scholarly journals The Computational Efficacy of Finite Field Arithmetic

1987 ◽  
Vol 16 (227) ◽  
Author(s):  
Gudmund Skovbjerg Frandsen ◽  
Carl Sturtivant
Keyword(s):  
Computational Complexity ◽  
Finite Field ◽  
Canonical Function ◽  
General Measure ◽  
Arithmetic Complexity ◽  
Finite Field Arithmetic ◽  
Modulo P ◽  
Prime Fields ◽  
Uniform Model ◽  
Characteristic Two

<p>We show that there exists an interesting non-uniform model of computational complexity within characteristic-two finite fields. This model regards all problems as families of functions whose domain and co-domain are characteristic-two fields. The model is both a <em>structured</em> and a <em>fully</em> <em>general</em> model of computation.</p><p>We ask if the same is true when the characteristics of the fields are unbounded. We show that this is equivalent to asking whether arithmetic complexity over the prime fields is a fully general measure of complexity.</p><p>We show that this reduces to whether or not a single canonical function is ''easy'' to compute using only modulo <em>p</em> arithmetic.</p><p>We show that the arithmetic complexity of the above function is divided between two other canonical functions, the first to be computed modulo <em>p</em> and the second with modulo p^2 arithmetic.</p><p>We thus have tied the efficacy of finite field arithmetic to specific questions about the arithmetic complexities of some fundamental functions.</p>

scholarly journals The Depth Efficacy of Unbounded: Characteristic Finite Field Arithmetic

1988 ◽  
Vol 17 (240) ◽  
Author(s):  
Gudmund Skovbjerg Frandsen ◽  
Carl Sturtivant
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Field Model ◽  
Irreducible Polynomial ◽  
Canonical Function ◽  
Full Generality ◽  
Input Size ◽  
Parallel Arithmetic ◽  
Prime Fields ◽  
Bounded Characteristic

We introduce an arithmetic model of parallel computation. The basic operations are ½ and Š gates over finite fields. Functions computed are unary and increasing input size is modelled by shifting the arithmetic base to a larger field. When only finite fields of bounded characteristic are used, then the above model is fully general for parallel computations in that size and depth of optimal arithmetic solutions are polynomially related to size and depth of general (boolean) solutions. In the case of finite fields of unbounded characteristic, we prove that the existence of a fast parallel (boolean) solution to the problem of powering an integer modulo a prime (and powering a polynomial modulo an irreducible polynomial) in combination with the existence of a fast parallel (arithmetic) solution for the problem of computing a single canonical function, f<em>(x)</em>, in the prime fields, guarantees the full generality of the finite field model of computation. We prove that the function f<em>(x)</em>, has a fast parallel arithmetic solution for any ''shallow'' class of primes, i.e. primes <em>p</em> such that any prime power divisor <em>q</em> of <em>p</em> -1 is bounded in value by a polynomial in log <em>p</em>.

Introduction

2012 ◽  
Author(s):  
Nicholas M. Katz
Keyword(s):  
Finite Field ◽  
Open Questions ◽  
Prime Fields

This introductory chapter sets out the book's focus, namely equidistribution results over larger and larger finite extensions of a given finite field. Emanuel Kowalski drew attention to the interest of having equidistribution results over, for example, prime fields 𝔽p, that become better and better as p grows. This question is addressed in Chapter 28, where the problem is to make effective the estimates, already given in the equicharacteristic setting of larger and larger extensions of a given finite field. Chapter 29 points out some open questions about “the situation over ℤ” and gives some illustrative examples. The chapter concludes by pointing out two potential ambiguities of notation.

Representations by Hermitian Forms in a Finite Field of Characteristic Two

1977 ◽  
Vol 29 (1) ◽  
pp. 169-179 ◽  
Author(s):  
John D. Fulton
Keyword(s):  
Positive Integer ◽  
Finite Field ◽  
Multiplicative Group ◽  
Group Homomorphism ◽  
Hermitian Forms ◽  
Characteristic Two ◽  
Field Automorphism

Throughout this paper, we let q = 2W,﹜ w a positive integer, and for u = 1 or 2, we let GF(qu) denote the finite field of cardinality qu. Let - denote the involutory field automorphism of GF(q2) with GF(q) as fixed subfield, where ā = aQ for all a in GF﹛q2). Moreover, let | | denote the norm (multiplicative group homomorphism) mapping of GF(q2) onto GF(q), where |a| — a • ā = aQ+1.

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