Cryptography and Lucas Sequence Discrete Logarithms

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.