Computationally hard algebraic problems

In this paper we present a simple geometric-like series of elements in a finite field F_q, and show that computing its sum is NP-hard. This problem is then reduced to the problem of counting mod p the number of zeroes in a linear recurrence sequence with elements in a finite F_p, where p is a small prime. Hence the latter problem is also NP-hard. In the lecture we shall also survey other computationally hard algebraic problems