Differential Properties of ${x\mapsto x^{2^{t}-1}}$
Céline Blondeau, Anne Canteaut, Pascale Charpin · IEEE Transactions on Information Theory · 2011
We provide an extensive study of the differential properties of the functionsx→x2t-1 over \BBF2n, for 1tn. We notably show that the differential spectra of these functions are determined by the number of roots of the linear polynomialsx2t+bx2+(b+1)xwherebvaries in \BBF2n. We prove a strong relationship between the differential spectra ofx→x2t-1 andx→x2s-1 fors=n-t+1. As a direct consequence, this result enlightens a connection between the differential properties of the cube function and of the inverse function. We also determine the complete differential spectra ofx→x7by means of the value of some Kloosterman sums, and ofx→x2t-1 fort∈ {[n/2], [n/2]+1,n-2}.