American Institute of Mathematical Sciences

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1930-5346

eISSN:
1930-5338

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May 2022 , Volume 16 , Issue 2

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2022, 16(2): 231-247 doi: 10.3934/amc.2020109 +[Abstract](1768) +[HTML](645) +[PDF](443.57KB)
Abstract:

The notion of an irredundant orthogonal array (IrOA) was introduced by Goyeneche and \begin{document}$\dot{Z}$\end{document}yczkowski who showed an IrOA\begin{document}$_{\lambda}(t, k, v)$\end{document} corresponds to a \begin{document}$t$\end{document}-uniform state of \begin{document}$k$\end{document} subsystems with local dimension \begin{document}$v$\end{document} (Physical Review A. 90 (2014), 022316). In this paper, we construct some kinds of 2-uniform states by establishing the existence of IrOA\begin{document}$_{\lambda}(2, 5, v)$\end{document} for any integer \begin{document}$v\geq 4$\end{document}, \begin{document}$v\neq 6$\end{document}; IrOA\begin{document}$_{\lambda}(2, 6, v)$\end{document} for any integer \begin{document}$v\geq 2$\end{document}; IrOA\begin{document}$_{\lambda}(2, q, q)$\end{document} and IrOA\begin{document}$_{\lambda}(2, q+1, q)$\end{document} for any prime power \begin{document}$q >3$\end{document}.

2022, 16(2): 249-267 doi: 10.3934/amc.2020110 +[Abstract](1620) +[HTML](680) +[PDF](434.8KB)
Abstract:

In this paper, we present two constructions of low-hit-zone frequen-cy-hopping sequence (LHZ FHS) sets. The constructions in this paper generalize the previous constructions based on \begin{document}$m$\end{document}-sequences and \begin{document}$d$\end{document}-form functions with difference-balanced property, and generate several classes of optimal LHZ FHS sets and LHZ FHS sets with optimal periodic partial Hamming correlation (PPHC).

2022, 16(2): 269-284 doi: 10.3934/amc.2020111 +[Abstract](1513) +[HTML](633) +[PDF](394.55KB)
Abstract:

Many generator matrices for constructing extremal binary self-dual codes of different lengths have the form \begin{document}$G = (I_n \ | \ A),$\end{document} where \begin{document}$I_n$\end{document} is the \begin{document}$n \times n$\end{document} identity matrix and \begin{document}$A$\end{document} is the \begin{document}$n \times n$\end{document} matrix fully determined by the first row. In this work, we define a generator matrix in which \begin{document}$A$\end{document} is a block matrix, where the blocks come from group rings and also, \begin{document}$A$\end{document} is not fully determined by the elements appearing in the first row. By applying our construction over \begin{document}$\mathbb{F}_2+u\mathbb{F}_2$\end{document} and by employing the extension method for codes, we were able to construct new extremal binary self-dual codes of length 68. Additionally, by employing a generalised neighbour method to the codes obtained, we were able to construct many new binary self-dual \begin{document}$[68, 34, 12]$\end{document}-codes with the rare parameters \begin{document}$\gamma = 7, 8$\end{document} and \begin{document}$9$\end{document} in \begin{document}$W_{68, 2}.$\end{document} In particular, we find 92 new binary self-dual \begin{document}$[68, 34, 12]$\end{document}-codes.

2022, 16(2): 285-302 doi: 10.3934/amc.2020112 +[Abstract](1478) +[HTML](632) +[PDF](437.78KB)
Abstract:

Quaternary sequences with optimal autocorrelation property are preferred in applications. Cyclotomic classes of order 4 are widely used in the constructions of quaternary sequences due to the convenience of defining a quaternary sequence with the cyclotomic classes of order 4 as its support set. Recently, several classes of optimal quaternary sequences of period \begin{document}$2p$\end{document}, which are all closely related to the cyclotomic classes of order 4 with respect to \begin{document}$\mathbb{Z}_p$\end{document} were introduced in the literature. However, less attention has been paid to the equivalence between these known results. In this paper, we introduce the unified form of this kind of quaternary sequences to classify these known results and then conclude the unified forms of these optimal quaternary sequences. By doing this, we disclose the relationship between the optimal quaternary sequences derived from different methods in the literature on one hand. And on the other hand, when the new obtained optimal quaternary sequence period is \begin{document}$2p$\end{document} and the cyclotomic classes of order 4 are involved, the methods and the results given in this paper can be used to identify if the sequence is new or not.

2022, 16(2): 303-348 doi: 10.3934/amc.2020113 +[Abstract](1790) +[HTML](655) +[PDF](675.24KB)
Abstract:

Elliptic curve cryptography is based upon elliptic curves defined over finite fields. Operations over such elliptic curves require arithmetic over the underlying field. In particular, fast implementations of multiplication and squaring over the finite field are required for performing efficient elliptic curve cryptography. The present work considers the problem of obtaining efficient algorithms for field multiplication and squaring. From a theoretical point of view, we present a number of algorithms for multiplication/squaring and reduction which are appropriate for different settings. Our algorithms collect together and generalize ideas which are scattered across various papers and codes. At the same time, we also introduce new ideas to improve upon existing works. A key theoretical feature of our work is that we provide formal statements and detailed proofs of correctness of the different reduction algorithms that we describe. On the implementation aspect, a total of fourteen primes are considered, covering all previously proposed cryptographically relevant (pseudo-)Mersenne prime order fields at various security levels. For each of these fields, we provide 64-bit assembly implementations of the relevant multiplication and squaring algorithms targeted towards two different modern Intel architectures. We were able to find previous 64-bit implementations for six of the fourteen primes considered in this work. On the Haswell and Skylake processors of Intel, for all the six primes where previous implementations are available, our implementations outperform such previous implementations.

Ye Wang and
2022, 16(2): 349-364 doi: 10.3934/amc.2020114 +[Abstract](1478) +[HTML](637) +[PDF](431.13KB)
Abstract:

The intersection of a linear code and its dual is called the hull of this code. The code is a linear complementary dual (LCD) code if the dimension of its hull is zero. In this paper, we develop a method to construct LCD codes and linear codes with one-dimensional hull by association schemes. One of constructions in this paper generalizes that of linear codes associated with Gauss periods given in [5]. In addition, we present a generalized construction of linear codes, which can provide more LCD codes and linear codes with one-dimensional hull. We also present some examples of LCD MDS, LCD almost MDS codes, and MDS, almost MDS codes with one-dimensional hull from our constructions.

2022, 16(2): 365-382 doi: 10.3934/amc.2020115 +[Abstract](2230) +[HTML](626) +[PDF](460.77KB)
Abstract:

Rotation symmetric Boolean functions constitute a class of cryptographically significant Boolean functions. In this paper, based on the theory of ordered integer partitions, we present a new class of odd-variable rotation symmetric Boolean functions with optimal algebraic immunity by modifying the support of the majority function. Compared with the existing rotation symmetric Boolean functions on odd variables, the newly constructed functions have the highest nonlinearity.

2022, 16(2): 383-411 doi: 10.3934/amc.2020116 +[Abstract](2190) +[HTML](629) +[PDF](927.58KB)
Abstract:

Since its proposal in Asiacrypt 2018, the commutative isogeny-based key exchange protocol (CSIDH) has spurred considerable attention to improving its performance and re-evaluating its classical and quantum security guarantees. In this paper we discuss how the optimal strategies employed by the Supersingular Isogeny Diffie-Hellman (SIDH) key agreement protocol can be naturally extended to CSIDH. Furthermore, we report a software library that achieves moderate but noticeable performance speedups when compared against state-of-the-art implementations of CSIDH-512, which is the most popular CSIDH instantiation. We also report an estimated number of field operations for larger instantiations of this protocol, namely, CSIDH-1024 and CSIDH-1792.

2022, 16(2): 413-437 doi: 10.3934/amc.2020117 +[Abstract](1368) +[HTML](629) +[PDF](555.24KB)
Abstract:

The concept of the signal-to-noise ratio (SNR) as a useful measure indicator of the robustness of \begin{document}$(n, m)$\end{document}-functions \begin{document}$F = (f_1, \ldots, f_m)$\end{document} (cryptographic S-boxes) against differential power analysis (DPA), has received extensive attention during the previous decade. In this paper, we give an upper bound on the SNR of balanced \begin{document}$(n, m)$\end{document}-functions, and a clear upper bound regarding unbalanced \begin{document}$(n, m)$\end{document}-functions. Moreover, we derive some deep relationships between the SNR of \begin{document}$(n, m)$\end{document}-functions and three other cryptographic parameters (the maximum value of the absolute value of the Walsh transform, the sum-of-squares indicator, and the nonlinearity of its coordinates), respectively. In particular, we give a trade-off between the SNR and the refined transparency order of \begin{document}$(n, m)$\end{document}-functions. Finally, we prove that the SNR of \begin{document}$(n, m)$\end{document}-functions is not affine invariant, and data experiments verify this result.

Addendum: The grant no. 2021ZYD0011 is added so it reads “Yu Zhou and Xinfeng Dong are supported in part by the National Key R & D Program of China (No. 2017YFB0802000, No. 2017YFB0802004), and in part by Sichuan Science and Technology Program (No. 2020JDJQ0076, 2021ZYD0011).” We apologize for any inconvenience this may cause.

2021 Impact Factor: 1.015
5 Year Impact Factor: 1.078
2021 CiteScore: 1.8