arXiv subjects

Yakov Sapozhnikov

Publications and source records attributed to Yakov Sapozhnikov.

3 recordsLinked to original sources

arXiv2026

Almost perfect nonlinear power functions with exponents expressed as fractions

Let $F$ be a finite field, let $f$ be a function from $F$ to $F$, and let $a$ be a nonzero element of $F$. The discrete derivative of $f$ in direction $a$ is $Δ_a f \colon F \to F$ with $(Δ_a f)(x)=f(x+a)-f(x)$. The differential spectrum of $f$ is the multiset of cardinalities of all the fibers of all the derivatives $Δ_a f$ as $a$ runs through $F^*$. An almost perfect nonlinear (APN) function is one for which the largest cardinality in its differential spectrum is $2$. Almost perfect nonlinear functions are of interest as cryptographic primitives. If $d$ is a positive integer, then the power function over $F$ with exponent $d$ is the function $f \colon F \to F$ with $f(x)=x^d$ for every $x \in F$. There is a small number of known infinite families of APN power functions. In this paper, we re-express the exponents for one such family in a more convenient form. This enables us not only to obtain the differential spectrum of each power function $f$ with an exponent in our family, but also to determine the elements that lie in an arbitrary fiber of the discrete derivative of $f$. This differential analysis, which is far more detailed than previous results, is achieved by composing the discrete derivative of $f$ with some permutations and a double covering of its domain to obtain a function whose fibers can more readily be analyzed.

cs.IT↗

arXiv2024

Book crossing numbers of the complete graph and small local convex crossing numbers

A $ k $-page book drawing of a graph $ G $ is a drawing of $ G $ on $ k $ halfplanes with common boundary $ l $, a line, where the vertices are on $ l $ and the edges cannot cross $ l $. The $ k $-page book crossing number of the graph $ G $, denoted by $ ν_k(G) $, is the minimum number of edge-crossings over all $ k $-page book drawings of $ G $. Let $G=K_n$ be the complete graph on $n$ vertices. We improve the lower bounds on $ ν_k(K_n) $ for all $ k\geq 14 $ and determine $ ν_k(K_n) $ whenever $ 2 < n/k \leq 3 $. Our proofs rely on bounding the number of edges in convex graphs with small local crossing numbers. In particular, we determine the maximum number of edges that a convex graph with local crossing number at most $ \ell $ can have for $ \ell\leq 4 $.

math.CO↗

arXiv2017

The $p$-Adic Valuations of Weil Sums of Binomials

We investigate the $p$-adic valuation of Weil sums of the form $W_{F,d}(a)=\sum_{x \in F} ψ(x^d -a x)$, where $F$ is a finite field of characteristic $p$, $ψ$ is the canonical additive character of $F$, the exponent $d$ is relatively prime to $|F^\times|$, and $a$ is an element of $F$. Such sums often arise in arithmetical calculations and also have applications in information theory. For each $F$ and $d$ one would like to know $V_{F,d}$, the minimum $p$-adic valuation of $W_{F,d}(a)$ as $a$ runs through the elements of $F$. We exclude exponents $d$ that are congruent to a power of $p$ modulo $|F^\times|$ (degenerate $d$), which yield trivial Weil sums. We prove that $V_{F,d} \leq (2/3)[F\colon{\mathbb F}_p]$ for any $F$ and any nondegenerate $d$, and prove that this bound is actually reached in infinitely many fields $F$. We also prove some stronger bounds that apply when $[F\colon{\mathbb F}_p]$ is a power of $2$ or when $d$ is not congruent to $1$ modulo $p-1$, and show that each of these bounds is reached for infinitely many $F$.

math.NT↗