New zero-knowledge proofs are given for some number-theoretic problems. All of the problems are in NP, but the proofs given here are much more efficient than the previously known proofs. In addition, these proofs do not require the prover to be super-polynomial in power. A probabilistic polynomial time prover with the appropriate trap-door knowledge is sufficient. The proofs are perfect or statistical zero-knowledge in all cases except one. 1 Introduction Many researchers have studied zero-knowledge proofs and the classes of problems which have such zero-knowledge proofs. Little attention, however, has been paid to the practicality of these proofs. It is known, for example, that, under certain cryptographic assumptions, all problems in NP have zero-knowledge proofs [19], [8], [10]. Although these proofs can be performed with probabilistic polynomial time provers who have the appropriate trapdoor information, these proofs may involve a transformation to a circuit or to an NP-complete p...

Abstract. Let E/K be an abelian extension of number fields, with E ̸ = Q. Let ∆ and n denote the absolute discriminant and degree of E. Letσdenote an element of the Galois group of E/K. Weprovethefollowingtheorems, assuming the Extended Riemann Hypothesis: () (1) There is a degree-1 prime p of K such that p = σ, satis-

We address the problem of constructing positive operator-valued measures (POVMs) in finite dimension n consisting of n 2 operators of rank one which have an inner product close to uniform. This is motivated by the related question of constructing symmetric informationally complete POVMs (SIC-POVMs) for which the inner products are perfectly uniform. However, SIC-POVMs are notoriously hard to construct and despite some success of constructing them numerically, there is no analytic construction known. We present two constructions of approximate versions of SIC-POVMs, where a small deviation from uniformity of the inner products is allowed. The first construction is based on selecting vectors from a maximal collection of mutually unbiased bases and works whenever the dimension of the system is a prime power. The second construction is based on perturbing the matrix elements of a subset of mutually unbiased bases. Moreover, we construct vector systems in C n which are almost orthogonal and which might turn out to be useful for quantum computation. Our constructions are based on results of analytic number theory. 1

This paper studies the descriptional complexity of (i) sequences over a finite alphabet; and (ii) subsets of N (the natural numbers). If (s(i)) i0 is a sequence over a finite alphabet \Delta, then we define the k-automaticity of s, A k s (n), to be the smallest possible number of states in any deterministic finite automaton that, for all i with 0 i n, takes i expressed in base-k as input and computes s(i). We give examples of sequences that have high automaticity in all bases k; for example, we show that the characteristic sequence of the primes has k- automaticity A k s (n) = \Omega\Gamma n 1=43 ) for all k 2, thus making quantitative the classical theorem of Minsky and Papert that the set of primes expressed in base-2 is not regular. We give examples of sequences with low automaticity in all bases k, and low automaticity in some bases and high in others. We also obtain bounds on the automaticity of certain sequences that are fixed points of homomorphisms, such as the Fibonac...

Abstract. For any ɛ>0 and any non-exceptional modulus q ≥ 3, we prove that, for x large enough (x ≥ αɛ log 2 q), the interval [ e x,e x+ɛ] contains a prime p in any of the arithmetic progressions modulo q. We apply this result to establish that every integer n larger than exp(71 000) is a sum of seven cubes. 1.

consider N as a continuous variable and seek N * so that aF,/aN = 0. The result is