Elliptic Curve Digital Signature Algorithm

The Elliptic Curve Digital Signature Algorithm (ECDSA).

Elliptic curves.

An elliptic curve E over ℤp (p ≥ 5) is defined by an equation of the form y^2 = x^3 + ax + b, where a, b ∈ ℤp and the discriminant ≢ 0 (mod p), together with a special point 𝒪 called the point at infinity. The set E(ℤp) consists of all points (x, y), with x, y ∈ ℤp, which satisfy the above defining equation, together with 𝒪.

There is a rule for adding two points on an elliptic curve to give a third point. This addition operation and the set of points E(ℤp) form a group with identity 𝒪. It is this group that is used in the construction of elliptic curve cryptosystems.

The addition rule — which can be explained geometrically — is summarized as follows:

1. P + 𝒪 = 𝒪 + P = P for all P ∈ E(ℤp).

2. If P = (x, y) ∈ E(ℤp), then inverse -P = (x,-y), and P + (-P) = 𝒪.

3. Let P = (xP, yP) and Q = (xQ, yQ), both ∈ E(ℤp), where P ≠ -Q.
   Then R = P + Q = (xR, yR), where

   xR = λ^2 - xP - xQ
   yR = λ·(xP - xR) - yP,

   with

   λ = (yP - yQ) / (xP - xQ) if P ≠ Q,
       (3·xP·xP + a) / 2·yP  if P = Q (point doubling).

Elliptic curve digital signature algorithm.

A digital signature is the electronic analogue of a hand-written signature that convinces the recipient that a message has been sent intact by the presumed sender. Anyone with access to the public key of the signer may verify this signature. Changing even a single bit of a signed message will cause the verification procedure to fail.

ECDSA key generation. Party A does the following:
1. Select an elliptic curve E defined over ℤp.
 The number of points in E(ℤp) should be divisible by a large prime r.
2. Select a base point G ∈ E(ℤp) of order r (which means that rG = 𝒪).
3. Select a random integer s in the interval [1, r - 1].
4. Compute W = sG.
 The public key is (E, G, r, W), the private key is s.

ECDSA signature computation. To sign a message m, A does the following:
1. Compute message representative f = H(m), using a cryptographic hash function.
 Note that f can be greater than r but not longer (measuring bits).
2. Select a random integer u in the interval [1, r - 1].
3. Compute V = uG = (xV, yV) and c ≡ xV mod r  (goto (2) if c = 0).
4. Compute d ≡ u^-1·(f + s·c) mod r  (goto (2) if d = 0).
 The signature for the message m is the pair of integers (c, d).

ECDSA signature verification. To verify A's signature, B should do the following:
1. Obtain an authentic copy of A's public key (E, G, r, W).
 Verify that c and d are integers in the interval [1, r - 1].
2. Compute f = H(m) and h ≡ d^-1 mod r.
3. Compute h1 ≡ f·h mod r and h2 ≡ c·h mod r.
4. Compute h1G + h2W = (x1, y1) and c1 ≡ x1 mod r.
 Accept the signature if and only if c1 = c.

To be cryptographically useful, the parameter r should at least have 250 bits. The basis for the security of elliptic curve cryptosystems is the intractability of the elliptic curve discrete logarithm problem (ECDLP) in a group of this size: given two points G, W ∈ E(ℤp), where W lies in the subgroup of order r generated by G, determine an integer k such that W = kG and 0 ≤ k < r.

Task.

The task is to write a toy version of the ECDSA, quasi the equal of a real-world implementation, but utilizing parameters that fit into standard arithmetic types. To keep things simple there's no need for key export or a hash function (just a sample hash value and a method to tamper with it). The program should be lenient where possible (for example: if it accepts a composite modulus N it will either function as expected, or demonstrate the principle of elliptic curve factorization), but strict where required (a point G that is not on E will always cause failure).
Toy ECDSA is of course completely useless for its cryptographic purpose. If this bothers you, feel free to add a multiple-precision version.

Reference.

Elliptic curves are in the IEEE Std 1363-2000 (Standard Specifications for Public-Key Cryptography), see:

7. Primitives based on the elliptic curve discrete logarithm problem (p. 27ff.)

7.1 The EC setting
7.1.2 EC domain parameters
7.1.3 EC key pairs

7.2 Primitives
7.2.7 ECSP-DSA (p. 35)
7.2.8 ECVP-DSA (p. 36)

Annex A. Number-theoretic background
A.9 Elliptic curves: overview (p. 115)
A.10 Elliptic curves: algorithms (p. 121)