same vein an implementation which
does include a particular option MUST be prepared to interoperate
with another implementation which does not include the option
(except, of course, for the feature the option provides.)
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Appendix B. Random Number Generation
It is easy to generate an integer uniformly at random between zero
and 2^t -1, inclusive, 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 all numbers that satisfy property P (plus some other
numbers, preferably not too many)
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, inclusive,
repeatedly generate integers between zero and 2^t - 1, inclusive,
stopping at the first integer that falls within that interval.
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Appendix C. 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 Fp.
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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