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DTSTART;TZID=America/Los_Angeles:20191105T121500
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DTSTAMP:20260519T164505
CREATED:20190910T234841Z
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UID:1522-1572956100-1572959400@colleges.claremont.edu
SUMMARY:Differential spectra of power permutations (Daniel Katz\, CSUN)
DESCRIPTION:If $F$ is a finite field and $d$ is a positive integer relatively prime to $|F^\times|$\, then the power map $x \mapsto x^d$ is a permutation of $F$\, and so is called a power permutation of $F$. For any function $f: F \to F$\, and $a\, b \in F$\, we define the differential multiplicity of $f$ with respect to $a$ and $b$\, written $\delta_f(a\,b)$\, to be the number of pairs $(x\,y) \in F^2$ with $x-y=a$ and $f(x)-f(y)=b$.  We usually insist that $a\not=0$\, since it is immediate that $\delta_f(0\,0)=|F|$ and $\delta_f(0\,b)=0$ for $b\not=0$.  The differential spectrum of $f$\, written $\Delta_f$\, is defined as $\Delta_f=\{\delta_f(a\,b): a \in F^\times\, b \in F\}$. Differential spectra of power permutations are of interest in applications to cryptography and digital communications.  We are especially interested in fields $F$ and exponents $d$ such $f(x)=x^d$ is a power permutation over $F$ whose differential spectrum contains at most three values. We present computational experiments that suggest conjectures as to which $(F\,d)$ pairs produce such spectra.  This is joint work with Kyle Pacheco and Yakov Sapozhnikov.
URL:https://colleges.claremont.edu/ccms/event/antc-talk-daniel-katz-csun-2/
LOCATION:Emmy Noether Room\, Millikan 1021\, Pomona College\, 610 N. College Ave.\, Claremont\, California\, 91711
CATEGORIES:Algebra / Number Theory / Combinatorics Seminar
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