Elliptic Curve Digital Signature Algorithm: Difference between revisions

An [[wp:Elliptic curveElliptic_curve|elliptic curve]] E over ℤp (p ≥ 5) is defined by an equation of the form '''y^2 = x^3 + ax + b''',

'''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,

12. If P += (x, y) � = E(�p), +then inverse -P = P for(x,-y), alland P ∈+ E(-P) = �p).

23. IfLet P = (xxP, yyP) and Q = (xQ, yQ), both ∈ E(ℤp), thenwhere inverse -P = (x,-y), and P + (-P) = � -Q.

Then RxR = Pλ^2 +- Q = (xR,xP yR),- wherexQ

A [[wp:digital signatureDigital_signature|digital signature]] is the electronic analogue of a hand-written signature that

'''[[wp:ECDSA|ECDSA]] key generation.''' Party A does the following:<br />

1. Compute message representative f = H(m), using a cryptographic hash function.<br />

To be cryptographically useful, the parameter r should have at least have 250 bits.

The basis for the security of [[wp:Elliptic-curve cryptographycurve_cryptography|elliptic curve cryptosystems]] is the intractability

is the intractability of the elliptic curve discrete logarithm problem (ECDLP) in a group of this size:

or demonstrate the principle of [[wp:Lenstra ellipticLenstra_elliptic-curve factorizationcurve_factorization|elliptic curve factorization]]) &mdash; but strict where required (a point G that is not on E will always cause failure).<br />

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