The researches of Golay complementary sequences (GCSs) over 16 and 64 quadrature amplitude modulation (QAM) from 2001 to 2008 were generalized to <inline-formula> <tex-math notation="LaTeX">4^{q} ...</tex-math></inline-formula>-QAM GCSs of length <inline-formula> <tex-math notation="LaTeX">2^{m} </tex-math></inline-formula> by Li (the generalized cases I-III for <inline-formula> <tex-math notation="LaTeX">q\ge 2 </tex-math></inline-formula>) in 2010 and Liu et al. (the generalized cases IV-V for <inline-formula> <tex-math notation="LaTeX">q\ge 3 </tex-math></inline-formula>) in 2013. Those sequences are presented by the combination of the quaternary standard GCSs and compatible offsets. By providing new compatible offsets based on the factorization of the integer <inline-formula> <tex-math notation="LaTeX">q </tex-math></inline-formula>, we propose two new constructions of <inline-formula> <tex-math notation="LaTeX">4^{q} </tex-math></inline-formula>-QAM GCSs, which have the generalized cases I-V as special cases. The numbers of the proposed GCSs are equal to the product of the number of the quaternary standard GCSs and the number of the compatible offsets. Denote the number of prime factors of <inline-formula> <tex-math notation="LaTeX">q </tex-math></inline-formula> counted with multiplicity by <inline-formula> <tex-math notation="LaTeX">\Omega (q) </tex-math></inline-formula>. The number of new offsets in our first construction is lower bounded by a polynomial of <inline-formula> <tex-math notation="LaTeX">m </tex-math></inline-formula> with degree <inline-formula> <tex-math notation="LaTeX">\Omega (q) </tex-math></inline-formula>, while the numbers of offsets in the generalized cases I-III and IV-V are a linear polynomial and a quadratic polynomial of <inline-formula> <tex-math notation="LaTeX">m </tex-math></inline-formula>, respectively. If <inline-formula> <tex-math notation="LaTeX">q </tex-math></inline-formula> has a prime factor larger than 2, the number of new offsets in our second construction is lower bounded by a polynomial of <inline-formula> <tex-math notation="LaTeX">m </tex-math></inline-formula> with degree <inline-formula> <tex-math notation="LaTeX">\Omega (q)+1 </tex-math></inline-formula>. As an example, the new offsets in our two constructions for <inline-formula> <tex-math notation="LaTeX">q=6 </tex-math></inline-formula>, whose number is bounded by a cubic polynomial, is also given. The proof in this paper implies that all the mentioned GCSs over QAM can be regarded as projections of Golay complementary arrays of size <inline-formula> <tex-math notation="LaTeX">2\times 2\times \cdots \times 2 </tex-math></inline-formula>.
Golay complementary pairs (GCPs) and complete complementary codes (CCCs) have found a wide range of practical applications in coding, signal processing and wire-less communication due to their ideal ...correlation properties. The binary CCCs have special advantages in spread spectrum communication for their simple modulo-2 arithmetic operation, modulation, and correlation simplicity; however, they are limited in length. In this paper, we present a direct construction of GCPs, mutually orthogonal Golay complementary sets (MOGCSs) and q-ary, including binary CCCs of non-power of two lengths to widen their application in the recent communication field. First, a generalised Boolean function (GBF) based truncation technique is used to construct GCPs of non-power of two lengths which has never been reported before. Then Golay complementary sets (GCSs) and MOGCSs of lengths of the form 2 m -1 + 2 m -3 ( m ≥ 5) and 2 m -1 + 2 m -2 + 2 m -4 ( m ≥ 6) are generated by higher order GBFs. The later length of MOGCSs with direct construction is not available in the literature. Finally, q -ary sequences including binary CCCs with non-power of two lengths are constructed using the union of MOGCSs. There is no such direct construction of binary CCC of non-power of two lengths available in the literature. The column sequence peak to mean envelope power ratio (PMEPR) have been investigated for GCSs and CCCs respectively, and compared with existing works. The column sequence PMEPR of resultant CCCs is effectively upper bounded by 2, which is much lower than that of paraunitary based CCC construction. The proposed construction is also compared with existing works.
When constructing block ciphers, it is necessary to use vector Boolean functions with special cryptographic properties as S-blocks for the cipher’s resistance to various types of cryptanalysis. In ...this paper, we investigate the following S-block construction: let
be a permutation on
elements, let
be the
-fold application of the permutation
, and let
be a Boolean function of
variables. Define a vector Boolean function
as
. We study the cryptographic properties of
such as high nonlinearity, balancedness, and low differential
-uniformity in the dependence on the properties of
and
for small
. Complete sets of Boolean functions
and vector Boolean functions
of few variables with maximum algebraic immunity are also obtained.
Permutation polynomials have important applications in cryptography, coding theory and combinatorial designs. In this letter, we construct four classes of permutation polynomials over 𝔽2n × 𝔽2n , ...where 𝔽2n is the finite field with 2n elements.
Minimal linear codes have important applications in secure communications, including in the framework of secret sharing schemes and secure multi-party computation. A lot of research have been carried ...out to derive codes with few weights (but more importantly, being minimal) using algebraic or geometric approaches. One of the main power and fructify algebraic methods is based on the design of those codes by employing functions over finite fields. K. Li, C. Li, T. Helleseth, and L. Qu have recently identified in Binary linear codes with few weights from two-to-one functions, IEEE Trans. Inf. Theory, 2021 some binary linear codes with few weights from two classes of two-to-one functions. In this paper, our ultimate objective is to expand the class of codes derived from the paper of Li et al . by proposing larger classes of binary linear codes with few weights via generic constructions involving other known families of two-to-one functions over the finite field F 2 n of order 2 n . We succeed in constructing such codes, and we also completely determine their weight distributions. The linear codes presented in this paper differ in parameters from those known in the literature. Besides, some of them are optimal concerning the well-known Griesmer bound. Notably, we prove that our codes are either optimal or almost optimal with respect to the online Database of Grassl. We next observe that the derived binary linear codes also have the minimality property for most cases. We then describe the access structures of the secret-sharing schemes based on their dual codes. Finally, we solve two problems left open in the paper by Li et al . (more specifically, a complete solution to Problem 2 and a partial solution to Problem 1).
Boolean functions are usually studied under the assumption that each input bit is considered independent and identically distributed. However, in the case of some stream ciphers, a keystream bit is ...generated by using a nonlinear Boolean function with inputs from a restricted domain. At Eurocrypt 2016, one such stream cipher (FLIP) has been proposed, where a Boolean function on n variables was exploited with inputs of weight n/2 only. Recently, Carlet et al. studied several properties of such functions and obtained certain bounds on linear approximations of direct sum in the restricted domain. In this paper, we observe that for a direct sum like f = f 1 + f 2 , the inputs to each sub-function f 1 , f 2 do not follow a uniform distribution in the restricted domain. In this regard, we study the properties of the Boolean functions by considering a general probability distribution on the inputs. We further obtain several bounds related to the biases of direct sums. Finally, we obtain a lower bound on the bias of the nonlinear filter function of FLIP. Our results provide a general framework to study security parameters of ciphers over restricted domain.
The autocorrelation properties of Boolean functions are closely related to the Shannon’s concept of diffusion and can be accompanied with other cryptographic criteria (such as high nonlinearity and ...algebraic degree) for ensuring an overall robustness to various cryptanalytic methods. In a series of recent articles 14,9,15, the design methods of n-variable balanced Boolean functions n is strictly even) with small absolute indicator Δf < 2n/2 have been considered. Whereas the two first articles managed to solve this problem for relatively large n⩾46, a recent approach 15 has introduced a generic design framework achieving Δf < 2n/2 for even n⩾22. Based on a suitable modification of the method of Rothaus, used to construct new bent functions from known ones, we provide a generic iterative framework for designing balanced functions satisfying the condition Δf < 2n/2 and having overall good cryptographic properties for any even n⩾12. Even though the problem of specifying functions having Δf < 2n/2 for smaller n has been considered in 14,9,15 using various search algorithms, our method for the first time provides relatively simple iterative framework for variable spaces of more practical interest. Moreover, our approach can be efficiently applied to certain classes of initial functions (derived from partial spread bent functions) for deriving balanced functions with Δf < 2n/2 for relatively large n, namely for n⩾48 satisfying n≡0 mod 4 and n⩾54 with n≡2 mod 4. In the latter case, our nonlinearity bound is better than the one presented in 14.
In this letter, a novel construction of binary cross Z-complementary pairs (CZCPs) based on Boolean functions is proposed. CZCPs have been proposed to be used as training sequences in the spatial ...modulation system. Due to their zero correlation zone (ZCZ) properties for autocorrelation sums and cross-correlation sums, optimal channel estimation performance over frequency-selective channels can be achieved. In the literature, most constructions of CZCPs are based on existing Golay complementary pairs (GCPs); only one method is based on Boolean functions. However, the length of CZCPs from Boolean functions is very limited. In this letter, the proposed construction can generate CZCPs with more flexible lengths and large ZCZ widths.
This paper aims to present novel constructions of Golay-ZCZ sequence sets with various set sizes and flexible lengths. A Golay-ZCZ sequence set is not only a Golay complementary set (GCS) but also a ...zero correlation zone (ZCZ) sequence set. Golay-ZCZ sequence sets have been employed in OFDM systems due to their low peak-to-average power ratios (PAPRs) which are upper bounded by the set sizes. Although several constructions of Golay-ZCZ sequence sets have been proposed in the literature, the known Golay-ZCZ sets cannot attain the theoretical upper bound on the ZCZ width for non-binary cases. Also, for the Golay-ZCZ sets constructed by generalized Boolean functions, parameters are limited to powers of two, such as set sizes, sequence lengths, and ZCZ widths. In this paper, constructions of Golay-ZCZ sequence sets based on the extended generalized Boolean functions are proposed, where set sizes, sequence lengths, and ZCZ widths are all flexible and not necessary to be power-of-two. Since the constructed Golay-ZCZ sets have various set sizes, their PAPR upper bounds can be tighter. In addition, the proposed Golay-ZCZ sequence sets can achieve the upper bound on the ZCZ width. Therefore, optimal and almost-optimal Golay-ZCZ sequence sets can be obtained by the proposed methods.
The notion of correlation immune functions has been introduced by Siegenthaler (1984) in symmetric cryptography in the framework of stream ciphers. At the conference CRYPTO'91 by Camion et al., it ...has been pointed out that this notion existed in statistics and combinatorics. It has recently been highlighted that such functions also play an important role in a new framework related to side-channel attack counter-measures. Since then, the interest in correlation immune Boolean functions has been renewed, and new challenges regarding these functions have appeared. Specifically, low Hamming weight correlation immune functions have been selected as useful for counter-measures to side-channel attacks. Despite their importance, the literature is not abundant in this research direction. Two very interesting articles in which such correlation immune functions were nicely explored, given this novel use of them. Carlet initiated the first one in 2013, and the second one is due to Carlet and Chen (2018). This paper deals with correlation immune Boolean functions aiming to produce more candidates of those processing low Hamming weights. We shall focus on correlation immune Boolean functions with Hamming weights power of 2 (which offer a flexibility to control the correlation immunity aspects) and present several methods of designing them. Some design methods are efficient and could be employed to derive such functions. Consequently, given two positive integers <inline-formula> <tex-math notation="LaTeX">n </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">m </tex-math></inline-formula>, we derive new effective constructions of correlation immune Boolean functions with Hamming weight power of 2. Furthermore, an upper bound on the correlation immunity of the newly constructed <inline-formula> <tex-math notation="LaTeX">n </tex-math></inline-formula>-variable Boolean functions with Hamming weight <inline-formula> <tex-math notation="LaTeX">2^{m} </tex-math></inline-formula> was determined for <inline-formula> <tex-math notation="LaTeX">n-m\ge 0 </tex-math></inline-formula>. Besides, exact values and lower bounds on the maximum correlation immunity of those functions are explored and discussed, mainly when the values of <inline-formula> <tex-math notation="LaTeX">n </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">m </tex-math></inline-formula> are very close. This paper also exhibits explicit examples of those correlation immune functions that illustrate our methods.
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