Low confusion coefficient values can make side-channel attacks harder for vector Boolean functions in Block cipher. In this paper, we give new results of confusion coefficient for f ⊞ g, f ⊡ g, f ⊕ g ...and fg for different Boolean functions f and g, respectively. And we deduce a relationship on the sum-of-squares of the confusion coefficient between one n-variable function and two (n - 1)-variable decomposition functions. Finally, we find that the confusion coefficient of vector Boolean functions is affine invariant.
Let f be a Boolean function in n variables. The Möbius transform and its converse of f can describe the transformation behaviors between the truth table of f and the coefficients of the monomials in ...the algebraic normal form representation of f. In this letter, we develop the Möbius transform and its converse into a more generalized form, which also includes the known result given by Reed in 1954. We hope that our new result can be used in the design of decoding schemes for linear codes and the cryptanalysis for symmetric cryptography. We also apply our new result to verify the basic idea of the cube attack in a very simple way, in which the cube attack is a powerful technique on the cryptanalysis for symmetric cryptography.
Mutually orthogonal complementary sets (MOCSs) have received significant research attention in recent years due to their wide applications in communications and radar. Existing MOCSs which are ...constructed based on generalized Boolean functions (GBFs) mostly have lengths of power-of-two. How to construct MOCSs with non-power-of-two lengths whilst having large set sizes is a largely open problem. With the aid of GBFs, in this paper, we present new constructions of such MOCSs and show that the maximal achievable set size is 1/2 of the flock size of an MOCS.
The existence of almost perfect nonlinear (APN) permutations operating on an even number of variables was a long-standing open problem, until an example with six variables was exhibited by Dillon et ...al. in 2009. However it is still unknown whether this example can be generalized to any even number of inputs. In a recent work, Perrin et al. described an infinite family of permutations, named butterflies, operating on (4k+2) variables and with differential uniformity at most 4, which contains the Dillon APN permutation. In this paper, we generalize this family, and we completely solve the two open problems raised by Perrin et al. Indeed we prove that all functions in this larger family have the best known nonlinearity. We also show that this family does not contain any APN permutation besides the Dillon permutation, implying that all other functions have differential uniformity exactly four.
The high peak-to-mean envelope power ratio (PMEPR) is a major problem in multicarrier code-division multiple-access (MC-CDMA) system. In this letter, we have proposed a new optimal Z-complementary ...code set (ZCCS) using higher-order (<inline-formula> <tex-math notation="LaTeX">\geq 2 </tex-math></inline-formula>) generalized Boolean functions (GBFs). The proposed ZCCS has maximum column sequence PMEPR of 2 and it achieves the theoretical upper bound of optimality.
In this letter, a new construction of Z-complementary pairs (ZCPs) with more flexible lengths based on generalized Boolean functions is proposed. Except for the powers of two, the proposed ZCPs exist ...for all lengths with various widths of the zero correlation zone (ZCZ). There exists a trade-off between the sequence lengths and the ZCZ widths of the proposed ZCPs. Moreover, the peak-to-average power ratio (PAPR) properties of the constructed ZCPs are investigated in this letter. Therefore, the new sequences are useful in applications in communications due to their flexible lengths, various possible ZCZ widths, and good PAPR properties.
Reed-Muller codes are error-correcting codes used in many areas related to coding theory, such as electrical engineering and computer science. The binary <inline-formula> <tex-math ...notation="LaTeX">r^{th} </tex-math></inline-formula>-order Reed-Muller code <inline-formula> <tex-math notation="LaTeX">RM(r,n) </tex-math></inline-formula> can be viewed as the set of all <inline-formula> <tex-math notation="LaTeX">n </tex-math></inline-formula>-variable Boolean functions of algebraic degree at most <inline-formula> <tex-math notation="LaTeX">r </tex-math></inline-formula>. Despite the intense work on these codes, many problems are known to be hard (notably, determining their covering radius) and remain open to this day. Fourteen years ago, Carlet and Mesnager improved in IEEE Transactions on Information Theory, "Improving the Upper Bounds on the Covering Radii of Binary Reed-Muller Codes", 53(1), 2007 the upper bound on the covering radius of the Reed-Muller code of order 2, and they deduced improved upper bounds on the covering radii of the Reed-Muller codes of higher orders. Until 2021, these upper bounds remained the best ones in the literature. The Reed-Muller code <inline-formula> <tex-math notation="LaTeX">RM(n-3,n) </tex-math></inline-formula>, which corresponds to the dual of the Reed-Muller code <inline-formula> <tex-math notation="LaTeX">RM(2,n) </tex-math></inline-formula>, has attracted much attention. One of the main reasons is that it is precisely the code that has been considered to get the upper bounds derived by Carlet and Mesnager. Those upper bounds have been obtained thanks to the characterization of the codewords of the Reed-Muller code, whose Hamming weights are strictly less than 2.5 times the minimum distance <inline-formula> <tex-math notation="LaTeX">2^{n-r} </tex-math></inline-formula> due to Kasami, Tokura, and Azumi. Despite their impressive work in the seventieth, a more refined study and profound description of those codewords of <inline-formula> <tex-math notation="LaTeX">{RM}({n-3},{n}) </tex-math></inline-formula> whose Hamming weight equals 16, and especially 18, seem necessary, as it could help us significantly in improving the covering radius of Reed-Muller codes. In this paper, we push further the known results on the Reed-Muller codes by focusing on the Reed-Muller code <inline-formula> <tex-math notation="LaTeX">{RM}({n-3},{n}) </tex-math></inline-formula>. We provide a classification of the codewords of weights 16 and 18 of the Reed-Muller code <inline-formula> <tex-math notation="LaTeX">RM(n-3,n) </tex-math></inline-formula>. Our algebraic descriptions allow us to count the number of such codewords and to enumerate all of them explicitly.
In this paper, we firstly introduce a kind of weightwise almost perfectly balanced Boolean functions. And then, a construction of weightwise perfectly balanced Boolean functions on 2q+2 variables is ...given by modifying the support of the weightwise almost perfectly balanced functions, where q is a non-negative integer. The algebraic degree, the weightwise nonlinearity, and the algebraic immunity of the newly constructed weightwise perfectly balanced functions are discussed at the end of this paper.
Non-orthogonal multiple access (NOMA) is a promising technology for massive machine-type communications (mMTC). This letter focuses on constructing spreading sequences with low coherence as well as ...low peak-to-average power ratio (PAPR) for uplink grant-free NOMA. Recently, a framework of spreading matrices from Boolean functions was provided by Yu. Exploiting this framework, we first show that optimum coherence of the constructed spreading matrices can be achieved if the difference of any two distinct Boolean functions is a bent (or almost bent) function. With this result, explicit constructions of Boolean functions are then developed, which lead to new binary Golay spreading matrices with optimum coherence and yielding PAPR of each spreading sequence at most 2. Simulation results demonstrate that such spreading sequences are suitable for uplink grant-free access. Moreover, the proposed Golay spreading sequences are capable of accommodating up to 400% overloaded devices, which can not be achieved from the existing algebraic methods.
Designing Boolean functions whose output can be computed with light means at high speed, and satisfying all the criteria necessary to resist all major attacks on the stream ciphers using them as ...nonlinear components, has been an open problem since the beginning of this century, when algebraic attacks were invented. Functions allowing a good resistance are known since 2008, but their output is a little too complex to compute. Functions with fast and easy to compute output are known which have good algebraic immunity, such as majority functions and the so-called hidden weight bit (HWB) functions, but they all have the same cryptographic weakness: their too small nonlinearity. In the present paper, we introduce a generalization of the HWB functions into a construction of <inline-formula> <tex-math notation="LaTeX">n </tex-math></inline-formula>-variable balanced functions <inline-formula> <tex-math notation="LaTeX">f </tex-math></inline-formula> from <inline-formula> <tex-math notation="LaTeX">(n-1) </tex-math></inline-formula>-variable Boolean functions <inline-formula> <tex-math notation="LaTeX">g </tex-math></inline-formula> having some property held by a large number of functions. Function <inline-formula> <tex-math notation="LaTeX">f </tex-math></inline-formula> is defined by its support, equal to the image set of a vectorial function depending on <inline-formula> <tex-math notation="LaTeX">g </tex-math></inline-formula>. This makes the function complex enough for allowing good cryptographic parameters, while its output is light to compute. The HWB function is what we obtain with <inline-formula> <tex-math notation="LaTeX">f </tex-math></inline-formula> when the initial function <inline-formula> <tex-math notation="LaTeX">g </tex-math></inline-formula> equals constant 1. Other well chosen functions <inline-formula> <tex-math notation="LaTeX">g </tex-math></inline-formula> provide functions <inline-formula> <tex-math notation="LaTeX">f </tex-math></inline-formula> having good cryptographic parameters. We analyze the constructed functions <inline-formula> <tex-math notation="LaTeX">f </tex-math></inline-formula>, we provide a fast way to compute their output, we determine their algebraic normal forms and we show that, most often, their algebraic degree is optimal. We study their Walsh transform and their nonlinearity and algebraic immunity. We observe with computer investigations that this generalization of the HWB function allows to keep its quality of being fast to compute and having good enough algebraic immunity, while significantly improving its nonlinearity. The functions already obtained in the investigations provide a quite good (and never reached before) trade-off between speed and security. Further (probably difficult) work should allow obtaining, among such generalized HWB functions whose number is huge, still better filter functions to be used in stream ciphers.
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