Almost perfect nonlinear power functions with exponents expressed as fractions
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866917175681351680 |
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| author | Katz, Daniel J. O'Connor, Kathleen R. Pacheco, Kyle Sapozhnikov, Yakov |
| author_facet | Katz, Daniel J. O'Connor, Kathleen R. Pacheco, Kyle Sapozhnikov, Yakov |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_15657 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Almost perfect nonlinear power functions with exponents expressed as fractions Katz, Daniel J. O'Connor, Kathleen R. Pacheco, Kyle Sapozhnikov, Yakov Information Theory Cryptography and Security Discrete Mathematics Combinatorics Number Theory 11T71, 11T06, 94A60, 94D10 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. |
| title | Almost perfect nonlinear power functions with exponents expressed as fractions |
| topic | Information Theory Cryptography and Security Discrete Mathematics Combinatorics Number Theory 11T71, 11T06, 94A60, 94D10 |
| url | https://arxiv.org/abs/2307.15657 |