2,056 research outputs found
Elliptic periods for finite fields
We construct two new families of basis for finite field extensions. Basis in
the first family, the so-called elliptic basis, are not quite normal basis, but
they allow very fast Frobenius exponentiation while preserving sparse
multiplication formulas. Basis in the second family, the so-called normal
elliptic basis are normal basis and allow fast (quasi linear) arithmetic. We
prove that all extensions admit models of this kind
Discrete logarithms in curves over finite fields
A survey on algorithms for computing discrete logarithms in Jacobians of
curves over finite fields
On the construction of elliptic Chudnovsky-type algorithms for multiplication in large extensions of finite fields
International audienceWe indicate a strategy in order to construct bilinear multiplication algorithms of type Chudnovsky in large extensions of any finite field. In particular, using the symmetric version of the generalization of Randriambololona specialized on the elliptic curves, we show that it is possible to construct such algorithms with low bilinear complexity. More precisely, if we only consider the Chudnovsky-type algorithms of type symmetric elliptic, we show that the symmetric bilinear complexity of these algorithms is in O(n(2q)^log * q (n)) where n corresponds to the extension degree, and log * q (n) is the iterated logarithm. Moreover, we show that the construction of such algorithms can be done in time polynomial in n. Finally, applying this method we present the effective construction, step by step, of such an algorithm of multiplication in the finite field F 3^57. Index Terms Multiplication algorithm, bilinear complexity, elliptic function field, interpolation on algebraic curve, finite field
Easy decision-Diffie-Hellman groups
The decision-Diffie-Hellman problem (DDH) is a central computational problem
in cryptography. It is known that the Weil and Tate pairings can be used to
solve many DDH problems on elliptic curves. Distortion maps are an important
tool for solving DDH problems using pairings and it is known that distortion
maps exist for all supersingular elliptic curves. We present an algorithm to
construct suitable distortion maps. The algorithm is efficient on the curves
usable in practice, and hence all DDH problems on these curves are easy. We
also discuss the issue of which DDH problems on ordinary curves are easy
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