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Abstract. We establish upper bounds for the number of smooth values of the Euler function. In particular, although the Euler function has a certain "smoothing" effect on its integer arguments, our results show that, in fact, most values produced by the Euler function are not smooth. We apply our results to study the distribution of "strong primes", which are commonly encountered in cryptography.We also consider the problem of obtaining upper and lower bounds for the number of positive integers n ≤ x for which the value of the Euler function ϕ(n) is a perfect square and also for the number of n ≤ x such that ϕ(n) is squarefull. We give similar bounds for the Carmichael function λ(n).

Abstract. Boneh and Venkatesan have recently proposed a polynomial time algorithm for recovering a "hidden" element α of a finite field IFp of p elements from rather short strings of the most significant bits of the remainder modulo p of αt for several values of t selected uniformly at random from IF * p . We use some recent bounds of exponential sums to generalize this algorithm to the case when t is selected from a quite small subgroup of IF * p . Namely, our results apply to subgroups of size at least p 1/3+ε for all primes p and to subgroups of size at least p ε for almost all primes p, for any fixed ε > 0. We also use this generalization to improve (and correct) one of the statements of the aforementioned work about the computational security of the most significant bits of the Diffie-Hellman key.

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 p 1/2+ε . 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 g x → g x 2 , where g is a primitive root of a finite field of q elements F q .These results can be used to obtain lower bounds on the parallel arithmetic and Boolean complexity of computing the discrete logarithm and breaking the Diffie-Hellman cryptosystem.The method is based on bounds of character sums and numbers of solutions of some polynomial equations.

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