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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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Cited by 27 (0 self)
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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...
Short Representation of Quadratic Integers
 PROCEEDINGS OF CANT
, 1992
"... Let O be a real quadratic order of discriminant \Delta. For elements ff in O we develop a compact representation whose binary length is polynomially bounded in log log H(ff), log N(ff) and log \Delta where H(ff) is the height of ff and N(ff) is the norm of ff. We show that using compact representa ..."
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Let O be a real quadratic order of discriminant \Delta. For elements ff in O we develop a compact representation whose binary length is polynomially bounded in log log H(ff), log N(ff) and log \Delta where H(ff) is the height of ff and N(ff) is the norm of ff. We show that using compact representations we can in polynomial time compute norms, signs, products, and inverses of numbers in O and principal ideals generated by numbers in O. We also show how to compare numbers given in compact represention in polynomial time.
Algorithms for Quadratic Orders
 PROCEEDINGS OF SYMPOSIUM ON MATHEMATICS OF COMPUTATION
, 1993
"... We describe deterministic algorithms for solving the following algorithmic problems in quadratic orders: Computing fundamental unit and regulator, principal ideal testing, solving prime norm equations, computing the structure of the class group, computing the order of an ideal class and determining ..."
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We describe deterministic algorithms for solving the following algorithmic problems in quadratic orders: Computing fundamental unit and regulator, principal ideal testing, solving prime norm equations, computing the structure of the class group, computing the order of an ideal class and determining discrete logarithms in the class group. We also prove upper bounds for the time and space complexity of the algorithms.
Approximating Euler products and class number computation in algebraic function fields
"... Abstract. We provide a number of results that can be used to derive approximations for the Euler product representation of the zeta function of an arbitrary algebraic function field. Three such approximations are given here. Our results have two main applications. They lead to a computationally suit ..."
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Abstract. We provide a number of results that can be used to derive approximations for the Euler product representation of the zeta function of an arbitrary algebraic function field. Three such approximations are given here. Our results have two main applications. They lead to a computationally suitable algorithm for computing the class number of an arbitrary function field. The ideas underlying the class number algorithms in turn can be used to analyze the distribution of the zeros of its zeta function. 1.
Interpolating Between Quantum and Classical Complexity Classes
, 2008
"... We reveal a natural algebraic problem whose complexity appears to interpolate between the wellknown complexity classes BQP and NP: ⋆ Decide whether a univariate polynomial with exactly m monomial terms has a padic rational root. In particular, we show that while (⋆) is doable in quantum randomized ..."
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We reveal a natural algebraic problem whose complexity appears to interpolate between the wellknown complexity classes BQP and NP: ⋆ Decide whether a univariate polynomial with exactly m monomial terms has a padic rational root. In particular, we show that while (⋆) is doable in quantum randomized polynomial time when m = 2 (and no classical randomized polynomial time algorithm is known), (⋆) is nearly NPhard for general m: Under a plausible hypothesis involving primes in arithmetic progression (implied by the Generalized Riemann Hypothesis for certain cyclotomic fields), a randomized polynomial time algorithm for (⋆) would imply the widely disbelieved inclusion NP⊆BPP. This type of quantum/classical interpolation phenomenon appears to new. 1 Introduction and Main Results Thanks to quantum computation, we now have exponential speedups for important practical problems such as Integer Factoring and Discrete Logarithm [Sho97]. However, a fundamental
A Number Theoretic Interpolation Between Quantum and Classical Complexity Classes
, 2008
"... We reveal a natural algebraic problem whose complexity appears to interpolate between the wellknown complexity classes BQP and NP: ⋆ Decide whether a univariate polynomial with exactly m monomial terms has a padic rational root. In particular, we show that while (⋆) is doable in quantum randomized ..."
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We reveal a natural algebraic problem whose complexity appears to interpolate between the wellknown complexity classes BQP and NP: ⋆ Decide whether a univariate polynomial with exactly m monomial terms has a padic rational root. In particular, we show that while (⋆) is doable in quantum randomized polynomial time when m = 2 (and no classical randomized polynomial time algorithm is known), (⋆) is nearly NPhard for general m: Under a plausible hypothesis involving primes in arithmetic progression (implied by the Generalized Riemann Hypothesis for certain cyclotomic fields), a randomized polynomial time algorithm for (⋆) would imply the widely disbelieved inclusion NP⊆BPP. This type of quantum/classical interpolation phenomenon appears to new. 1 Introduction and Main Results Thanks to quantum computation, we now have exponential speedups for important practical problems such as Integer Factoring and Discrete Logarithm [Sho97]. However, a fundamental
On the Complexity of Computing Units in a Number Field
, 2008
"... Given an algebraic number field K, such that [K: Q] is constant, we show that the problem of computing the units group O∗K is in the complexity class SPP. As a consequence, we show that principal ideal testing for an ideal in OK is in SPP. Furthermore, assuming the GRH, the class number of K, and a ..."
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Given an algebraic number field K, such that [K: Q] is constant, we show that the problem of computing the units group O∗K is in the complexity class SPP. As a consequence, we show that principal ideal testing for an ideal in OK is in SPP. Furthermore, assuming the GRH, the class number of K, and a presentation for the class group of K can also be computed in SPP. A corollary of our result is that solving PELL′S EQUATION, recently shown by Hallgren [12] to have a quantum polynomialtime algorithm, is also in SPP. 1