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Abstract: In this paper we have studied repeated root cyclic codes of length pk over \(R=\mathbb {Z}_{p^{2}}+u\mathbb {Z}_{p^{2}}\) , u2 = 0, where p is a prime and k is a positive integer. We have determined a unique set of generators for these codes and obtained some results on their Lee distances. A minimal spanning set for them has been obtained and their ranks are determined. Further, we have determined the complete algebraic structure of principally generated cyclic codes in this class. An upper bound for the Lee distance of linear codes over R is presented. We have considered two Gray maps \(\psi :R \rightarrow \mathbb {Z}_{p}^{4}\) and \(\phi _{1}:R \rightarrow \mathbb {Z}_{p^{2}}^{2}\) , and using them, we have obtained some optimal binary linear codes as well as some quaternary linear codes from cyclic codes of length 4 over \(\mathbb {Z}_{4}+u\mathbb {Z}_{4}\) . Three of the quaternary linear codes obtained are new, and the remaining of them have the best known parameters for their lengths and types. We have also obtained some optimal ternary codes of length 12 as Gray images of repeated root cyclic codes of length 3 over \(\mathbb {Z}_{9}+u\mathbb {Z}_{9}\) . PubDate: 2021-10-08
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Abstract: Several constructions of Mutually Unbiased Bases (MUBs) borrow tools from combinatorial objects. In this paper we focus on how one can construct Approximate Real MUBs (ARMUBs) with improved parameters using results from the domain of Resolvable Block Designs (RBDs). We first explain the generic idea of our strategy in relating the RBDs with MUBs/ARMUBs, which are sparse (the basis vectors have small number of non-zero co-ordinates). Then specific parameters are presented, for which we can obtain new classes and improve the existing results. To be specific, we present an infinite family of \(\lceil \sqrt {d}\rceil \) many ARMUBs for dimension d = q(q + 1), where q ≡ 3 mod 4 and it is a prime power, such that for any two vectors v1,v2 belonging to different bases, \( \langle {v_{1} v_{2}}\rangle < \frac {2}{\sqrt {d}}\) . We also demonstrate certain cases, such as d = sq2, where q is a prime power and sq ≡ 0 mod 4. These findings subsume and improve our earlier results in [Cryptogr. Commun. 13, 321-329, January 2021]. This present construction idea provides several infinite families of such objects, not known in the literature, which can find efficient applications in quantum information processing for the sparsity, apart from suggesting that parallel classes of RBDs are intimately linked with MUBs/ARMUBs. PubDate: 2021-09-22
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Abstract: In this paper we study the linear congruential generator on elliptic curves from the cryptographic point of view. We show that if sufficiently many of the most significant bits of the composer and of three consecutive values of the sequence are given, then one can recover the seed and the composer (even in the case where the elliptic curve is private). The results are based on lattice reduction techniques and improve some recent approaches of the same security problem. We also estimate limits of some heuristic approaches, which still remain much weaker than those known for nonlinear congruential generators. Several examples are tested using implementations of ours algorithms. PubDate: 2021-09-12 DOI: 10.1007/s12095-021-00535-6
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Abstract: Construction and equivalence of APN functions play a significant role in the research of cryptographic functions. On finite fields of characteristic 2, six infinite families of APN power functions and thirteen infinite families of APN polynomial functions have been constructed in the literature. Yoshiara showed that CCZ-equivalence between APN power functions in characteristic 2 reduces to cyclotomic equivalence. After that, in 2018, Dempwolff generalized Yoshira’s result to any characteristic p. Moreover, some results on equivalence between known families of APN polynomial functions and APN power functions were obtained. In this paper, we show that two other known infinite families of APN polynomial functions are CCZ-inequivalent to APN power functions. PubDate: 2021-09-09 DOI: 10.1007/s12095-021-00533-8
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Abstract: In this paper, first we consider the linear complexity of quaternary sequences over the finite ring of order four and the finite field of order four. These sequences are constructed from new generalized cyclotomic classes modulo pn. Second, we study the linear complexity of new generalized cyclotomic binary sequences of period 2pn recently proposed by Ouyang et al. (Des. Codes Cryptogr. 87(5), 1–12 2019). We generalized results presented in this work and discuss the author’s conjecture. In conclusion, we again derive the linear complexity of quaternary sequences with period 2pn over the finite ring of order four and the finite field of order four. We use generalized cyclotomic classes modulo 2pn to define these sequences. PubDate: 2021-09-07 DOI: 10.1007/s12095-021-00534-7
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Abstract: Nowadays, ciphers have been widely used in high-end platforms, resource-constrained, and side-channel attacks vulnerable environments. This motivates various S-boxes aimed at providing a good trade-off between security and efficiency. For small S-boxes, the most natural approach of constructing such S-boxes is a comprehensive search in the space of permutations, which inevitably becomes more challenging when the size grows. For large S-boxes (e.g., 8-bit), previous works concentrated on creations from finite fields or smaller ones (e.g., 4-bit). This paper proposes a new algorithm with a layered structure to search for 8-bit SKINNY-like S-boxes. We compare our new S-box with the original 8-bit SKINNY S-box by analyzing its security properties. Besides, due to our searching algorithm’s rules and constraints, SKINNY-like S-boxes have other features of lightweight implementation, low multiplicative complexity, low AND depth, and an effective inverse. Eventually, the searching algorithm outputs 224000 8-bit SKINNY-like S-boxes. The cipher designers can use these new S-boxes to construct lightweight block ciphers with easy-to-mask property and efficient implementation performance. PubDate: 2021-09-01
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Abstract: Three constructions of binary sequences with period 4N and optimal autocorrelation value or optimal autocorrelation magnitude have been presented by Tang and Gong based on interleaving technique. In this paper, the 2-adic complexity of the sequences with optimal autocorrelation magnitude constructed from the Legendre sequence pair or the twin-prime sequence pair is investigated. With the method proposed by Hu, we completely determine the 2-adic complexity of the sequences by calculating the exact autocorrelation distribution of the sequences and discussing the greatest common divisors. Results show that the 2-adic complexity of these sequences is either maximum or very close to maximum. PubDate: 2021-09-01
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Abstract: Motivated by the impact of fast algebraic attacks on stream ciphers, and recent constructions using a threshold function as main part of the filtering function, we study the fast algebraic immunity of threshold functions. As a first result, we determine exactly the fast algebraic immunity of all majority functions in more than 8 variables. Then, For all n ≥ 8 and all threshold value between 1 and n we exhibit the fast algebraic immunity for most of the thresholds, and we determine a small range for the value related to the few remaining cases. Finally, provided m ≥ 2, we determine exactly the fast algebraic immunity of all threshold functions in 3 ⋅ 2m or 3 ⋅ 2m + 1 variables. PubDate: 2021-09-01
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Abstract: Minimal binary linear codes are a special class of binary codes with important applications in secret sharing and secure two-party computation. These codes are characterized by the property that none of the nonzero codewords is covered by any other codeword. Denoting by \(w_{{\min \limits }}\) and \(w_{{\max \limits }}\) the minimum and maximum weights of the codewords, respectively, such codes are relatively easy to design when the ratio \(w_{{\min \limits }}/w_{{\max \limits }}\) is larger than 1/2 (known as the Ashikhmin-Barg bound). On the other hand, a few known classes of minimal codes violate this bound, hence having the property \(w_{{\min \limits }}/w_{{\max \limits }} \leq 1/2\) . In this article, we provide several explicit classes of minimal binary linear codes that violate the Ashikhmin-Barg bound while achieving a great variety of the ratio \(w_{{\min \limits }}/w_{{\max \limits }}\) . Our first generic method employs suitable characteristic functions with relatively low weights within the range [n + 1,2n− 2]. The second approach specifies characteristic functions with weights in [2n− 2 + 1,2n− 2 + 2n− 3 − 1], whose supports contain a skewed (removing one element) affine subspace of dimension n − 2. Finally, we also characterize an infinite family of minimal codes based on the class of so-called root Boolean functions of weight 2n− 1 − (n − 1), useful in specific hardware testing applications. Consequently, many infinite classes of minimal codes crossing the Ashikhmin-Barg bound are derived from an ample range of characteristic functions. In certain cases, we completely specify the weight distributions of the resulting codes. PubDate: 2021-09-01
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Abstract: Nikova et al. investigated the decomposition problem of power permutations over finite fields \(\mathbb {F}_{2^{n}}\) in (Cryptogr. Commun. 11:379–384, 2019). In particular, they provided an algorithm to give a decomposition of a power permutation into quadratic power permutations. Their algorithm has a precomputation step that finds all cyclotomic classes of \(\mathbb {F}_{2^{n}}\) and then use the quadratic ones. In this paper, we provide an efficient and systematic method to generate the representatives of quadratic cyclotomic classes and hence reduce the complexity of the precomputation step drastically. We then apply our method to extend their results on shortest quadratic decompositions of \(x^{2^{n}-2}\) from 3 ≤ n ≤ 16 to 3 ≤ n ≤ 24 and correct a typo (for n = 11). We also give two explicit formulas for the time complexity of the adaptive search to understand its efficiency with respect to the parameters. PubDate: 2021-09-01
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Abstract: For any positive integer m > 2 and an odd prime p, let \(\mathbb {F}_{p^{m}}\) be the finite field with pm elements and let \( \text {Tr}^{m}_{e}\) be the trace function from \(\mathbb {F}_{p^{m}}\) onto \(\mathbb {F}_{p^{e}}\) for a divisor e of m. In this paper, for the defining set \(D=\{x\in \mathbb {F}_{p^{m}}:\text {Tr}^{m}_{e}(x)=1\text { and } \text {Tr}^{m}_{e}(x^{2})=0\}=\{d_{1}, d_{2}, \ldots , d_{n}\}\) (say), we define a pe-ary linear code \(\mathcal {C}_{D}\) by $$ \mathcal{C}_{D}=\{\textbf{c}_{a} =\left( \text{Tr}^{m}_{e}(ad_{1}), \text{Tr}^{m}_{e}(ad_{2}),\ldots,\text{Tr}^{m}_{e}(ad_{n})\right) : a\in \mathbb{F}_{p^{m}}\}. $$ Then we determine the complete weight enumerator and weight distribution of the linear code \(\mathcal {C}_{D}\) . The presented code is optimal with respect to the Griesmer bound provided that \(\frac {m}{e}=3\) . In fact, it is MDS when \(\frac {m}{e}=3\) . This paper gives the results of S. Yang, X. Kong and C. Tang (Finite Fields Appl. 48 (2017)) if we take e = 1. In addition to the generalization of the results of Yang et al., we study the dual code \(\mathcal {C}_{D}^{\perp }\) of the code \(\mathcal {C}_{D}\) as well as find some optimal constant composition codes. PubDate: 2021-09-01
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Abstract: Asymmetric quantum error-correcting codes (AQECCs) are an extremely efficient coding scheme against the different occurring probabilities of qubit-flip and phase-shift errors in quantum asymmetric channel. It is generally known that good quantum codes play a decisive role in ensuring the reliability and the authenticity of quantum communication. In this paper, our main objective is to obtain good AQECCs from quasi-cyclic (QC) codes over small fields, which will effectively fill some gaps of the constructions of AQECCs. At first, a suitable family of r-generator QC codes is proposed and their parameters are determined. Moreover, we show that their dual codes are 1-generator QC codes. In particular, when r = 2 and 3, we embed these dual codes into 2-generator and 3-generator QC codes, and obtain two pairs of nested codes. Then two explicit constructions of AQECCs from QC codes are presented, respectively. As for computational results, many new binary and ternary AQECCs with good parameters are constructed, which all can not be deduced by the asymmetric quantum Gilbert-Varshamov (GV) bound in Matsumoto (Quantum Inf. Process. 16, 1–7, 2017). PubDate: 2021-09-01
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Abstract: In this work, we study a new family of rings, \({\mathscr{B}}_{j,k}\) , whose base field is the finite field \({\mathbb {F}}_{p^{r}}\) . We study the structure of this family of rings and show that each member of the family is a commutative Frobenius ring. We define a Gray map for the new family of rings, study G-codes, self-dual G-codes, and reversible G-codes over this family. In particular, we show that the projection of a G-code over \({\mathscr{B}}_{j,k}\) to a code over \({\mathscr{B}}_{l,m}\) is also a G-code and the image under the Gray map of a self-dual G-code is also a self-dual G-code when the characteristic of the base field is 2. Moreover, we show that the image of a reversible G-code under the Gray map is also a reversible \(G^{2^{j+k}}\) -code. The Gray images of these codes are shown to have a rich automorphism group which arises from the algebraic structure of the rings and the groups. Finally, we show that quasi-G codes, which are the images of G-codes under the Gray map, are also Gs-codes for some s. PubDate: 2021-09-01
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Abstract: We study the notion of formal self duality in finite abelian groups. Formal duality in finite abelian groups has been proposed by Cohn, Kumar, Reiher and Schürmann. In this paper we give a precise definition of formally self dual sets and discuss results from the literature in this perspective. Also, we discuss the connection to formally dual codes. We prove that formally self dual sets can be reduced to primitive formally self dual sets similar to a previously known result on general formally dual sets. Furthermore, we describe several properties of formally self dual sets. Also, some new examples of formally self dual sets are presented within this paper. Lastly, we study formally self dual sets of the form \(\{(x,F(x)) \ : \ x\in {\mathbb {F}}_{2^{n}}\}\) where F is a vectorial Boolean function mapping \({\mathbb {F}}_{2^{n}}\) to \({\mathbb {F}}_{2^{n}}\) . PubDate: 2021-09-01
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Abstract: Quantum synchronizable codes (QSCs) are special quantum error-correcting codes that can correct the effects of both quantum noise on qubits and misalignment in block synchronization. In this paper, using the factorizations of cyclotomic polynomials \({{\varPhi }}_{p_{1}p_{2}}(x)\) , where p1 and p2 are distinct odd primes, we give a new method to construct QSCs whose synchronization capabilities can reach the best attainable tolerance against misalignment. PubDate: 2021-09-01
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Abstract: A weighing matrix W = (wi,j) is a square matrix of order n and entries wi,j in {0,± 1} such that WWT = kIn. In his thesis, Strassler gave a table of existence results for circulant weighing matrices with n ≤ 200 and k ≤ 100. In the latest version of Strassler’s table given by Tan, there are 34 open cases remaining. In this paper we give nonexistence proofs for 12 of these cases, report on preliminary searches outside Strassler’s table, and characterize the known proper circulant weighing matrices. PubDate: 2021-09-01
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Abstract: Espresso cipher is designed targeting 5G wireless communication systems. To achieve high efficiency, a maximum period Galois NLFSR is used as the only building block. The Galois NLFSR is constructed by a scalable method which converts a maximum LFSR to a Galois NLFSR. Based on this method, a new class of stream ciphers, namely maximum period Galois NLFSR-based stream ciphers can be designed. However, we identify a conditional equivalence problem in the design method and adopt the Type-II-to-Fibonacci transformation algorithm. We apply the algorithm to the Espresso cipher and successfully transform the Galois NLFSR to a Fibonacci LFSR with a nonlinear output function. The Espresso cipher is transformed to an LFSR filter generator. We break it by the fast algebraic attack and the Rønjom-Helleseth attack with complexity of 268.50 and 248.59 logical operations respectively. Moreover, we show that the entire class of maximum period Galois NLFSR-based stream ciphers can be transformed to LFSRs. Therefore, this kind of cipher is always equivalent to an LFSR filter generator. We discuss other related attacks and give suggestions for future design. PubDate: 2021-09-01
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Abstract: A 3-phase Golay complementary triad (GCT) is a set of three sequences over the 3-phase alphabet {1, ω, ω2}, where \(\omega =e^{\frac {2\pi \sqrt {-1}}{3}}\) is a 3rd root of unity, whose aperiodic autocorrelations sum up to zero for each out-of-phase non-zero time-shifts. Recent results by Avis and Jedwab proved the non-existence of 3-phase GCTs of length N ≡ 4 (mod 6). In this paper, we introduce 3-phase Z-complementary triads (ZCTs) and almost-complementary triads (ACTs). We present systematic constructions of 3-phase ZCTs for various lengths including the case when N ≡ 4 (mod 6). We also analyse the peak-to-mean envelope power ratio (PMEPR) upper-bounds of the proposed ZCTs and ACTs. PubDate: 2021-09-01
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Abstract: A formula discovered by L. Carlitz in 1935 finds an interesting application in permutation rational functions of finite fields. It allows us to determine all rational functions of degree three that permute the projective line \(\mathbb {P}^{1}(\mathbb {F}_{q})\) over \(\mathbb {F}_{q}\) , a result previously obtained by Ferraguti and Micheli through a different method. It also allows us to determine all rational functions of degree four that permute \(\mathbb {P}^{1}(\mathbb {F}_{q})\) under a certain condition. (A complete determination of all rational functions of degree four that permute \(\mathbb {P}^{1}(\mathbb {F}_{q})\) without any condition will appear in a separate forthcoming paper.) PubDate: 2021-09-01
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