Prime (order theory)

About: Prime (order theory) is a research topic. Over the lifetime, 15324 publications have been published within this topic receiving 157801 citations.

Algebraic methods in the theory of lower bounds for Boolean circuit complexity

Abstract: We use algebraic methods to get lower bounds for complexity of different functions based on constant depth unbounded fan-in circuits with the given set of basic operations. In particular, we prove that depth k circuits with gates NOT, OR and MODp where p is a prime require Exp(O(n1/2k)) gates to calculate MODr functions for any r ≠ pm. This statement contains as special cases Yao's PARITY result [ Ya 85 ] and Razborov's new MAJORITY result [Ra 86] (MODm gate is an oracle which outputs zero, if the number of ones is divisible by m).

Pairing-Friendly elliptic curves of prime order

Abstract: Previously known techniques to construct pairing-friendly curves of prime or near-prime order are restricted to embedding degree $k \leqslant 6 $. More general methods produce curves over ${\mathbb F}_{p}$ where the bit length of p is often twice as large as that of the order r of the subgroup with embedding degree k; the best published results achieve ρ ≡ log(p)/log(r) ~ 5/4. In this paper we make the first step towards surpassing these limitations by describing a method to construct elliptic curves of prime order and embedding degree k = 12. The new curves lead to very efficient implementation: non-pairing operations need no more than ${\mathbb F}_{p^4}$ arithmetic, and pairing values can be compressed to one third of their length in a way compatible with point reduction techniques. We also discuss the role of large CM discriminants D to minimize ρ; in particular, for embedding degree k = 2q where q is prime we show that the ability to handle log(D)/log(r) ~ (q–3)/(q–1) enables building curves with ρ ~ q/(q–1).

Derivations in prime rings

Abstract: We prove two theorems that are easily conjectured, namely: (1) In a prime ring of characteristics not 2, if the iterate of two derivations is a derivation, then one of them is zero; (2) If d is a derivation of a prime ring such that, for all elements a of the ring, ad(a) -d(a)a is central, then either the ring is commutative or d is zero. DEFINITION. A ring R is called prime if and only if xay= 0 for all aER implies x=O or y=O. From this definition it follows that no nonzero element of the centroid has nonzero kernel, so that we can divide by the prime p, unless px = 0 for all x in R, in which case we call R of characteristic p. A known result that will be often used throughout this paper is given in

Monte Carlo methods for index computation ()

Abstract: We describe some novel methods to compute the index of any integer relative to a given primitive root of a prime p. Our flrst method avoids the use of stored tables and apparently requires O(p 1/2) operations. Our second algorithm, which may be regarded as a method of catching kangaroos, is applicable when the index is known to lie in a certain interval; it requires O(w/2) operations for an interval of width w, but does not have complete certainty of success. It has several possible areas of application, including the f1actorization of integers. 1. A Rho Method for Index Computation. The concept of a random mapping of a finite set is used by Knuth [1, pp. 7-8] to explain the behavior of a type of random number generator. A sequence obtained by iterating such a function in a set of p elements is 'rho-shaped' with a tail and cycle which are random variables with expectation close to (1) /(irp/8) 0.6267 N/p, (as shown first in [2], [3]). Recently [4], we proposed that this theory be applied to recurrence relations such as (2) xi l x ? 1 (mod p), and showed how a very simple factorization method results, in which a prime factor p of a number can be found in only 0(pW2) operations. The method has been further discussed by Guy [5] and Devitt [6], who have found it suitable for use in programmable calculators. We now suggest that the same theory can be applied to sequences such as xo= 1, i1 qx1 0< x1 < jp 13~~~ (3)~SXi 2? 2i3 ()Xi+ 1 xi' (mod p) for 3' p

Prime Numbers: A Computational Perspective

Abstract: Prime numbers beckon to the beginner, the basic notion of primality being accessible to a child. Yet, some of the simplest questions about primes have stumped humankind for millennia. In this book, the authors concentrate on the computational aspects of prime numbers, such as recognizing primes and discovering the fundamental prime factors of a given number. Over 100 explicit algorithms cast in detailed pseudocode are included in the book. Applications and theoretical digressions serve to illuminate, justify, and underscore the practical power of these algorithms. The 2nd edition adds new material on primality and algorithms and updates all the numerical records, such as the largest prime, etc. It has been revised throughout.