Abstract. New probabilistic algorithms are presented for factoring univariate polynomials over finite fields. The algorithms factor a polynomial of degree n over a finite field of constant cardinality in time O(n 1.815). Previous algorithms required time Θ(n 2+o(1)). The new algorithms rely on fast matrix multiplication techniques. More generally, to factor a polynomial of degree n over the finite field Fq with q elements, the algorithms use O(n 1.815 log q) arithmetic operations in Fq. The new “baby step/giant step ” techniques used in our algorithms also yield new fast practical algorithms at super-quadratic asymptotic running time, and subquadratic-time methods for manipulating normal bases of finite fields. 1.

We study the complexity of the circuit minimization problem: given the truth table of a Boolean function f and a parameter s, decide whether f can be realized by a Boolean circuit of size at most s. We argue why this problem is unlikely to be in P (or even in P=poly) by giving a number of surprising consequences of such an assumption. We also argue that proving this problem to be NP-complete (if it is indeed true) would imply proving strong circuit lower bounds for the class E, which appears beyond the currently known techniques. Keywords: hard Boolean functions, derandomization, natural properties, NP-completeness. 1 Introduction An n-variable Boolean function f n : f0; 1g n ! f0; 1g can be given by either its truth table of size 2 n , or a Boolean circuit whose size may be significantly smaller than 2 n . It is well known that most Boolean functions on n variables have circuit complexity at least 2 n =n [Sha49], but so far no family of sufficiently hard functions has ...

Abstract. We study the complexity of computing one or several terms (not necessarily consecutive) in a recurrence with polynomial coefficients. As applications, we improve the best currently known upper bounds for factoring integers deterministically and for computing the Cartier–Manin operator of hyperelliptic curves.

Si l’on se bornait à demander les intégrales entières, le problème n’offrirait aucune difficulté. 1 Joseph Liouville, 1833. We investigate polynomial solutions of homogeneous linear differential equations with coefficients that are polynomials with integer coefficients. The problems we consider are the existence of nonzero polynomial solutions, the determination of the dimension of the vector space of polynomial solutions, the computation of a basis of this space. Previous algorithms have a bit complexity that is at least quadratic in an integer N (that can be computed from the equation), even for merely detecting the existence of nonzero polynomial solutions. We give a deterministic algorithm that computes a compact representation of a basis of polynomial solutions in O(N log 3 N) bit operations. We also give a probabilistic algorithm that computes the dimension of the space of polynomial solutions in O ( √ N log 2 N) bit operations. In general, the integer N is not bounded polynomially in the bit size of the input differential equation. We isolate a class of equations for which detecting nonzero polynomial solutions can be performed in polynomial complexity. We discuss implementation issues and possible extensions.

this paper, two more O(n

Abstract not found

Abstract. An odd prime p is called a Wieferich prime if 2 p−1 ≡ 1 (mod p 2); alternatively, a Wilson prime if (p − 1)! ≡−1 (mod p 2). To date, the only known Wieferich primes are p = 1093 and 3511, while the only known Wilson primes are p =5,13, and 563. We report that there exist no new Wieferich primes p<4×1012, and no new Wilson primes p<5×108. It is elementary that both defining congruences above hold merely (mod p), and it is sometimes estimated on heuristic grounds that the “probability ” that p is Wieferich (independently: that p is Wilson) is about 1/p. We provide some statistical data relevant to occurrences of small values of the pertinent Fermat and Wilson quotients (mod p). Wieferich primes figure strongly in classical treatments of the first case of Fermat’s Last Theorem (“FLT(I)”). For an odd prime p not dividing xyz, Wieferich

It has long been observed that certain factorization algorithms provide a way to write product of a lot of integers succinctly. In this paper, we study the problem of representing the product of all integers from 1 to n (n!) by straight-line programs. Formally, we say that a sequence of integers a n is ultimately f(n)-computable, if there exists a nonzero integer sequence m n such that for any n, a n m n can be computed by a straight-line program (using only additions, subtractions and multiplications) of length at most f(n). Shub and Smale [12] showed that if n! is ultimately hard to compute, then algebraic version of NP P is true.

Abstract A d-gem is a {+, −, ×}-circuit having very few ×-gates and computing from {x} ∪ Z a univariate polynomial of degree d having d distinct integer roots. We introduce d-gems because they could help factoring integers and because their existence for infinitely many d would blatantly disprove a variant of the Blum-Cucker-Shub-Smale conjecture. A natural step towards validating the conjecture would thus be to rule out d-gems for large d. Here we construct d-gems for several values of d up to 55. Our 2 n-gems for n ≤ 4 are skew, that is, each {+, −}-gate adds an integer. We prove that skew 2 n-gems if they exist require n {+, −}-gates, and that these for n ≥ 5 would imply new solutions to the Prouhet-Tarry-Escott problem in number theory. By contrast, skew d-gems over the real numbers are shown to exist for every d. 1