By Lilya Budaghyan
This booklet covers novel study on building and research of optimum cryptographic capabilities similar to virtually excellent nonlinear (APN), virtually bent (AB), planar and bent services. those features have optimum resistance to linear and/or differential assaults, that are the 2 strongest assaults on symmetric cryptosystems. in addition to cryptographic functions, those features are major in lots of branches of arithmetic and data idea together with coding conception, combinatorics, commutative algebra, finite geometry, series layout and quantum details concept. the writer analyzes equivalence kinfolk for those services and develops a number of new tools for development in their countless households. moreover, the publication bargains options to 2 longstanding open difficulties, together with the matter on characterization of APN and AB capabilities through Boolean, and the matter at the relation among periods of bent functions.
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Extra resources for Construction and Analysis of Cryptographic Functions
CCZ-equivalence of single and multi output Boolean functions. , AMS, v. 518, pp. 43–54, 2010. 21. L. Budaghyan and T. Helleseth. New perfect nonlinear multinomials over Fp2k for any odd prime p. Proceedings of the International Conference on Sequences and Their Applications SETA 2008, Lecture Notes in Computer Science 5203, pp. 403–414, Lexington, USA, Sep. 2008. 22. L. Budaghyan and T. Helleseth. On Isotopisms of Commutative Presemifields and CCZ-Equivalence of Functions. Special Issue on Cryptography of International Journal of Foundations of Computer Science, v.
Then, since L is bijective, we have l(b) = 1 (otherwise, the vector (b, 0) would belong to the kernel of L). By Lemma 1, we have g(x + 1) = g(x) + 1 for any x ∈ F2n , that is, f (bx + b) = f (bx) + 1 for any x ∈ F2n , that is, f (x + b) = f (x) + 1 for any x ∈ F2n . By Lemma 1, the inverse of the function x 2 + x + ag(x) equals φ(x) + tr n (x) + g(φ(x)) for a certain linear permutation φ of F2n . 6): F1−1 (x) = b(φ(x) + tr n (x) + f (b φ(x))) and therefore, since f = F2 ◦ F1−1 : f (x) = l b(φ(x) + tr n (x) + f (b φ(x))) + f b(φ(x) + tr n (x) + f (b φ(x))) = l(b φ(x)) + tr n (x) + f (b φ(x)) + f (b φ(x)) + tr n (x) + f (b φ(x)) = l(b φ(x)) + f (b φ(x)).
11, pp. 1–8, 1961. 3. T. Bending, D. Fon-Der-Flaass. Crooked functions, bent functions and distance-regular graphs. Electron. J. , 5 (R34), 14, 1998. 4. T. Beth and C. Ding. On almost perfect nonlinear permutations. Advances in CryptologyEUROCRYPT’93, Lecture Notes in Computer Science, 765, Springer-Verlag, New York, pp. 65–76, 1993. 5. J. Bierbrauer. New semifields, PN and APN functions. Designs, Codes and Cryptography, v. 54, pp. 189–200, 2010. 6. J. Bierbrauer. Commutative semifields from projection mappings.
Construction and Analysis of Cryptographic Functions by Lilya Budaghyan