Discrete Logarithm Problem In Group Research Articles

О генерической сложности проблемы дискретного логарифма в группах точек эллиптических кривых над конечными полями

О генерической сложности проблемы дискретного логарифма в группах точек эллиптических кривых над конечными полями

A non-uniform birthday problem with applications to discrete logarithms

We consider a generalisation of the birthday problem that arises in the analysis of algorithms for certain variants of the discrete logarithm problem in groups. More precisely, we consider sampling coloured balls and placing them in urns, such that the distribution of assigning balls to urns depends on the colour of the ball. We determine the expected number of trials until two balls of different colours are placed in the same urn. As an aside we present an amusing “paradox” about birthdays.

On the discrete logarithm problem in class groups of curves

We study the discrete logarithm problem in degree 0 class groups of curves over finite fields, with particular emphasis on curves of small genus. We prove that for every fixed g ≥ 2, the discrete logarithm problem in degree 0 class groups of curves of genus g can be solved in an expected time of O(q 2-2/g ), where F q is the ground field. This result generalizes a corresponding result for hyperelliptic curves given in imaginary quadratic representation with cyclic degree 0 class group, and just as this previous result, it is obtained via an index calculus algorithm with double large prime variation. Generalizing this result, we prove that for fixed g 0 > 2 the discrete logarithm problem in class groups of all curves C/F q of genus g ≥ g 0 can be solved in an expected time of O((q g ) 2/g 0 (1-1/g 0 ) ) and in an expected time of O( #Cl 0 (C) 2/g 0 (1- 1/g0) ). As a complementary result we prove that for any fixed n ∈ ℕ with n > 2 the discrete logarithm problem in the groups of rational points of elliptic curves over finite fields F q n, q a prime power, can be solved in an expected time of O(q 2-2/n ). Furthermore, we give an algorithm for the efficient construction of a uniformly randomly distributed effective divisor of a specific degree, given the curve and its L-polynomial.