Differential Properties of {$x\mapsto x^{2^{t}-1}$}

Céline Blondeau (1) , Anne Canteaut (1) , Pascale Charpin (1)

1 SECRET - Security, Cryptology and Transmissions

We provide an extensive study of the differential properties of the functions x→ x2t-1 over BBF 2n, for 1 <; t <; n. We notably show that the differential spectra of these functions are determined by the number of roots of the linear polynomials x2t+bx2+(b+1)x where b varies in BBF 2n. We prove a strong relationship between the differential spectra of x→ x2t-1 and x→ x2s-1 for s = 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 of x → x7 by means of the value of some Kloosterman sums, and of x → x2t-1 for t ∈ {[n/2], [n /2]+1, n-2}.

Domaines: Théorie de l'information et codage [math.IT]; Théorie de l'information [cs.IT]

Céline Blondeau, Anne Canteaut, Pascale Charpin. Differential Properties of {$x\mapsto x^{2^{t}-1}$}. IEEE Transactions on Information Theory, 2011, 57 (12), pp.8127 - 8137. ⟨10.1109/TIT.2011.2169129⟩. ⟨hal-00610099⟩