Integer Complexity Generalizations in Various Rings (original) (raw)

On the reduction in multiplicative complexity achieved by the polynomial residue number system

IEEE Transactions on Signal Processing, 1992

The polynomial residue number system (PRNS) is known to reduce the complexity of polynomial m iltiplication from O(N2) to O (N). A new interpretation of this complexity reduction is given in the context of associative algt,bras over a finite field. The new point of view provides a clearer understanding of the Chinese remainder theorem.

On algorithms to calculate integer complexity

arXiv (Cornell University), 2017

We consider a problem first proposed by Mahler and Popken in 1953 and later developed by Coppersmith, Erdős, Guy, Isbell, Selfridge, and others. Let f (n) be the complexity of n ∈ Z + , where f (n) is defined as the least number of 1's needed to represent n in conjunction with an arbitrary number of +'s, * 's, and parentheses. Several algorithms have been developed to calculate the complexity of all integers up to n. Currently, the fastest known algorithm runs in time O(n 1.230175 ) and was given by J. Arias de Reyna and J. van de Lune in 2014. This algorithm makes use of a recursive definition given by Guy and iterates through products, f (d) + f n d , for d | n, and sums, f (a) + f (na), for a up to some function of n. The rate-limiting factor is iterating through the sums. We discuss potential improvements to this algorithm via a method that provides a strong uniform bound on the number of summands that must be calculated for almost all n. We also develop code to run J. Arias de Reyna and J. van de Lune's analysis in higher bases and thus reduce their runtime of O(n 1.230175 ) to O(n 1.222911236 ). All of our code can be found online at: .

Some Complexity Results for Polynomial Ideals

Journal of Complexity, 1997

In this paper, we survey some of our new results on the complexity of a number of problems related to polynomial ideals. We consider multivariate polynomials over some ring, like the integers or the rationals. For instance, a polynomial ideal membership problem is a (w + 1)-tuple P = ( f, g 1 , g 2 , . . . , g w ) where f and the g i are multivariate polynomials, and the problem is to determine whether f is in the ideal generated by the g i . For polynomials over the integers or rationals, this problem is known to be exponential space complete. We discuss further complexity results for problems related to polynomial ideals, like the word and subword problems for commutative semigroups, a quantitative version of Hilbert's Nullstellensatz in a complexity theoretic version, and problems concerning the computation of reduced polynomials and Gröbner bases. © 1997 Academic Press

On the linear complexity of bounded integer sequences over different moduli

Information Processing Letters, 2005

We give a relation between the linear complexity over the integers and over the residue rings modulo m of a bounded integer sequence. This relation can be used to obtain a variety of new results for several sequences widely studied in the literature. In particular we apply it to Sidelnikov sequences.

Lower bounds on the linear complexity of the discrete logarithm in finite fields

IEEE Transactions on Information Theory, 2001

Let be a prime, a positive integer, = , and a divisor of ( 1). We derive lower bounds on the linear complexity over the residue class ring of a ( -periodic) sequence representing the residues modulo of the discrete logarithm in . Moreover, we investigate a sequence over representing the values of a certain polynomial over introduced by Mullen and White which can be identified with the discrete logarithm in via -adic expansions and representations of the elements of with respect to some fixed basis.

Integer Complexity: Experimental and Analytical Results II

Lecture Notes in Computer Science, 2015

We consider representing of natural numbers by expressions using 1's, addition, multiplication and parentheses. n denotes the minimum number of 1's in the expressions representing n. The logarithmic complexity n log is defined as n /log 3 n. The values of n log are located in the segment [3, 4.755], but almost nothing is known with certainty about the structure of this "spectrum" (are the values dense somewhere in the segment etc.). We establish a connection between this problem and another difficult problem: the seemingly "almost random" behaviour of digits in the base 3 representations of the numbers 2 n. We consider also representing of natural numbers by expressions that include subtraction, and the so-called P-algorithms-a family of "deterministic" algorithms for building representations of numbers.

Complexity Degrees of Algebraic Structures

2014

We aim at studying collections of algebraic structures defined over a commutative ring and investigating the complexity of significant constructions carried out on these objects. The assignment of measures of size, via a multiplicity theory, to the algebras and to the construction itself is a novel aspect to the subject.

On possible growths of arithmetical complexity

RAIRO - Theoretical Informatics and Applications, 2006

The arithmetical complexity of infinite words, defined by Avgustinovich, FonDer -Flaass and the author in 2000, is the number of words of length n which occur in the arithmetical subsequences of the infinite word. This is one of the modifications of the classical function of subword complexity, which is equal to the number of factors of the infinite word of length n. In this paper, we show that the orders of growth of the arithmetical complexity can behave as many sub-polynomial functions. More precisely, for each sequence u of subword complexity fu(n) and for each prime p ≥ 3 we build a Toeplitz word on the same alphabet whose arithmetical complexity is a(n) = Θ(nfu(log p n)).

Complexity of Ring Morphism Problems

Computational Complexity, 2006

We study the complexity of the isomorphism and automorphism problems for finite rings. We show that both integer factorization and graph isomorphism reduce to the problem of counting automorphisms of a ring. This counting problem is shown to be in the functional version of the complexity class AM ∩ coAM and hence is not NP-complete unless the polynomial hierarchy collapses. As a “positive” result we show that deciding whether a given ring has a non-trivial automorphism can be done in deterministic polynomial time. Finding such an automorphism is, however, shown to be randomly equivalent to integer factorization.