Double-Error-Correcting Cyclic Codes and Absolutely Irreducible Polynomials over GF(2)

We consider exceptional APN functions on F 2 m , which by definition are functions that are APN on infinitely many extensions of F 2 m . Our main result is that polynomial functions of odd degree are not exceptional, provided the degree is not a Gold number (2 k + 1) or a Kasami-Welch number (4 k − 2 k + 1). We also have partial results on functions of even degree, and functions that have degree 2 k + 1.

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“…One counterexample is with g(x) = x 5 and k ≥ 4 and even. Remark: It is known that φ j is irreducible in the following cases (see [4]):…”

Section: ⊔ ⊓mentioning

confidence: 99%

A few more functions that are not APN infinitely often

Aubry¹,

McGuire²,

Rodier³

2010

Finite Fields: Theory and Applications

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“…Indeed, absolute irreducibility is a necessary condition in the applications of the bounds of Weil, Bombieri, Deligne, Lang-Weil, Ghorpade-Lachaud, and others that estimate the number of rational points on the corresponding varieties, or give bounds on exponential sums along curves. Except for the well known Eisenstein criterion (applicable in very restrictive cases), only few scattered results were known for proving absolute irreducibility (see Schmidt [19]-mostly applicable to Kummer and Artin-Schreier type of extensions), before the major breakthrough in [14]. Therefore, our techniques and results are of independent interest.…”

Section: Introductionmentioning

confidence: 93%

“…The best known examples of APN functions are the Gold function f (x) = x 2 k +1 , and the Kasami-Welch function f (x) = x 2 2k −2 k +1 ; they are APN on any field F 2 n when k and n are relatively prime. The Welch function f (x) = x 2 r +3 is also APN on F 2 n when n = 2r + 1 (see [14] and [15]). …”

Section: Introductionmentioning

confidence: 99%

“…Up until now, the main tool used by most researchers in the study of exceptional APN functions, has been the method of [14] to prove the absolute irreducibility of multivariate polynomials. The algorithmic approach in [14] is based on intersection multiplicity theory and Bezout's theorem, and computations initiated in [12].…”

Section: Introductionmentioning

confidence: 99%

“…The algorithmic approach in [14] is based on intersection multiplicity theory and Bezout's theorem, and computations initiated in [12].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On the conjecture on APN functions and absolute irreducibility of polynomials

Delgado

Janwa

2016

Des. Codes Cryptogr.

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An almost perfect nonlinear (APN) function (necessarily a polynomial function) on a finite field F is called exceptional APN, if it is also APN on infinitely many extensions of F. In this article we consider the most studied case of F = F 2 n . A conjecture of Janwa-Wilson and McGuire-Janwa- Wilson (1993Wilson ( /1996, settled in 2011, was that the only monomial exceptional APN functions are the monomials x n , where n = 2 k + 1 or n = 2 2k − 2 k + 1 (the Gold or the Kasami exponents, respectively). A subsequent conjecture states that any exceptional APN function is one of the monomials just described. One of our results is that all functions of the form f (x) = x 2 k +1 + h(x) (for any odd degree h(x), with a mild condition in few cases), are not exceptional APN, extending substantially several recent results towards the resolution of the stated conjecture. We also show absolute irreducibility of a class of multivariate polynomials over finite fields (by repeated hyperplane sections, linear transformations, and reductions) and discuss their applications.

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Absolute Irreducibility of Polynomials via Newton Polytopes

Gao

2001

Journal of Algebra

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A multivariable polynomial is associated with a polytope, called its Newton polytope. A polynomial is absolutely irreducible if its Newton polytope is indecomposable in the sense of Minkowski sum of polytopes. Two general constructions of indecomposable polytopes are given, and they give many simple irreducibility criteria including the well-known Eisenstein criterion. Polynomials from these criteria are over any field and have the property of remaining absolutely irreducible when their coefficients are modified arbitrarily in the field, but keeping a certain collection of them nonzero. ᮊ

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Double-Error-Correcting Cyclic Codes and Absolutely Irreducible Polynomials over GF(2)

Cited by 49 publications

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A few more functions that are not APN infinitely often

A few more functions that are not APN infinitely often

On the conjecture on APN functions and absolute irreducibility of polynomials

Absolute Irreducibility of Polynomials via Newton Polytopes

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