Digital Elliptic curve signature algorithm

Elliptic Curve DIGITAL Signature Algorithm ( ECDSA ) is a numerical algorithm of Signature.

It is an alternative of the standard DSA which with the difference of the algorithm of origin uses the Cryptographie on the elliptic curves. The advantages of ECDSA on DSA and RSA are lengths of shorter keys and operations of signature and faster coding.

ECDSA east defines by the standard X9.62-1998, Public Key Cryptography For The Financial Services Industry: The Elliptic Curve DIGITAL Signature Algorithm (ECDSA) .

Outline

That is to say an element G of a elliptic Curve of order N with N the largest prime number than 2¹⁶⁰. The curve is also defined by two elements has and B which is elements of a Welshman field of cardinality Q . That is to say the message m to be signed.

Preparation of the keys

To choose an entirety S between 1 and N -1.
To calculate Q = sG by using the element of the elliptic curve.
the public key is Q and the private key is S .

Signature

To choose in a random way a number K between 1 and N -1
To calculate $(I, J) = kG$
To calculate $x = integer (I) ~mod~n$
To calculate $y= \ frac {H (m) + sx} {K} ~mod~n$ where H ( m ) is the result of a cryptographic chopping with SHA-1 on the message m
If X or is null there, to start again
the signature is the pair ( X , there ).

Checking

To control that X and is there well between 1 and N -1
Vérifier that $x = integer (I) ~mod~n$ knowing that $(I, J) = (\ frac {H (m)}{there} ~mod~n) G + (\ frac {X} {there} ~mod~n) Q$ .
Vérifier that Q is different from (0,0) and that Q belongs well to the elliptic curve
Vérifier that nQ gives (0,0)

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