Bundles, presemifields and nonlinear functions

• Published in: Designs, codes, and cryptography codes, and cryptography, 2008-12, Vol.49 (1-3), p.79-94
• Main Authors: Horadam, K. J., Farmer, D. G.
• Format: Article
• Language: English
• Subjects: Circuits; Coding and Information Theory; Computer Science; Cryptology; Data Structures and Information Theory; Discrete Mathematics in Computer Science; Information and Communication
• ISSN: 0925-1022; 1573-7586
• DOI: 10.1007/s10623-008-9172-z

Bundles are equivalence classes of functions derived from equivalence classes of transversals. They preserve measures of resistance to differential and linear cryptanalysis. For functions over GF (2 n ), affine bundles coincide with EA-equivalence classes. From equivalence classes (“bundles”) of presemifields of order p n , we derive bundles of functions over GF ( p n ) of the form λ ( x )* ρ ( x ), where λ, ρ are linearised permutation polynomials and * is a presemifield multiplication. We prove there are exactly p bundles of presemifields of order p 2 and give a representative of each. We compute all bundles of presemifields of orders p n ≤ 27 and in the isotopism class of GF (32) and we measure the differential uniformity of the derived λ ( x )* ρ ( x ). This technique produces functions with low differential uniformity, including PN functions ( p odd), and quadratic APN and differentially 4-uniform functions ( p  = 2).