]>
Fundamental Elliptic Curve Cryptography Algorithms
Cisco Systems
510 McCarthy Blvd.
Milpitas
CA
`95035`

US
(408) 525 8651
mcgrew@cisco.com
http://www.mindspring.com/~dmcgrew/dam.htm
Network
Elliptic Curve Cryptography
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.
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 and an
Elliptic Curve version of the ElGamal Signature Algorithm
. 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. ,
, and furnish the
necessary terminology and notation from modular arithmetic, group
theory and the theory of finite fields, respectively. defines the groups based on elliptic curves over finite
fields of characteristic greater than three.
and present the fundamental ECDH and ECES
algorithms, respectively. 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 . reviews
intellectual property issues. summarizes
security considerations. describes random number
generation and provides an example of an
Elliptic Curve group.
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 .
This section reviews mathematical preliminaries and establishes
terminology and notation that is used below.
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 , for
example).

Algorithms for these operations are well known; for instance, see Chapter
4 of .
This section establishes some terminology and notation for
mathematical groups, which is needed later on. Background references
abound; see , 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.
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 (see also Knuth, Vol. 2, Section
4.6.3.), or other methods.
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

is equivalent to the statement that x, y, and z are in the set { 0, 1, ..., p-1 } and
z = x + y mod p.

This note only covers elliptic curves over fields with characteristic
greater than three; these are the curves used in Suite B .
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 . 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 . 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
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.
An alternative way to implement the group operation is to use
homogeneous coordinates (see also ). 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
coordinates to affine coordinates, it is necessary to perform
a modular inverse to find 1/Z mod p.
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 .
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 for further information.
Security is highly dependent on the choice of these parameters.
This section gives normative guidance on acceptable choices.
See also 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
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)
. These parameter choices are inferior for
cryptographic purposes and SHOULD NOT be used.
The Diffie-Hellman (DH) key exchange protocol
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 observed that it can be easily
generalized so that it is defined in terms of an arbitrary
mathematical group. Miller and Koblitz 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 regarding generation of random numbers.
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 .
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.
As described in the final paragraph of , 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 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.
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.
The ElGamal signature algorithm was introduced in 1984
. 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
and .
This signature method is called Elliptic Curve ElGamal Signatures (ECES).
The algorithm uses an elliptic curve group, as described in , with prime field order p, curve equation
parameters a and b. We follow 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.
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.
To sign message m, using the private key z:
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
regarding random integers.)
Next, compute the group element r = alpha^k.
Finally, compute the integer s as
s = (h(m) + z * g(r)) / k (mod n).

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.
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.

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:
H(m) is evaluated; the result is a fixed-length bit string.
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.
After conversion, reduce i modulo n, where n is the group order.

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
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 .
As described in the final paragraph of , 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.
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 . 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 .
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 and , 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 and for a discussion of
these alternatives; the security of the methods is equivalent.) We
refer to this scheme as Abbreviated ECES, or AECES.
Keypairs are the same as for ECES and are as described in .
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:
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.
Given the message m, the public key Q, and the signature (s1, s)
verification is as follows:
Compute the inverse of s modulo n. We denote this value as w.
Compute the non-negative integers u and v, where
u = w * h(m) mod n, and
v = w * s1 mod n.

Compute the elliptic curve point R' = alpha^u * Q^v
If the x-coordinate of R' is equal to s1, then the signature and
message pass the verification; otherwise, they fail.

The algorithms in this note can be used to interoperate with
some other ECC specifications. This section provides details
for each algorithm.
can be used with the Internet Key Exchange (IKE)
versions one or two . These algorithms are compatible
with the ECP groups for the defined by
, , and . 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 .
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 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 can be used to interoperate
with the IEEE and ANSI
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).
The Digital Signature Algorithm (DSA) is based on the discrete
logarithm problem over the multiplicative subgroup of the finite field
large prime order .
The Elliptic Curve Digital Signature Algorithm (ECDSA) is an elliptic curve version
of DSA.
AECES can interoperate with the IEEE and ANSI
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 , an ECDSA (or AECES) 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
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 . However, it is not as
efficient to convert an ECDSA signature (or an AECES signature) to an
ECES signature.
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 ) 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.
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 and
and at
https://datatracker.ietf.org/ipr/about/.
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
. 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 computes a discrete
logarithm in a group of order n using O(sqrt(n)) operations and
O(sqrt(n)) storage. The Pollard rho algorithm
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 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
and .
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" . Each elliptic curve group (e.g.
each parameter set as in ) is associated with a
particular cofactor.
It is possible and desirable to use a cofactor equal to 1.
Note that the key exchange protocol as defined in
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 and the Internet Key Exchange .
When the cofactor of a group is not equal to 1, there are a number of
attacks that are possible against ECDH. See ,
, and .
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 . 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.
Elliptic curve parameters should only be used if they come from a
trusted source; otherwise, some attacks are possible
, .
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
:
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 .
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.
This note has no actions for IANA. This section should be removed by
the RFC editor before publication as an RFC.
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.
Abstract Algebra
The Art of Computer Programming, Vol. 2: Seminumerical Algorithms
New Directions in Cryptography
Logarithms in finite cyclic groups - cryptographic issues
Cryptography and logarithms over finite fields
Cryptography and logarithms over finite fields
A public key cryptosystem and a signature scheme
based on discrete logarithms
Use of elliptic curves in cryptography
Elliptic Curve Cryptosystems
On the Implementation of Elliptic Curve Cryptosystems
Improved Digital Signature Scheme based on Discrete Exponentiation
Response to the proposed DSS
Response to the proposed DSS
Elliptic Curve Cryptosystems and Their Implementation
Reducing Elliptic Curve Logarithms to
Logarithms in a Finite Field
A remark concerning m-divisibility and the discrete
logarithm in the divisor class group of curves.
Digital signature system based on elliptic curve
and signer device and verifier device for said system
Submission to the IEEE P1363 Working Group (Part 6:
Elliptic Curve Systems, Draft 2)
&rfc2119;
NSA Suite B Cryptography
An Improved Algorithm for Computing Logarithms over
GF(p) and its Cryptographic Significance
The Art of Computer Programming, Vol. 3:
Sorting and Searching
Monte Carlo methods for index computation mod p
New Public-Key Schemes Based on Elliptic Curves over the Ring Zn
Minding Your P's and Q's
Parallel Collision Search with Application to Hash Functions and Discrete Logarithms
On Diffie-Hellman key agreement with short exponents
Hidden Collisions on DSS
A Key Recovery Attack on Discrete Log-based Schemes
Using a Prime Order Subgroup
Differential fault analysis on elliptic curve cryptosystems
Public Key Cryptography for the Financial Services
Industry: The Elliptic Curve Digital Signature
Algorithm (ECDSA)
Standard Specifications for Public Key Cryptography
DIGITAL SIGNATURE STANDARD
DIGITAL SIGNATURE STANDARD
SECURE HASH STANDARD
&rfc2409;
&rfc2412;
&rfc4306;
&rfc4753;
&rfc3979;
&rfc4879;
The definitions of these key words are quoted from and are
commonly used in Internet standards. They are reproduced in this note in order
to avoid a normative reference from after 1994.
MUST - This word, or the terms "REQUIRED" or "SHALL", mean that the
definition is an absolute requirement of the specification.
MUST NOT - This phrase, or the phrase "SHALL NOT", mean that the
definition is an absolute prohibition of the specification.
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.
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.
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.)

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:
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)
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.
For concreteness, we recall an elliptic curve defined by Solinas and
Yu in and referred to as P-256, which is
believed to provide a 128-bit security level. We use the notation of
, 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.