Using number fields to compute logarithms in finite fields

Using number fields to compute logarithms in finite fields

We describe an adaptation of the number field sieve to the problem of computing logarithms in a finite field. We conjecture that the running time of the algorithm, when restricted to finite fields of an arbitrary but fixed degree, is Lq[1/3; (64/9)1/3 + o(1)], where q is the cardinality of the field, Lq [s; c] = exp(c(log q)s(log log q)1−s), and the o(1) is for q →∞. The number field sieve fact...

Discrete logarithms over finite fields

Discrete exponentiation in a finite field is a direct analog of ordinary exponentiation. The exponent can only be an integer, say n, but for w in a field F , w is defined except when w = 0 and n ≤ 0, and satisfies the usual properties, in particular w = ww and (for u and v in F ) (uv) = uv. The discrete logarithm is the inverse function, in analogy with the ordinary logarithm for real numbers. ...

Discrete logarithms in curves over finite fields

The discrete logarithm problem in finite groups is one of the supposedly difficult problems at the foundation of asymmetric or public key cryptography. The first cryptosystems based on discrete logarithms were implemented in the multiplicative groups of finite fields, in which the discrete logarithm problem turned out to be easier than one would wish, just as the factorisation problem at the he...

Finite Fields and Ze h's Logarithms

Reducing Logarithms in Totally Non-maximal Imaginary Quadratic Orders to Logarithms in Finite Fields

Since nobody can guarantee that the computation of discrete logarithms in elliptic curves or IF p remains intractible for the future it is important to study cryptosystems based on alternative groups. A promising candidate, which was proposed by Buch-mann and Williams 8], is the class group C l(() of an imaginary quadratic order O. This ring is isomorphic to the endomorphism ring of a non-super...