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Open Problems in Number Theoretic Complexity, II
"... 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 ..."
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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...
Factoring into Coprimes in Essentially Linear Time
"... . Let S be a nite set of positive integers. A \coprime base for S" means a set P of positive integers such that (1) each element of P is coprime to every other element of P and (2) each element of S is a product of powers of elements of P . There is a natural coprime base for S. This paper introduc ..."
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. Let S be a nite set of positive integers. A \coprime base for S" means a set P of positive integers such that (1) each element of P is coprime to every other element of P and (2) each element of S is a product of powers of elements of P . There is a natural coprime base for S. This paper introduces an algorithm that computes the natural coprime base for S in essentially linear time. The best previous result was a quadratictime algorithm of Bach, Driscoll, and Shallit. This paper also shows how to factor S into elements of P in essentially linear time. The algorithms apply to any free commutative monoid with fast algorithms for multiplication, division, and greatest common divisors; e.g., monic polynomials over a eld. They can be used as a substitute for prime factorization in many applications. 1.
Are `Strong' Primes Needed for RSA?
 In The 1997 RSA Laboratories Seminar Series, Seminars Proceedings
, 1999
"... We review the arguments in favor of using socalled "strong primes" in the RSA publickey cryptosystem. There are two types of such arguments: those that say that strong primes are needed to protect against factoring attacks, and those that say that strong primes are needed to protect against "cy ..."
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We review the arguments in favor of using socalled "strong primes" in the RSA publickey cryptosystem. There are two types of such arguments: those that say that strong primes are needed to protect against factoring attacks, and those that say that strong primes are needed to protect against "cycling" attacks (based on repeated encryption).
The complexity of generating functions for integer points in polyhedra and beyond
 In Proceedings of the International Congress of Mathematicians
, 2006
"... Abstract. Motivated by the formula for the sum of the geometric series, we consider various classes of sets S ⊂ Z d of integer points for which an a priori “long ” Laurent series or polynomial � m∈S xm can be written as a “short ” rational function f(S; x). Examples include the sets of integer point ..."
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Abstract. Motivated by the formula for the sum of the geometric series, we consider various classes of sets S ⊂ Z d of integer points for which an a priori “long ” Laurent series or polynomial � m∈S xm can be written as a “short ” rational function f(S; x). Examples include the sets of integer points in rational polyhedra, integer semigroups, and Hilbert bases of rational cones, among others. We discuss applications to efficient counting and optimization and open questions.
Lattice points, polyhedra, and complexity
 Park City Math Institute Lecture Notes
"... The central topic of these lectures is efficient counting of integer points in polyhedra. ..."
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The central topic of these lectures is efficient counting of integer points in polyhedra.