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SUMMARY:Niho's last conjecture (Daniel Katz\, Cal State Northridge)
DESCRIPTION:A power permutation of a finite field F is a permutation of F whose functional form is x -> x^d for some exponent d.  Power permutations are used in cryptography\, and the exponent d must be chosen so that the permutation is highly nonlinear\, that is\, not easily approximated by linear functions.  The Walsh spectrum of a power permutation is a list of numbers measuring the correlation of our power permutation with the various linear functions. The last conjecture in Niho’s 1972 thesis considers a particular infinite family of highly nonlinear power permutations\, and states that each permutation in this family has a Walsh spectrum with at most five distinct values. Niho’s own techniques show that there are at most eight distinct values. Each of the eight candidate values corresponds to a possible number of distinct roots of a seventh degree polynomial on a subset of the finite field F called the unit circle. We use symmetry arguments to show that it is impossible to have four\, six\, or seven roots on the unit circle: this proves Niho’s last conjecture. This is joint work with Tor Helleseth and Chunlei Li.
URL:https://colleges.claremont.edu/ccms/event/antc-talk-daniel-katz-cal-state-northridge/
LOCATION:On Zoom
CATEGORIES:Algebra / Number Theory / Combinatorics Seminar
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