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Quadratic zero-difference balanced functions, APN functions and strongly regular graphs

机译：二次零差平衡函数，APN函数和强正则图

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Let be a function from to itself and a positive integer. is called zero-difference -balanced if the equation has exactly solutions for all nonzero . As a particular case, all known quadratic planar functions are zero-difference 1-balanced; and some quadratic APN functions over are zero-difference 2-balanced. In this paper, we study the relationship between this notion and differential uniformity; we show that all quadratic zero-difference -balanced functions are differentially -uniform and we investigate in particular such functions with the form , where and where the restriction of to the set of all nonzero th powers in is an injection. We introduce new families of zero-difference -balanced functions. More interestingly, we show that the image set of such functions is a regular partial difference set, and hence yields strongly regular graphs; this generalizes the constructions of strongly regular graphs using planar functions by Weng et al. Using recently discovered quadratic APN functions on , we obtain new negative Latin square type strongly regular graphs.
机译：设为到本身的函数，为正整数。如果方程式对所有非零都有精确的解，则称为零差分平衡。在特定情况下，所有已知的二次平面函数都是零差1平衡的；以及一些二次APN函数是零差2平衡的。在本文中，我们研究了该概念与微分均匀性之间的关系。我们证明了所有二次零差平衡函数都是微分一致的，并且我们特别研究了这样的函数，其形式为，其中对所有非零次幂的集合的限制是注入。我们介绍了零差动平衡函数的新系列。更有趣的是，我们证明了这类函数的图像集是一个规则的偏差集，因此产生了很强的规则图。这概括了Weng等人使用平面函数构造的强正则图的构造。使用上最近发现的二次APN函数，我们获得了新的负拉丁方型强正则图。

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