Strongly regular graphs from weakly regular plateaued functions

The paper provides the first constructions of strongly regular graphs and association schemes from weakly regular plateaued functions over finite fields of odd characteristic. We generalize the construction method of strongly regular graphs from weakly regular bent functions given by Chee et al. in [Journal of Algebraic Combinatorics, 34(2), 251-266, 2011] to weakly regular plateaued functions. In this framework, we construct strongly regular graphs with three types of parameters from weakly regular plateaued functions with some homogeneous conditions. We also construct a family of association schemes of class p from weakly regular p-ary plateaued functions

A new class of three-weight linear codes from weakly regular plateaued functions

Linear codes with few weights have many applications in secret sharing schemes, authentication codes, communication and strongly regular graphs. In this paper, we consider linear codes with three weights in arbitrary characteristic. To do this, we generalize the recent contribution of Mesnager given in [Cryptography and Communications 9(1), 71-84, 2017]. We first present a new class of binary linear codes with three weights from plateaued Boolean functions and their weight distributions. We next introduce the notion of (weakly) regular plateaued functions in odd characteristic $p$ and give concrete examples of these functions. Moreover, we construct a new class of three-weight linear $p$-ary codes from weakly regular plateaued functions and determine their weight distributions. We finally analyse the constructed linear codes for secret sharing schemes.Comment: The Extended Abstract of this work was submitted to WCC-2017 (the Tenth International Workshop on Coding and Cryptography

New self-orthogonal codes from weakly regular plateaued functions and their application in LCD codes

A linear code with few weights is a significant code family in coding theory. A linear code is considered self-orthogonal if contained within its dual code. Self-orthogonal codes have applications in linear complementary dual codes, quantum codes, etc. The construction of linear codes is an interesting research problem. There are various methods to construct linear codes, and one approach involves utilizing cryptographic functions defined over finite fields. The construction of linear codes (in particular, self-orthogonal codes) from functions has been studied in the literature. In this paper, we generalize the construction method given by Heng et al. in [Des. Codes Cryptogr. 91(12), 2023] to weakly regular plateaued functions. We first construct several families of p-ary linear codes with few weights from weakly regular plateaued unbalanced (resp. balanced) functions over the finite fields of odd characteristics. We observe that the constructed codes are self-orthogonal codes when p = 3. Then, we use the constructed ternary self-orthogonal codes to build new families of ternary LCD codes. Consequently, we obtain (almost) optimal ternary self-orthogonal codes and LCD codes

Simetrik kriptografi ve kodlama teorisi için (vektörel) plato fonksiyonlar üzerine katkılar.

Plateaued functions, used to construct nonlinear functions and linear codes, play a significant role in cryptography and coding theory. They can possess various desirable cryptographic properties such as high nonlinearity, low autocorrelation, resiliency, propagation criteria, balanced-ness and correlation immunity. In fact, they provide the best possible compromise between resiliency order and nonlinearity. Besides they resist against linear cryptanalysis and fast correlation attacks due to their low Walsh-Hadamard transform values. Indeed, cryptographic algorithms are usually designed by appropriate composition of nonlinear functions, hence plateaued functions have a great effect on the security of these algorithms. Additionally, plateaued functions are closely related to linear codes, the most significant class of codes in coding theory, which have diverse applications in secret sharing schemes, authentication codes, communication, data storage devices and consumer electronics. The main objectives of this thesis are twofold: to study in detail the explicit characterizations for plateaued-ness of functions over finite fields from a cryptographic point of view, and to construct linear codes from weakly regular plateaued functions in coding theory. In this thesis, we first analyse characterizations of plateaued (vectorial) functions over a finite field F_p with p a prime number. More precisely, we obtain a large number of their characterizations in terms of their Walsh power moments, derivatives and autocorrelation functions, with the aim of both clarifying their structure and obtaining information about their construction. In particular, we observe the non-existence of a homogeneous cubic bent function (and in some cases a (homogeneous) cubic plateaued function) over F_p with p an odd prime. Moreover, we show the non-existence of a function whose absolute Walsh transform takes exactly three distinct values (one being zero), and introduce a new class of functions whose absolute Walsh transform takes exactly four distinct values (one being zero). Furthermore, we study partially bent and plateaued functions over a finite field F_q, with q a prime power, and obtain some of their characterizations in order to understand their behaviour over this field. In addition, we introduce the notion of (non)-weakly regular plateaued functions over F_p, with p an odd prime, and provide the secondary constructions of these functions. We then construct three-weight linear p-ary (resp. binary) codes from weakly regular p-ary plateaued (resp. Boolean plateaued) functions and determine their weight distributions. Finally, we show that the constructed linear codes can be used to construct secret sharing schemes with ``nice'' access structures. To the best of our knowledge, the construction of linear codes from plateaued functions over F_p, with p an odd prime, is studied in this thesis for the first time in the literaturePh.D. - Doctoral Progra

Vektör boole fonksiyonlarının kısıtlı genişletilmiş afin denkliği üzerine.

Vectorial Boolean functions are used as S-boxes in cryptosystems. To design inequivalent vectorial Boolean functions resistant to known attacks is one of the challenges in cryptography. Verifying whether two vectorial Boolean functions are equivalent or not is the final step in this challenge. Hence, finding a fast technique for determining whether two given vectorial Boolean functions are equivalent is an important problem. A special class of the equivalence called restricted extended affine (REA) equivalence is studied in this thesis. We study the verification complexity of REA-equivalence of two vectorial Boolean functions for some types, namely types I to VI. We first review the verification of the REA-equivalence types I to IV given in the recent work of Budaghyan and Kazymyrov (2012). Furthermore, we present the complexities of the verification of REA-equivalence types I and IV in the case basic simultaneous Gaussian elimination method is used. Next, we present two new REA-equivalence types V and VI with their complexities. Finally, we give the algorithms of each type I to VI with their MAGMA codes.M.S. - Master of Scienc

Free storage basis conversion over finite fields

WOS:000392340700010Representation of a field element plays a crucial role in the efficiency of field arithmetic. If an efficient representation of a field element in one basis exists, then field arithmetic in the hardware and/or software implementations becomes easy. Otherwise, a basis conversion to an efficient one is searched for easier arithmetic. However, this conversion often brings a storage problem for transition matrices associated with these bases. In this paper, we study this problem for conversion between normal and polynomial bases in the extension field Fqp over Fq where q p n . We construct transition matrices that are of a special form. This provides free storage basis conversion algorithms between normal and polynomial bases, which is crucial from the implementation point of view