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Abstract-In this paper, error-correcting codes from perfect nonlinear mappings are constructed, and then employed to construct secret sharing schemes. The error-correcting codes obtained in this paper are very good in general, and many of them are optimal or almost optimal. The secret sharing schemes obtained in this paper have two types of access structures. The first type is democratic in the sense that every participant is involved in the same number of minimal-access sets. In the second type of access structures, there are a few dictators who are in every minimal access set, while each of the remaining participants is in the same number of minimal-access sets.

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Abstract. Algebraic attacks on LFSR-based stream ciphers recover the secret key by solving an overdefined system of multivariate algebraic equations. They exploit multivariate relations involving key bits and output bits and become very efficient if such relations of low degrees may be found. Low degree relations have been shown to exist for several well known constructions of stream ciphers immune to all previously known attacks. Such relations may be derived by multiplying the output function of a stream cipher by a well chosen low degree function such that the product function is again of low degree. In view of algebraic attacks, low degree multiples of Boolean functions are a basic concern in the design of stream ciphers as well as of block ciphers. This paper investigates the existence of low degree multiples of Boolean functions in several directions: The known scenarios under which low degree multiples exist are reduced and simplified to two scenarios, that are treated differently in algebraic attacks. A new algorithm is proposed that allows to successfully decide whether a Boolean function has low degree multiples. This represents a significant step towards provable security against algebraic attacks. Furthermore, it is shown that a recently introduced class of degree optimized Maiorana-McFarland functions immanently has low degree multiples. Finally, the probability that a random Boolean function has a low degree multiple is estimated.

Boolean functions are essential to systems for secure and reliable communication. This comprehensive survey of Boolean functions for cryptography and coding covers the whole domain and all important results, building on the author's influential articles with additional topics and recent results. A useful resource for researchers and graduate students, the book balances detailed discussions of properties and parameters with examples of various types of cryptographic attacks that motivate the consideration of these parameters. It provides all the necessary background on mathematics, cryptography, and coding, and an overview on recent applications, such as side channel attacks on smart cards, cloud computing through fully homomorphic encryption, and local pseudo-random generators. The result is a complete and accessible text on the state of the art in single and multiple output Boolean functions that illustrates the interaction between mathematics, computer science, and telecommunications.

We recall why linear codes with complementary duals (LCD codes) play a role in countermeasures to passive and active side-channel analyses on embedded cryptosystems. The rate and the minimum distance of such LCD codes must be as large as possible. We recall the known primary construction of such codes with cyclic codes, and investigate other constructions, with expanded Reed-Solomon codes and generalized residue codes, for which we study the idempotents. These constructions do not allow to reach all the desired parameters. We study then those secondary constructions which preserve the LCD property, and we characterize conditions under which codes obtained by direct sum, direct product, puncturing, shortening, extending codes, or obtained by the Plotkin sum, can be LCD.

We construct infinite classes of almost bent and almost perfect nonlinear polynomials, which are affinely inequivalent to any sum of a power function and an affine function.

Linear codes with complementary duals (abbreviated LCD) are linear codes whose intersection with their dual are trivial. When they are binary, they play an important role in armoring implementations against side-channel attacks and fault injection attacks. Non-binary LCD codes in characteristic 2 can be transformed into binary LCD codes by expansion. In this paper, we introduce a general construction of LCD codes from any linear codes. Further, we show that any linear code over Fq(q > 3) is equivalent to an Euclidean LCD code and any linear code over F q 2 (q > 2) is equivalent to a Hermitian LCD code. Consequently an [n, k, d]-linear Euclidean LCD code over Fq with q > 3 exists if there is an [n, k, d]-linear code over Fq and an [n, k, d]-linear Hermitian LCD code over F q 2 with q > 2 exists if there is an [n, k, d]-linear code over F q 2 . Hence, when q > 3 (resp.q > 2) q-ary Euclidean (resp. q 2 -ary Hermitian) LCD codes possess the same asymptotical bound as q-ary linear codes (resp. q 2 -ary linear codes). Finally, we present an approach of constructing LCD codes by extending linear codes.