this paper contains a list of 36 open problems in numbertheoretic complexity. We expect that none of these problems are easy; we are sure that many of them are hard. This list of problems reflects our own interests and should not be viewed as definitive. As the field changes and becomes deeper, new problems will emerge and old problems will lose favor. Ideally there will be other `open problems' papers in future ANTS proceedings to help guide the field. It is likely that some of the problems presented here will remain open for the forseeable future. However, it is possible in some cases to make progress by solving subproblems, or by establishing reductions between problems, or by settling problems under the assumption of one or more well known hypotheses (e.g. the various extended Riemann hypotheses, NP 6= P; NP 6= coNP). For the sake of clarity we have often chosen to state a specific version of a problem rather than a general one. For example, questions about the integers modulo a prime often have natural generalizations to arbitrary finite fields, to arbitrary cyclic groups, or to problems with a composite modulus. Questions about the integers often have natural generalizations to the ring of integers in an algebraic number field, and questions about elliptic curves often generalize to arbitrary curves or abelian varieties. The problems presented here arose from many different places and times. To those whose research has generated these problems or has contributed to our present understanding of them but to whom inadequate acknowledgement is given here, we apologize. Our list of open problems is derived from an earlier `open problems' paper we wrote in 1986 [AM86]. When we wrote the first version of this paper, we feared that the problems presented were so difficult...

We present an unconditional deterministic polynomial-time algorithm that determines whether an input number is prime or composite. 1

Deutsch proposed two sorts of models of quantum computers, quantum Turing machines (QTMs) and quantum circuit families (QCFs). At present quantum algorithms are represented by these two models. This paper shows the equivalence of the computational powers of these two models. For this purpose, we introduce two notions of uniformity for QCFs and complexity classes based on uniform QCFs. For Monte Carlo algorithms, it is proved that the complexity classes based on uniform QCFs are identical with the corresponding classes based on QTMs. For Las Vegas algorithms, various complexity classes are introduced for QTMs and QCFs according to constraints on the algorithms and their interrelations are investigated in detail. In addition, we generalize Yao’s construction of quantum circuits simulating single tape QTMs to multi-tape QTMs and give a complete proof of the existence of a universal QTM simulating multi-tape QTMs efficiently.

, 44 equivalent, 48 admissible, 19 associated, 48 binary addition chain, 45 binary method, 43, 63 Carmichael function, 4 Carmichael number, 16, 29 Chinese Remainder Theorem, 5 complex extension, 3 conjugate, 3 CRT, 5 Dickson polynomials, 11 doubling step, 63 dual, 48 Fermat test, 15, 16 graph reduced, 48 group of units, 3 in-degree, 45 Jacobi symbol, 6 Legendre symbol, 5 Lucas chain, 62 composite, 63 degenerate, 63 simple, 63 Lucas sequence, 8 Mathematica, 23, 41 Miller-Rabin test, 18 norm, 3 order of a group element, 7 out-degree, 45 Pocklington, 25 probable prime, 15 pseudo-primality, 2 BIBLIOGRAPHY 85 [R'ed48] L. R'edei. Uber eindeutig umkehrbare Polynome in endlichen Korpern. Acta Sci. Math., 11:71--76, 1946--48. [Rie85] H. Riesel. Prime Numbers and Computer Methods for Factorization. Birkhauser, 1985. [RLS + 93] R. A. Rueppel, A. K. Lenstra, M. E. Smid, K. S. McCurley, Y. Desmedt, A. Odlyzko, and P. Landrock. Panel

We present algorithms that are deterministic primality tests for a large family of integers, namely, integers n ≡ 1 (mod 4) for which an integer a is given such that the Jacobi symbol ( a) = −1, and n integers n ≡ −1 (mod 4) for which an integer a is given such that ( a 1−a) = ( ) = −1. The algorithms n n we present run in 2 − min(k,[2 log log n]) Õ(log n) 6 time, where k = ν2(n − 1) is the exact power of 2 dividing n − 1 when n ≡ 1 (mod 4) and k = ν2(n + 1) if n ≡ −1 (mod 4). The complexity of our algorithms improves up to Õ(log n)4 when k ≥ [2 log log n]. We also give tests for more general family of numbers and study their complexity.

F13.54> k = 100, the algorithm answers correctly with an overwhelming probability: 1 \Gamma 2 \Gamma100 . Also observe that the running time is O(kn 3 ), since for each of the k a's, we compute each of the at most n u i 's by a simple squaring. The idea behind the proof of claim 1 is to distinguish two special cases: 1. N is a prime power, N = p b , b 2 and 2-1 2-2 Lecture 2 : March 8, 1995 2. N has at least two distinct prime factors. Proof: (In case 1.) Here, we are almost home free, since this case won't pass the criterion in Fermat's little theorem, i.e. for most a we have a N \Gamma1 6j 1 (mod N ). Assume N = p b

We begin with a (very brief) review of P and NP. Pointers are given to the appropriate sections of Motwani & Raghavan, denoted M&R, where appropriate: for complexity classes, one can consult Section 1.5.2 of the book. A nice complexity resource is on the web at

Abstract not found