Cryptography and Lucas Sequence Discrete Logarithms

Abstract

A number of public key cryptosystems such as El Gamal [2] are based on the difficulty of solving the discrete logarithm problem in certain groups, that is, solving g ^x = h for x, where g and h are given group elements. The computational difficulty of the discrete logarithm depends on the representation of the group. The additive version in ℤ _m is essentially trivial, involving only the solution of a linear congruence. However, in ${\Bbb F}_{{p^d}}^*$, the multiplicative group of the finite field with p ^d elements, the problem is intractable if p _d is large enough, even though it is isomorphic to the cyclic group ${{\Bbb Z}_{{p^d} - 1}}$. The computation appears even somewhat more difficult in groups based on elliptic and hyperelliptic curves, with some exceptions such as supersingular curves.