The weights of the orthogonals of the extended quadratic binary Goppa codes

Carlet

2014

Des. Codes Cryptogr.

Block ciphers use Substitution boxes (S-boxes) to create confusion into the cryptosystems. Functions used as S-boxes should have low differential uniformity, high nonlinearity and algebraic degree larger than 3 (preferably strictly larger). They should be fastly computable; from this viewpoint, it is better when they are in even number of variables. In addition, the functions should be bijections in a Substitution-Permutation Network. Almost perfect nonlinear (APN) functions have the lowest differential uniformity 2 and the existence of APN bijections over F 2 n for even n ≥ 8 is a big open problem. In the present paper, we focus on constructing differentially 4-uniform bijections suitable for designing S-boxes for block ciphers. Based on the idea of permuting the inverse function, we design a construction providing a large number of differentially 4-uniform bijections with maximum algebraic degree and high nonlinearity. For every even n ≥ 12, we mathematically prove that the functions in a subclass of the constructed class are CCZ-inequivalent to known differentially 4-uniform power functions and to quadratic functions. This is the first mathematical proof that an infinite class of differentially 4-uniform bijections is CCZ-inequivalent to known differentially 4-uniform power functions and to quadratic functions. We also get a general lower bound on the nonlinearity of our functions, which can be very high in some cases, and obtain three improved lower bounds on the nonlinearity for three special subcases of functions which are extremely large.

Section: Theoremmentioning

confidence: 93%

Differentially 4-uniform bijections by permuting the inverse function

Carlet

2014

Des. Codes Cryptogr.

“…https://doi.org/10.1017/S0004972708001366 [7] Codes associated with Sp(4, q) 433 THEOREM 3.2 (Lachaud and Wolfman [6]). Let q = 2 r , with r ≥ 2.…”

Section: Proof Of Main Resultsmentioning

confidence: 99%

“…J. H. Kim [6] PROOF. From Proposition 2.9 and using the Weil bound |K (χ 1 ; a)| ≤ 2 √ q(a ∈ F * q ), we see that…”

Section: Preliminariesmentioning

confidence: 99%

CODES ASSOCIATED WITH Sp(4,q) AND EVEN-POWER MOMENTS OF KLOOSTERMAN SUMS

Kim¹

2009

Bull. Aust. Math. Soc.

“…In a remarkable paper, P. Lachaud and J. Wolfman made a tremendous advance in the computation of cross-correlations [118]. Their result is based on the following simple observation.…”

Section: 5d Other Decimations Especially D = −1mentioning

confidence: 99%

Algebraic Shift Register Sequences

Goresky

Klapper

2012

Pseudo-random sequences are essential ingredients of every modern digital communication system including cellular telephones, GPS, secure internet transactions and satellite imagery. Each application requires pseudo-random sequences with specific statistical properties. This book describes the design, mathematical analysis and implementation of pseudo-random sequences, particularly those generated by shift registers and related architectures such as feedback-with-carry shift registers. The earlier chapters may be used as a textbook in an advanced undergraduate mathematics course or a graduate electrical engineering course; the more advanced chapters provide a reference work for researchers in the field. Background material from algebra, beginning with elementary group theory, is provided in an appendix.