This paper aims to construct optimal Z-complementary code set (ZCCS) with non-power-of-two (NPT) lengths to enable interference-free multicarrier code-division multiple access (MC-CDMA) systems. The ...existing ZCCSs with NPT lengths, which are constructed from generalized Boolean functions (GBFs), are sub-optimal only with respect to the set size upper bound. For the first time in the literature, we advocate the use of pseudo-Boolean functions (PBFs) (each of which transforms a number of binary variables to a real number as a natural generalization of GBF) for direct constructions of optimal ZCCSs with NPT lengths.
Cryptographic Boolean functions play an important role in the design of symmetric ciphers. Many cryptographic criteria such as balancedness, nonlinearity, correlation immunity and transparency order ...are connected with the Walsh support of a Boolean function. However, we still know little about the possible structure of the Walsh supports of Boolean functions. In 2005, Carlet and Mesnager studied the Walsh supports of Boolean functions and constructed a class of n-variable Boolean functions whose Walsh support is F2n∖{0}, for n≥10. For n≤6, it can be verified using the computer that there is no Boolean function with the Walsh Support F2n∖{0}. However, concerning the values of n=7,8,9, it has been an open problem for many years. In this paper, we construct two classes of balanced Boolean functions with the maximum possible Walsh support F2n∖{0}, and partially solve this problem. The first class of functions are of odd variables with n≥9, and the second class of functions are constructed based on the Maiorana–McFarland bent functions, which are of even variables with n≥8. As a result, the above open problem has been settled for n=8,9, and the only unsolved case is n=7.
This paper is to design substitution boxes (S-Boxes) using innovative I-Ching operators (ICOs) that have evolved from ancient Chinese I-Ching philosophy. These three operators-intrication, turnover, ...and mutual- inherited from I-Ching are specifically designed to generate S-Boxes in cryptography. In order to analyze these three operators, identity, compositionality, and periodicity measures are developed. All three operators are only applied to change the output positions of Boolean functions. Therefore, the bijection property of S-Box is satisfied automatically. It means that our approach can avoid singular values, which is very important to generate S-Boxes. Based on the periodicity property of the ICOs, a new network is constructed, thus to be applied in the algorithm for designing S-Boxes. To examine the efficiency of our proposed approach, some commonly used criteria are adopted, such as nonlinearity, strict avalanche criterion, differential approximation probability, and linear approximation probability. The comparison results show that S-Boxes designed by applying ICOs have a higher security and better performance compared with other schemes. Furthermore, the proposed approach can also be used to other practice problems in a similar way.
Minimal Binary Linear Codes Ding, Cunsheng; Heng, Ziling; Zhou, Zhengchun
IEEE transactions on information theory,
2018-Oct., 2018-10-00, Volume:
64, Issue:
10
Journal Article
Peer reviewed
In addition to their applications in data communication and storage, linear codes also have nice applications in combinatorics and cryptography. Minimal linear codes, a special type of linear codes, ...are preferred in secret sharing. In this paper, a necessary and sufficient condition for a binary linear code to be minimal is derived. This condition enables us to obtain three infinite families of minimal binary linear codes with <inline-formula> <tex-math notation="LaTeX">w_{\min }/w_{\max } \leq 1/2 </tex-math></inline-formula> from a generic construction, where <inline-formula> <tex-math notation="LaTeX">w_{\min } </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">w_{\max } </tex-math></inline-formula>, respectively, denote the minimum and maximum nonzero weights in a code. The weight distributions of all these minimal binary linear codes are also determined.
Boolean functions with high algebraic immunity are important cryptographic primitives in some stream ciphers. In this paper, two methodologies for constructing minimal binary codes from sets, Boolean ...functions and vectorial Boolean functions with high algebraic immunity, are proposed. More precisely, a general construction of new minimal codes using minimal codes contained in Reed-Muller codes and sets without nonzero low degree annihilators is presented. The other construction allows us to yield minimal codes from certain subcodes of Reed-Muller codes and vectorial Boolean functions with high algebraic immunity. Via these general constructions, infinite families of minimal binary linear codes of dimension <inline-formula> <tex-math notation="LaTeX">m </tex-math></inline-formula> and length less than or equal to <inline-formula> <tex-math notation="LaTeX">m(m+1)/2 </tex-math></inline-formula> are obtained. Besides, a lower bound on the minimum distance of the proposed minimal linear codes is established. Conjectures and open problems are also presented. The results of this paper show that Boolean functions with high algebraic immunity have nice applications in several fields additionally to symmetric cryptography, such as coding theory and secret sharing schemes.
As the only nonlinear module, S-Box (substitution box) is widely used in stream cipher, hash function, and so on. However, there exist some weaknesses in existing S-Boxes, such as fixed point, ...reverse fixed point and short iteration period rings. Furthermore, the S-Boxes in AES and SM4.0 were fixed, which makes them vulnerable to attack. To address these weaknesses, first, a non-degenerate 2D chaotic map (2D-NDCM) with ergodicity in phase space was constructed. Second, based on 2D-NDCM, the seed S-Boxes with high nonlinearity were constructed in batches by affine transformation and Boolean function, and then three possible weaknesses were eliminated to obtain strong keyed S-Boxes. Statistical analysis demonstrated the effectiveness and practicability of the proposed S-Box batch generating algorithm.
<inline-formula> <tex-math notation="LaTeX">Z </tex-math></inline-formula>-complementary code set (ZCCS), an extension of perfect CCs, refers to a set of 2-D matrices having zero correlation zone ...properties. ZCCS can be used in various multi-channel systems to support, for example, quasi-synchronous interference-free multicarrier code-division multiple access communication and optimal channel estimation in multiple-input multiple-output systems. Traditional constructions of ZCCS heavily rely on a series of sequence operations which may not be feasible for rapid hardware generation particularly for long ZCCSs. In this paper, we propose a direct construction of ZCCS using the second-order Reed-Muller codes with efficient graphical representation. Our proposed construction, valid for any number of isolated vertices present in the graph, is capable of generating optimal ZCCS meeting the set size upper bound.
Boolean functions play an important role in symmetric ciphers. One of important open problems on Boolean functions is determining the maximum possible resiliency order of n-variable Boolean functions ...with optimal algebraic immunity. In this letter, we search Boolean functions in the rotation symmetric class, and determine the maximum possible resiliency order of 9-variable Boolean functions with optimal algebraic immunity. Moreover, the maximum possible nonlinearity of 9-variable rotation symmetric Boolean functions with optimal algebraic immunity-resiliency trade-off is determined to be 224.
Nonlinear feedback shift registers (NFSRs) are used in many stream ciphers as their main building blocks. In particular, Galois NFSRs with terminal bits are used in typical stream ciphers Grain and ...Trivium. Seven types of Galois NFSRs have been found equivalent to Fibonacci ones, among which three types are particular cases of another type of lower triangular Galois NFSRs. This paper continues the research of equivalence between Galois NFSRs and Fibonacci ones. It first enumerates the Galois NFSRs with terminal bits that are equivalent to a Fibonacci NFSR. It then discloses n-stage (n−2)-terminal-bit Galois NFSRs equivalent to Fibonacci ones, must be lower triangular Galois NFSRs. Finally, it presents two new types of Galois NFSRs that are equivalent to Fibonacci NFSRs. Some examples show that, compared to a Fibonacci NFSR, its equivalent Galois NFSR from our new types may decrease the area and increase the throughput of the circuits implementing feedback functions.
•Two NFSRs are said to be equivalent if their sets of output sequences are equal.•The Galois NFSRs with terminal bits equivalent to a given Fibonacci NFSR are counted.•n-stage (n−2)-terminal-bit Galois NFSRs equivalent to n-stage Fibonacci ones are lower triangular Galois NFSRs.•Two new types of Galois NFSRs equivalent to Fibonacci NFSRs are given.•Both new types may decrease the area and increase the throughput compared to their equivalent Fibonacci NFSRs.
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