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Abstract We study combinatorial complexity characteristics of a Boolean function related to a natural number theoretic problem. In particular we obtain a linear lower bound on the average sensitivity of the Boolean function deciding whether a given integer is square-free. This result allows us to derive a quadratic lower bound for the formula size complexity of testing square-free numbers and a linear lower bound on the average decision tree depth. We also obtain lower bounds on the degrees of exact and approximative polynomial representations of this function. \Lambda Supported by DFG grant Me 1077/14-1.

Abstract. We say a polynomial P over ZM strongly M-represents a Boolean function F if F(x) ≡ P(x) (mod M) for all x ∈ {0, 1} n. Similarly, P one-sidedly M-represents F if F(x) = 0 ⇐ ⇒ P(x) ≡ 0 (mod M) for all x ∈ {0, 1} n. Lower bounds are obtained on the degree and the number of monomials of polynomials over Z M, which strongly or one-sidedly M-represent the Boolean function deciding if a given nbit integer is square-free. Similar lower bounds are also obtained for polynomials over the reals which provide a threshold representation of the above Boolean function. 1

Abstract We study combinatorial complexity characteristics of a Boolean function related to a natural number theoretic problem. In particular we obtain an asymtotic formula, having a linear main term, for the average sensitivity of the Boolean function deciding whether a given integer is square-free. This result allows us to derive a quadratic lower bound for the formula size complexity of testing square-free numbers and a linear lower bound on the average decision tree depth. We also obtain lower bounds on the degrees of exact and approximative polynomial representations of this function. *Supported by DFG grant Me 1077/14-1.#

Abstract. We show that for any fixed base a, a positive proportion of primes have the property that they become composite after altering any one of their digits in the base a expansion; the case a = 2 was already established by Cohen-Selfridge and Sun, using some covering congruence ideas of Erdős. Our method is slightly different, using a partially covering set of congruences followed by an application of the Selberg sieve upper bound. As a consequence, it is not always possible to test whether a number is prime from its base a expansion without reading all of its digits. We also present some slight generalisations of these results. 1.

Suppose a 1 ! a 2 ! : : : ! aZ are distinct integers in f1; : : : ; Ng. If a 0 +" 1 a 1 +" 2 a 2 + \Delta \Delta \Delta +" Z aZ is prime for all choices of " 1 ; " 2 ; : : : ; " Z 2 f0; 1g, then Z (9=2 + o(1)) log N= log log N . The argument uses a modified form of Gallagher's Larger Sieve, and shows that if Z ? (9=2 + o(1)) log N= log log N , then for some prime p ! Z 2 , the subset sums " 1 a 1 + " 2 a 2 + \Delta \Delta \Delta + " Z aZ represent every residue class modulo p. Consequently, the smallest \Sigma 2 3 Boolean circuit which tests primality for any number given by n binary digits has size 2 n\Gammao(n) . Let A be a set of distinct positive integers a 1 ! a 2 ! : : : ! aZ . The subset sums of A are the integers " 1 a 1 + " 2 a 2 + \Delta \Delta \Delta + " Z aZ , where all " i 2 f0; 1g. The set of all such subset sums will be denoted by A + so A + = n " 1 a 1 + " 2 a 2 + \Delta \Delta \Delta + " Z aZ : " i 2 f0; 1g o : A cube a 0 +A + = n a 0 + " 1 a 1...

Let p be a prime and let g be a primitive root of the field IFp of p elements. In the paper we show that the communication complexity of the last bit of the Diffie-Hellman key g xy , is at least n/24 + o(n) where x and y are n-bit integers where n is defined by the inequalities 2 n # p # 2 n+1 - 1. We also obtain a nontrivial upper bound on the Fourier coefficients of the last bit of g xy . The results are based on some new bounds of exponential sums with g xy .

In this paper, we study various combinatorial complexity characteristics of Boolean functions related to some natural arithmetic problems about polynomials over IF 2 . In particular, we consider the Boolean function deciding whether a given polynomial over IF 2 is squarefree. We obtain an exponential lower bound on the size of a decision tree for this function, and derive an asymptotic formula, having a linear main term, for its average sensitivity. This allows us to estimate other complexity characteristics such as the formula size, the average decision tree depth and the degrees of exact and approximative polynomial representations of this function. Finally, using a di#erent method, we show that squarefree testing and deciding irreducibility of polynomials over IF 2 are not in AC

Abstract. We show that for all sufficiently large integers n, a positive fraction of the primes p between 2 n−1 and 2 n have the property that p − 2 i and p+2 i are composite for every 0 ≤ i < n − 1. As a consequence, it is not possible to test whether a number is prime from its binary expansion without reading all of its digits. The objective of this note is to establish 1.