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>.

scholarly journals Constructing irreducible polynomials with prescribed level curves over finite fields

2001 ◽  
Vol 27 (4) ◽  
pp. 197-200
Author(s):  
Mihai Caragiu
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Irreducible Polynomial ◽  
Irreducible Polynomials ◽  
Level Curves ◽  
Curves Over Finite Fields ◽  
Irreducibility Criterion ◽  
Absolutely Irreducible Polynomial

We use Eisenstein's irreducibility criterion to prove that there exists an absolutely irreducible polynomialP(X,Y)∈GF(q)[X,Y]with coefficients in the finite fieldGF(q)withqelements, with prescribed level curvesXc:={(x,y)∈GF(q)2|P(x,y)=c}.

The Sato-Tate distribution in thin families of elliptic curves over high degree extensions of finite fields

2019 ◽  
Vol 15 (03) ◽  
pp. 469-477
Author(s):  
Igor E. Shparlinski
Keyword(s):  
Finite Field ◽  
Elliptic Curves ◽  
Finite Fields ◽  
Function Field ◽  
Prime Fields ◽  
High Degree

Over the last two decades, there has been a wave of activity establishing the Sato-Tate kind of distribution in various families of elliptic curves over prime fields. Typically the goal here is to prove this for families which are as thin as possible. We consider a function field analogue of this question, that is, for high degree extensions of a finite field where new effects allow us to study families, which are much thinner that those typically investigated over prime fields.

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>

Isometry groups of additive codes over finite fields

2018 ◽  
Vol 17 (10) ◽  
pp. 1850198 ◽  
Author(s):  
Jay A. Wood
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Linear Code ◽  
Necessary Conditions ◽  
Isometry Groups ◽  
Additive Codes ◽  
Prime Fields

When [Formula: see text] is a linear code over a finite field [Formula: see text], every linear Hamming isometry of [Formula: see text] to itself is the restriction of a linear Hamming isometry of [Formula: see text] to itself, i.e. a monomial transformation. This is no longer the case for additive codes over non-prime fields. Every monomial transformation mapping [Formula: see text] to itself is an additive Hamming isometry, but there may exist additive Hamming isometries that are not monomial transformations.The monomial transformations mapping [Formula: see text] to itself form a group [Formula: see text], and the additive Hamming isometries form a larger group [Formula: see text]: [Formula: see text]. The main result says that these two subgroups can be as different as possible: for any two subgroups [Formula: see text], subject to some natural necessary conditions, there exists an additive code [Formula: see text] such that [Formula: see text] and [Formula: see text].

Construction of irreducible polynomials over finite fields

2021 ◽  
Author(s):  
P. L. Sharma ◽  
Ashima
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Irreducible Polynomial ◽  
Irreducible Polynomials ◽  
Polynomials Over Finite Fields

Irreducible polynomials over finite fields and their applications have been quite well studied. Here, we discuss the construction of the irreducible polynomials of degree [Formula: see text] over the finite field [Formula: see text] for a given irreducible polynomial of degree [Formula: see text]. Furthermore, we construct the irreducible polynomials of degree [Formula: see text] over the finite field [Formula: see text] for a given irreducible polynomial of degree [Formula: see text] by using the method of composition of polynomials with some conditions on coefficients and degree of a given irreducible polynomial.

Maximal Sets of Pairwise Orthogonal Vectors in Finite Fields

2012 ◽  
Vol 55 (2) ◽  
pp. 418-423 ◽  
Author(s):  
Le Anh Vinh
Keyword(s):  
Positive Integer ◽  
Finite Field ◽  
Finite Fields ◽  
Bilinear Form ◽  
Symmetric Bilinear Form

AbstractGiven a positive integern, a finite fieldofqelements (qodd), and a non-degenerate symmetric bilinear formBon, we determine the largest possible cardinality of pairwiseB-orthogonal subsets, that is, for any two vectorsx,y∈ Ε, one hasB(x,y) = 0.

scholarly journals Construction of elliptic curves with cyclic groups over prime fields

2006 ◽  
Vol 73 (2) ◽  
pp. 245-254 ◽  
Author(s):  
Naoya Nakazawa
Keyword(s):  
Elliptic Curves ◽  
Finite Fields ◽  
Function Field ◽  
Modular Group ◽  
Modular Function ◽  
Invariant Function ◽  
Rational Points ◽  
Modular Invariant ◽  
Cyclic Groups ◽  
Prime Fields

The purpose of this article is to construct families of elliptic curves E over finite fields F so that the groups of F-rational points of E are cyclic, by using a representation of the modular invariant function by a generator of a modular function field associated with the modular group Γ0(N), where N = 5, 7 or 13.

scholarly journals Short Kloosterman Sums for Polynomials over Finite Fields

2003 ◽  
Vol 55 (2) ◽  
pp. 225-246 ◽  
Author(s):  
William D. Banks ◽  
Asma Harcharras ◽  
Igor E. Shparlinski
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Small Degree ◽  
Kloosterman Sums ◽  
Degree Relative ◽  
Irreducible Polynomials ◽  
Arbitrary Polynomial ◽  
Polynomials Over Finite Fields ◽  
Titchmarsh Theorem

AbstractWe extend to the setting of polynomials over a finite field certain estimates for short Kloosterman sums originally due to Karatsuba. Our estimates are then used to establish some uniformity of distribution results in the ring [x]/M(x) for collections of polynomials either of the form f−1g−1 or of the form f−1g−1 + afg, where f and g are polynomials coprime to M and of very small degree relative to M, and a is an arbitrary polynomial. We also give estimates for short Kloosterman sums where the summation runs over products of two irreducible polynomials of small degree. It is likely that this result can be used to give an improvement of the Brun-Titchmarsh theorem for polynomials over finite fields.

scholarly journals Monomial Generalized Almost Perfect Nonlinear Functions

2020 ◽  
Vol 31 (03) ◽  
pp. 411-419
Author(s):  
Masamichi Kuroda
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Algebraic Degree ◽  
Apn Functions ◽  
Nonlinear Functions ◽  
Almost Perfect Nonlinear ◽  
Basic Properties ◽  
Almost Perfect Nonlinear Functions

Generalized almost perfect nonlinear (GAPN) functions were defined to satisfy some generalizations of basic properties of almost perfect nonlinear (APN) functions for even characteristic. In particular, on finite fields of even characteristic, GAPN functions coincide with APN functions. In this paper, we study monomial GAPN functions for odd characteristic. We give monomial GAPN functions whose algebraic degree are maximum or minimum on a finite field of odd characteristic. Moreover, we define a generalization of exceptional APN functions and give typical examples.

scholarly journals Restriction Operators Acting on Radial Functions on Vector Spaces over Finite Fields

2014 ◽  
Vol 57 (4) ◽  
pp. 834-844
Author(s):  
Doowon Koh
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Algebraic Varieties ◽  
Test Functions ◽  
Radial Functions ◽  
Zero Radius ◽  
Intersection Points ◽  
Field Setting

AbstractWe study Lp → Lr restriction estimates for algebraic varieties V in the case when restriction operators act on radial functions in the finite field setting. We show that if the varieties V lie in odd dimensional vector spaces over finite fields, then the conjectured restriction estimates are possible for all radial test functions. In addition, assuming that the varieties V are defined in even dimensional spaces and have few intersection points with the sphere of zero radius, we also obtain the conjectured exponents for all radial test functions.

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