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首页> 外文期刊>Advances in mathematics of communications >SOME NEW RESULTS ON THE CONJECTURE ON EXCEPTIONAL APN FUNCTIONS AND ABSOLUTELY IRREDUCIBLE POLYNOMIALS: THE GOLD CASE
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SOME NEW RESULTS ON THE CONJECTURE ON EXCEPTIONAL APN FUNCTIONS AND ABSOLUTELY IRREDUCIBLE POLYNOMIALS: THE GOLD CASE

机译：关于特殊APN功能的猜想和绝对不可缩短多项式的一些新结果：金案

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摘要

An almost perfect nonlinear (APN) function f : F-2n -> F-2n, (necessarily polynomial) is called exceptional APN if it is APN on infinitely many extensions of F-2n. Aubry, McGuire and Rodier conjectured that the only exceptional APN functions are the Gold and the Kasami-Welch monomial functions. They established that a polynomial function of odd degree is not exceptional APN provided the degree is not a Gold number (2(k) + 1) or a Kasami-Welch number (2(2k) - 2(k) + 1). When the degree of the polynomial function is a Gold number or a Kasami-Welch number, several partial results have been obtained by several authors including us. In this article we address these exceptions. We almost prove the exceptional APN conjecture in the Gold degree case when deg (h(x)) is odd. We also show exactly when the corresponding multivariate polynomial phi(x, y, z) is absolutely irreducible. Also, there is only one result known when f (x) = x2(k) +1 vertical bar h(x), and deg(h(x)) is even. Here, we extend this result as well, thus making progress in this case that seems more difficult.
机译：如果IT在F-2N的无限许多延伸部分上，则几乎完美的非线性（APN）函数F：F-2N - > F-2N（必然多项式）被称为异常APN。 Aubry，McGuire和Rodier推测，唯一的卓越APN功能是黄金和Kasami-Welch单体功能。他们确定奇数程度的多项式函数不是特殊的APN，所以提供程度不是金数（2（k）+ 1）或Kasami-welch数（2（2k） - 2（k）+ 1）。当多项式函数的程度为金数或Kasami-Welch数量时，几个作者都获得了几个部分结果，包括我们。在本文中，我们解决了这些例外。当DEG（H（x））是奇数时，我们几乎证明了金度案例中的特殊APN猜想。我们还恰好显示了相应的多变量多项式PHI（x，y，z）绝对不可缩短。此外，当F（x）= x2（k）+1垂直条H（x）和deg（h（x））是偶数时，只有一个结果。在这里，我们也扩展了这个结果，从而在这种情况下取​​得进展似乎更加困难。

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