By Igor Shparlinski
The e-book introduces new options which suggest rigorous decrease bounds at the complexity of a few quantity theoretic and cryptographic difficulties. those tools and methods are according to bounds of personality sums and numbers of strategies of a few polynomial equations over finite fields and residue jewelry. It additionally features a variety of open difficulties and recommendations for additional study. We receive a number of reduce bounds, exponential by way of logp, at the de grees and orders of • polynomials; • algebraic features; • Boolean services; • linear routine sequences; coinciding with values of the discrete logarithm modulo a main p at suf ficiently many issues (the variety of issues should be as small as pI/He). those capabilities are thought of over the residue ring modulo p and over the residue ring modulo an arbitrary divisor d of p - 1. The case of d = 2 is of distinct curiosity because it corresponds to the illustration of the perfect such a lot little bit of the discrete logarithm and defines even if the argument is a quadratic residue. We additionally receive non-trivial top bounds at the de gree, sensitivity and Fourier coefficients of Boolean features on bits of x determining even if x is a quadratic residue. those effects are used to procure reduce bounds at the parallel mathematics and Boolean complexity of computing the discrete logarithm. for instance, we end up that any unbounded fan-in Boolean circuit. of sublogarithmic intensity computing the discrete logarithm modulo p has to be of superpolynomial size.