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 % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznf % gDOjdaryqr1ngBPrginfgDObcv39gaiuaacqWFfcVrdaqhaaWcbaGa % amiCamaaCaaameqabaGaamizaaaaaSqaaiaacQcaaaaaaa!44C9! $${\Bbb F}_{{p^d}}^*$$ , 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 % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSijHi6aaS % baaSqaaiaadchadaahaaadbeqaaiaadsgaaaWccqGHsislcaaIXaaa % beaaaaa!3B57! $${{\Bbb Z}_{{p^d} - 1}}$$ . The computation appears even somewhat more difficult in groups based on elliptic and hyperelliptic curves, with some exceptions such as supersingular curves.