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.

The present study proposes a scheme in which variable-length orthogonal codes generated by combining inverse discrete Fourier transform matrices over a finite field multiplex user data into a multiplexed sequence and its sequence forms one or a plural number of codewords for Reed-Solomon coding. The proposed scheme realizes data multiplexing, error correction coding, and multi-rate transmitting at the same time. This study also shows a design example and its performance analysis of the proposed scheme.

We introduce a new type of exponentiation algorithm in GF(2m) using Euclidean inversion. Our approach is based on the fact that Euclidean inversion cost much less logic gates than ordinary multiplication in GF(2m). By applying signed binary form of the exponent instead of classic binary form, the proposed algorithm can reduce the number of operations further compared with the classic algorithms.

A unified construction for yielding optimal and balanced quaternary sequences from ideal/optimal balanced binary sequences was proposed by Zeng et al. In this paper, the linear complexity over finite field 𝔽2, 𝔽4 and Galois ring ℤ4 of the quaternary sequences are discussed, respectively. The exact values of linear complexity of sequences obtained by Legendre sequence pair, twin-prime sequence pair and Hall's sextic sequence pair are derived.

In this paper, we present a scheme to compute either AB or AB2 multiplications over GF(2m) and propose a bit-parallel systolic architecture based on the proposed algorithm. The AB multiplication algorithm is derived in the same form as the formula of AB2 multiplication algorithm, and an architecture that can perform AB multiplication by adding very little extra hardware to AB2 multiplier is designed. Therefore, the proposed architecture can be effectively applied to hardware constrained applications that cannot deploy AB2 multiplier and AB multiplier separately.

The present paper proposes orthogonal variable spreading factor codes over finite fields for multi-rate communications. The proposed codes have layered structures that combine sequences generated by discrete Fourier transforms over finite fields, and have various code lengths. The design method for the proposed codes and examples of the codes are shown.

A construction method of self-orthogonal and self-dual quasi-cyclic codes is shown which relies on factorization of modulus polynomials for cyclicity in this study. The smaller-size generator polynomial matrices are used instead of the generator matrices as linear codes. An algorithm based on Chinese remainder theorem finds the generator polynomial matrix on the original modulus from the ones constructed on each factor. This method enables us to efficiently construct and search these codes when factoring modulus polynomials into reciprocal polynomials.

In this paper, we propose a notion for high-dimensional generalizations of mutually orthogonal Latin squares (MOLS) and mutually orthogonal diagonal Latin squares (MODLS), called mutually dimensionally orthogonal d-cubes (MOC) and mutually dimensionally orthogonal diagonal d-cubes (MODC). Systematic constructions for MOC and MODC by using polynomials over finite fields are investigated. In particular, for 3-dimensional cubes, the results for the maximum possible number of MODC are improved by adopting the proposed construction.

In DNA data storage and computation, DNA strands are required to meet certain combinatorial constraints. This paper shows how some of these constraints can be achieved simultaneously. First, we use the algebraic structure of irreducible cyclic codes over finite fields to generate cyclic DNA codes that satisfy reverse and complement properties. We show how such DNA codes can meet constant guanine-cytosine content constraint by MacWilliams-Seery algorithm. Second, we consider fulfilling the run-length constraint in parallel with the above constraints, which allows a maximum predetermined number of consecutive duplicates of the same symbol in each DNA strand. Since irreducible cyclic codes can be represented in terms of the trace function over finite field extensions, the linearity of the trace function is used to fulfill a predefined run-length constraint. Thus, we provide an algorithm for constructing cyclic DNA codes with the above properties including run-length constraint. We show numerical examples to demonstrate our algorithms generating such a set of DNA strands with all the prescribed constraints.

Hadamard matrix is defined as a square matrix where any components are -1 or +1, and where any pairs of rows are mutually orthogonal. In this work, we consider the similar matrix on finite field GF(p) where p is an odd prime. In such a matrix, every component is one of the integers on GF(p){0}, that is, {1,2,...,p-1}. Any additions and multiplications should be executed under modulo p. In this paper, a method to generate such matrices is proposed. In addition, the paper includes the applications to generate n-shift orthogonal sequences and complete complementary codes. The generated complete complementary code is a family of multi-valued sequences on GF(p){0}, where the number of sequence sets, the number of sequences in each sequence set and the sequence length depend on the various divisors of p-1. Such complete complementary codes with various parameters have not been proposed in previous studies.

Pseudo-random sequences with good statistical property, such as low autocorrelation, high linear complexity and 2-adic complexity, have been widely applied to designing reliable stream ciphers. In this paper, we explicitly determine the 2-adic complexities of two classes of generalized cyclotomic binary sequences with order 4. Our results show that the 2-adic complexities of both of the sequences attain the maximum. Thus, they are large enough to resist the attack of the rational approximation algorithm for feedback with carry shift registers. We also present some examples to illustrate the validity of the results by Magma programs.

In this letter, a construction of sparse measurement matrices is presented. Based on finite fields, a base matrix is obtained. Then a Hadamard matrix or a discrete Fourier transform (DFT) matrix is nested in the base matrix, which eventually formes a new deterministic measurement matrix. The coherence of the proposed matrices is low, which meets the Welch bound asymptotically. Thus these matrices could satisfy the restricted isometry property (RIP). Simulation results demonstrate that the proposed matrices give better performance than Gaussian counterparts.

As a generalization of perfect nonlinear (PN) functions, zero-difference balanced (ZDB) functions play an important role in coding theory, cryptography and communications engineering. Inspired by a foregoing work of Liu et al. [1], we present a class of ZDB functions with new parameters based on the cyclotomy in finite fields. Employing these ZDB functions, we obtain simultaneously optimal constant composition codes and perfect difference systems of sets.

We introduce a new type of Montgomery-like square root formulae in GF(2m) defined by an arbitrary irreducible trinomial, which is more efficient compared with classic square root operation. By choosing proper Montgomery factors for different kind of trinomials, the space and time complexities of such square root computations match or outperform the best results. A practical application of the Montgomery-like square root in inversion computation is also presented.

Compressed sensing theory provides a new approach to acquire data as a sampling technique and makes sure that a sparse signal can be reconstructed from few measurements. The construction of compressed sensing matrices is a main problem in compressed sensing theory (CS). In this paper, the deterministic constructions of compressed sensing matrices based on affine singular linear space over finite fields are presented and a comparison is made with the compressed sensing matrices constructed by DeVore based on polynomials over finite fields. By choosing appropriate parameters, our sparse compressed sensing matrices are superior to the DeVore's matrices. Then we use a new formulation of support recovery to recover the support sets of signals with sparsity no more than k on account of binary compressed sensing matrices satisfying disjunct and inclusive properties.

MICKEY-family ciphers are lightweight cryptographic primitives and include a register R determined by two related maximal-period linear transformations. Provided that primitivity is efficiently decided in finite fields, it is shown by quantitative analysis that potential parameters for R can be found in probabilistic polynomial time.

Multivariate Public Key Cryptosystems (MPKCs) are often touted as future-proofing against Quantum Computers. In 2009, it was shown that hardware advances do not favor just “traditional” alternatives such as ECC and RSA, but also makes MPKCs faster and keeps them competitive at 80-bit security when properly implemented. These techniques became outdated due to emergence of new instruction sets and higher requirements on security. In this paper, we review how MPKC signatures changes from 2009 including new parameters (from a newer security level at 128-bit), crypto-safe implementations, and the impact of new AVX2 and AESNI instructions. We also present new techniques on evaluating multivariate polynomials, multiplications of large finite fields by additive Fast Fourier Transforms, and constant time linear solvers.

In this paper, we consider to develop a recovery algorithm of a sparse signal for a compressed sensing (CS) framework over finite fields. A basic framework of CS for discrete signals rather than continuous signals is established from the linear measurement step to the reconstruction. With predetermined priori distribution of a sparse signal, we reconstruct it by using a message passing algorithm, and evaluate the performance obtained from simulation. We compare our simulation results with the theoretic bounds obtained from probability analysis.

In this paper, we present a new three-way split formula for binary polynomial multiplication (PM) with five recursive multiplications. The scheme is based on a recently proposed multievaluation and interpolation approach using field extension. The proposed PM formula achieves the smallest space complexity. Moreover, it has about 40% reduced time complexity compared to best known results. In addition, using developed techniques for PM formulas, we propose a three-way split formula for Toeplitz matrix vector product with five recursive products which has a considerably improved complexity compared to previous known one.

We propose subquadratic space complexity multipliers for any finite field $mathbb{F}_{q^n}$ over the base field $mathbb{F}_q$ using the Dickson basis, where q is a prime power. It is shown that a field multiplication in $mathbb{F}_{q^n}$ based on the Dickson basis results in computations of Toeplitz matrix vector products (TMVPs). Therefore, an efficient computation of a TMVP yields an efficient multiplier. In order to derive efficient $mathbb{F}_{q^n}$ multipliers, we develop computational schemes for a TMVP over $mathbb{F}_{q}$. As a result, the $mathbb{F}_{2^n}$ multipliers, as special cases of the proposed $mathbb{F}_{q^n}$ multipliers, have lower time complexities as well as space complexities compared with existing results. For example, in the case that n is a power of 3, the proposed $mathbb{F}_{2^n}$ multiplier for an irreducible Dickson trinomial has about 14% reduced space complexity and lower time complexity compared with the best known results.