Cryptography on the elliptic curves

In Cryptography, the elliptic curved , mathematical objects, can be used for asymmetrical operations like exchanges of keys on a not-protected channel or an asymmetrical coding, one then speaks about cryptography on the elliptic curves or ECC (of the Acronyme English Elliptic curve cryptography ). The use of the elliptic curves in cryptography was suggested, in an independent way, by Neal Koblitz and Victor Miller in 1985.

The key S employed for a coding by elliptic curve are shorter than with a system based on the problem of factorization like RSA. Moreover the ECC a security level equivalent or higher than the other methods gets. Another attraction of the ECC is that a bilinear Opérateur can be defined between the groups. This operator bases himself on the Couplage of Weil or the Couplage of Touches. The bilinear operators recently saw themselves applied in many ways in cryptography, for example for the Chiffrement based on the identity. A negative point is that the operations of Chiffrement and Déchiffrement can have larger complexity than for other methods.

The resistance of a system based on the elliptic curves rests on the problem of the discrete Logarithme in the group corresponding to the elliptic curve. The theoretical developments on the curves being relatively recent, cryptography on elliptic curve is not very known and suffers from a great number of patents which prevent its development.

Exchange keys

Alice and Bob agree (publicly) on an elliptic curve

E (has, B, p) ~

, i.e. they choose an elliptic curve

y^2~mod~p = (x^3+ax+b)~mod~p~

. They put also agreement, publicly, on a point

P~

located on the curve.

Secretly, Alice chooses an entirety $d_A~$ , and Bob an entirety $d_B~$ . Alice sends to Bob the point $d_A P~$ , and Bob sends to Alice $d_B P~$ . Each one on their side, they are able to calculate $d_A (d_B P) = (d_A d_B) P~$ which is a point of the curve, and constitutes their common secret key.

If Eve has spy their exchanges, she knows $E (has, B, p), P, d_A P, d_B P~$ . To be able to calculate $d_A d_B P~$ , it is necessary to be able to calculate $d_A~$ knowing $P~$ and $d_A P~$ . It is what one invites to solve the discrete logarithm on an elliptic curve. However, currently, if the numbers are sufficiently large, one does not know effective method to solve this problem in a reasonable time.

Implementation

Parameters of the field

Algorithm of Schoof

Longor of the keys

For a safety of 128 bits, one uses a curve on the body $\ mathbb {F} _q$ , where $q \ approx 2^ {256}$ .

References

Neal Koblitz, " Elliptic curve cryptosystems" , Mathematics off Computation 48,1987, pp203-209
V. Miller, " Use elliptic curves in cryptography" off; , CRYPTO 85,1985.
Blake, Seroussi, Smart, " Elliptic Curves in Cryptography" , Cambridge University Near, 1999
Hankerson, Menezes, Vanstone, " Guide to Elliptic Curve Cryptography" , Springer-Verlag, 2004
L. Washington, " Elliptic Curves: Number Theory and Cryptography", Chapman & Hall/CRC, 2003

See too

Cryptography on the hyperelliptic curves
Cryptography based on the identity

Companies

Certicom

External bonds

To quantify with elliptic curves
Collection of bonds on articles on the elliptic curves and their applications

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