The book introduces new strategies that recommend rigorous lower bounds on the com plexity of some amount-theoretic and cryptographic points. It moreover establishes positive partaking pseudorandom properties of quite a few cryptographic primitives. These methods and strategies are based mostly totally on bounds of character sums and num bers of choices of some polynomial equations over finite fields and residue rings. Totally different amount theoretic strategies similar to sieve methods and lattice low cost algorithms are used as correctly. The book moreover incorporates a amount of open points and proposals for extra evaluation. The emphasis is on buying unconditional rigorously proved statements. The sensible facet of this technique is that the outcomes do not depend upon any assumptions or conjectures. On the draw again, the outcomes are quite a bit weaker than these which are extensively believed to be true. We get maintain of a quantity of lower bounds, exponential in phrases of logp, on the degrees and orders of o polynomials; o algebraic options; o Boolean options; o linear recurrence sequences; coinciding with values of the discrete logarithm modulo a serious p at sufficiently many elements (the amount of elements might be as small as pI/2+O:). These options are thought-about 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 specific curiosity since it corresponds to the illustration of the rightmost bit of the discrete logarithm and defines whether or not or not the argument is a quadratic residue.