On Polynomial Approximation of the Discrete Logarithm and the Diffie—Hellman Mapping

We obtain several lower bounds, exponential in terms of lg p , on the degrees of polynomials and algebraic functions coinciding with values of the discrete logarithm modulo a prime p at sufficiently many points; the number of points can be as little as p1/2 + ɛ . We also obtain improved lower bounds on the degree and sensitivity of Boolean functions on bits of x deciding whether x is a quadratic residue. Similar bounds are also proved for the Diffie—Hellman mapping gx→ gx2 , where g is a primitive root of a finite field of q elements Fq .