Differential spectra of power permutations (Daniel Katz, CSUN)

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.