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A new method is used to resolve a long-standing conjecture of Niho concerning the crosscorrelation spectrum of a pair of maximum length linear recursive sequences of length $2^{2m}-1$ with relative decimation $d=2^{m+2}-3$ , where $m$ is even. The result indicates that there are at most five distinct crosscorrelation values. Equivalently, the result indicates that there are at most five distinct values in the Walsh spectrum of the power permutation $f(x)=x^{d}$ over a finite field of order $2^{2m}$ and at most five distinct nonzero weights in the cyclic code of length $2^{2m}-1$ with two primitive nonzeros $\alpha $ and $\alpha ^{d}$. The method used to obtain this result proves constraints on the number of roots that certain seventh degree polynomials can have on the unit circle of a finite field. The method also works when $m$ is odd, in which case the associated crosscorrelation and Walsh spectra have at most six distinct values. [ABSTRACT FROM AUTHOR]

In this paper, a class of functions from Niho exponents with four-valued Walsh transform is obtained for any prime by a uniform method, and the distribution of the Walsh transform values is also completely determined. In particular, this class of functions is proven to be bent for a special case. Although it is shown that the obtained bent functions are equivalent to the Leander–Kholosha's class of bent functions, a direct and much simpler proof for the bentness of this kind of Niho functions is provided. [ABSTRACT FROM AUTHOR]

The Walsh transform of two-level autocorrelation sequences has played an important role in the construction of the set of sequences in which any two sequences are orthogonal. For p-ary sequences, there are only two basic classes of two-level autocorrelation sequences with no subfield structures for an arbitrary odd prime p. One is the class of p-ary m-sequences and the other is the class of p-ary Helleseth–Gong sequences. In this paper, the Walsh transform of a subclass of p-ary Helleseth–Gong sequences and the Walsh transform of their particular decimations are completely determined and are shown to be three-valued. [ABSTRACT FROM AUTHOR]

In this paper, a coding-theory construction of Cartesian authentication codes is presented. The construction is a generalization of some known constructions. Within the framework of this generic construction, several classes of authentication codes using certain classes of error-correcting codes are described. The authentication codes presented in this paper are better than known ones with comparable parameters. It is demonstrated that the construction is related to certain combinatorial designs, such as difference matrices and generalized Hadamard matrices. [ABSTRACT FROM AUTHOR]

Linear equations have always been powerful tools in crypt-analysis. In this correspondence, we present a general linear equation of minimum weight 3 in F2 that holds for all state lengths n and all shifts i of sequences generated by the T-function xi = xi-1² V C + xi-1 rood 2n proposed by Klimov and Shamir. It is surprising that these linear properties exist, and they indicate that the sequences generated by the T-functions have more structures than claimed by Klimov and Shamir. [ABSTRACT FROM AUTHOR]

Considered are p-ary bent functions having the form (multiple line equation(s) cannot be represented in ASCII text). A new class of ternary monomial regular bent function with the Dillon exponent is discovered. The existence of Dillon bent functions in the general case is an open problem of deciding whether a certain Kloosterman sum can take on the value -1. Also described is the general Gold-like form of a bent function that covers all the previously known monomial quadratic cases. The (weak) regularity of the new as well as of known monomial bent functions is discussed and the first example of a not weakly regular bent function is given. Finally, some criteria for an arbitrary quadratic function to be bent are proven. [ABSTRACT FROM AUTHOR]

In this paper, we apply two-to-one functions over F2n in two generic constructions of binary linear codes. We consider two-to-one functions in two forms: (1) generalized quadratic functions; and (2) (x2t + x)e with gcd (t, n) = gcd(e, 2n−1) = 1. Based on the study of the Walsh transforms of those functions or their variants, we present many classes of linear codes with few nonzero weights, including one weight, three weights, four weights, and five weights. The weight distributions of the proposed codes with one weight and with three weights are determined. In addition, we discuss the minimum distance of the dual of the constructed codes and show that some of them achieve the sphere packing bound. Moreover, examples show that some codes in this paper have best-known parameters. [ABSTRACT FROM AUTHOR]

Zero correlation zone (ZCZ) sequences are a class of spreading sequences having ideal auto-correlation and cross correlation in a zone around the origin. They have been extensively studied in recent years due to their important applications in quasi-synchronous code division multiple access systems. In this paper, a construction of ZCZ sequence sets is proposed based on perfect nonlinear functions. It generates multiple ZCZ sequence sets with the properties: 1) each sequence is perfect in the sense that its out-of-phase auto-correlation is always zero; 2) each ZCZ sequence set is optimal with respect to the Tang–Fan–Matsufuji bound in which all the sequences are pairwise cyclically distinct; and 3) the maximum inter-set cross correlation of multiple sequence sets achieves the well-known Sarwate bound. [ABSTRACT FROM AUTHOR]

Let $n$ be a positive integer and $p$ a prime. The power mapping $x^{p^{n}-3}$ over ${\mathbb {F}}_{p^{n}}$ has desirable differential properties, and its differential spectra for $p=2,\,3$ have been determined. In this paper, for any odd prime $p$ , by investigating certain quadratic character sums and some equations over ${\mathbb {F}}_{p^{n}}$ , we determine the differential spectrum of $x^{p^{n}-3}$ with a unified approach. The obtained result shows that for any given odd prime $p$ , the differential spectrum can be expressed explicitly in terms of $n$. Compared with previous results, a special elliptic curve over ${\mathbb {F}}_{p}$ plays an important role in our computation for the general case $p \ge 5$. [ABSTRACT FROM AUTHOR]

Let \(m\geq 3\) be an odd integer and \(p\) be an odd prime. In this paper, a number of classes of three-weight cyclic codes \(\mathcal {C}_{(1,e)}\) over \(\mathbb {F}_{p}\) , which have parity-check polynomial \(m_1(x)m_e(x)\) , are presented by examining general conditions on the parameters \(p\) , \(m\) , and \(e\) , where \(m_i(x)\) is the minimal polynomial of \(\pi ^{-i}\) over \(\mathbb {F}_{p}\) for a primitive element \(\pi \) of \(\mathbb {F}_{p^m}\) . Furthermore, for \(p\equiv 3 \pmod {4}\) and a positive integer \(e\) satisfying \((p^k+1)\cdot e\equiv 2 \pmod {p^m-1}\) for some positive integer \(k\) with \(\gcd (m,k)=1\) , the value distributions of the exponential sums \(T(a,b)=\sum \nolimits _{x\in \mathbb {F}_{p^m}}\omega ^{ \text {Tr}(ax+bx^e)}\) and \(S(a,b,c)=\sum \nolimits _{x\in \mathbb {F}_{p^m}}\omega ^{ \text {Tr}(ax+bx^e+cx^s)},\) where \(s=(p^m-1)/2\) , are determined. As an application, the value distribution of \(S(a,b,c)\) is utilized to derive the weight distribution of the cyclic codes \(\mathcal {C}_{(1,e,s)}\) with parity-check polynomial \(m_1(x)m_e(x)m_s(x)\) . In the case of \(p=3\) and even \(e\) satisfying the above condition, the dual of the cyclic code \(\mathcal {C}_{(1,e,s)}\) has optimal minimum distance. [ABSTRACT FROM PUBLISHER]

In this paper, some new results are presented on the selective discrete Fourier spectra attack introduced first as the Rønjom–Helleseth attack and the modifications due to Rønjom, Gong, and Helleseth. The first part of this paper fills some gaps in the theory of analysis in terms of the discrete Fourier transform (DFT). The second part introduces the new fast selective DFT attacks, which are closely related to the fast algebraic attacks in the literature. However, in contrast to the classical view that successful algebraic cryptanalysis of LFSR-based stream cipher depends on the degree of certain annihilators, the analysis in terms of the DFT spectral properties of the sequences generated by these functions is far more refined. It is shown that the selective DFT attack is more efficient than known methods for the case when the number of observed consecutive bits of a filter generator is less than the linear complexity of the sequence. Thus, by utilizing the natural representation imposed by the underlying LFSRs, in certain cases, the analysis in terms of DFT spectra is more efficient and has more flexibility than classical and fast algebraic attacks. Consequently, the new attack imposes a new criterion for the design of cryptographic strong Boolean functions, which is defined as the spectral immunity of a sequence or a Boolean function. [ABSTRACT FROM PUBLISHER]

Presents information on a study which examined almost difference sets and their applications in cryptography and coding theory. Relations between some difference sets and some almost difference sets; Classes of binary sequences with optimal autocorrelation and almost difference sets; Conclusions.

Functions with low differential uniformity have important applications in cryptography, coding theory, and sequence design. The differential spectrum of a cryptographic function is of great interest for estimating its resistance to some variants of differential cryptanalysis. Finding power permutations (i.e., monomial bijective mappings) over finite fields with low differential uniformity and determining their differential spectra have received a lot of attention over the past two decades. The objective of this paper is to study the differential properties of the well-known Kasami power permutations $x^{p^{2k}-p^{k}+1}$ over $ {\mathrm {GF}}(p^{n})$ , where $p$ is an odd prime and $k$ is an integer with $\gcd (n,k)=1$. It turns out that this family of monomials is differentially $(p+1)$ -uniform. Our result in the case of $p=3$ gives an affirmative solution to a recent conjecture by Xu, Cao, and Xu. Most notably, the differential spectrum of this family of power permutations is completely determined. [ABSTRACT FROM AUTHOR]

As a class of optimal combinatorial objects, bent functions have important applications in cryptography, sequence design, and coding theory. Bent idempotents are a subclass of bent functions and of great interest, since they can be stored in less space and allow faster computation of the Walsh-Hadamard transform. The objective of this paper is to present a generic construction of bent functions from known ones. It includes the previous constructions of bent functions by Mesnager and Xu et al. as special cases, and produces new bent functions, which cannot be produced by earlier ones. In particular, it also generates infinite families of bent idempotents over F2^{2m} of any algebraic degree between 2 and $m$ . This together with a recent construction by Su and Tang gives a positive answer to an open problem on bent idempotents proposed by Carlet. In addition, an infinite family of anti-self-dual bent functions is obtained in which the sum of any three distinct functions is again an anti-self-dual bent function in this family. This solves an open problem recently proposed by Mesnager. [ABSTRACT FROM PUBLISHER]

The nonlinear complexity of a periodic sequence s is the length of the shortest feedback shift register that can generate s, and its value is upper bounded by the least period of s minus 1. In this paper, a recursive approach that generates all periodic sequences with maximum nonlinear complexity is presented, and the total number of such sequences is determined. The randomness properties of these sequences are also examined. [ABSTRACT FROM PUBLISHER]

Linear codes with a few weights have applications in consumer electronics, communication, data storage system, secret sharing, authentication codes, association schemes, and strongly regular graphs. This paper first generalizes the method of constructing two-weight and three-weight linear codes of Ding et al. and Zhou et al. to general weakly regular bent functions and determines the weight distributions of these linear codes. It solves an open problem proposed by Ding et al. Furthermore, this paper constructs new linear codes with two or three weights and presents their weight distributions. They contain some optimal codes meeting certain bound on linear codes. [ABSTRACT FROM AUTHOR]

In 1998, Lin presented a conjecture on a class of ternary sequences with ideal two-level autocorrelation. Those sequences have a very simple structure, i.e., their trace representation has two trace monomial terms. In this paper, we present a proof for the conjecture. The mathematical tools employed are the second-order multiplexing decimation-Hadamard transform, Stickelberger’s theorem, the Teichmüller character, and combinatorial techniques for enumerating the Hamming weights of ternary numbers. As a by-product, we also prove that the ternary sequences conjectured by Lin are Hadamard equivalent to ternary \(m\) -sequences. [ABSTRACT FROM PUBLISHER]