Field GF Research Articles

The Subfield Codes of Some [q + 1, 2, q] MDS Codes

Recently, subfield codes of geometric codes over large finite fields GF(q) with dimension 3 and 4 were studied and distance-optimal subfield codes over GF(p) were obtained, where q = pm. The key idea for obtaining very good subfield codes over small fields is to choose very good linear codes over an extension field with small dimension. This paper first presents a general construction of [q + 1,2,q] MDS codes over GF(q), and then studies the subfield codes over GF(p) of some of the [q+1,2,q] MDS codes over GF(q). Two families of dimensionoptimal codes over GF(p) are obtained, and several families of nearly optimal codes over GF(p) are produced. Several open problems are also proposed in this paper.

Image encryption based on a novel memristive chaotic system, Grain-128a algorithm and dynamic pixel masking

In this paper, we first propose a memristive chaotic system and implement it by circuit simulation. The chaotic dynamics and various attractors are analysed by using phase portrait, bifurcation diagram, and Lyapunov exponents. In particular, the system has robust chaos in a wide parameter range and the initial value space, which is favourable to the security communication application. Consequently, we further explore its application in image encryption and present a new scheme. Before image processing, the external key is protected by the Grain-128a algorithm and the initial values of the memristive system are updated with the plain image. We not only perform random pixel extraction and masking with the chaotic cipher, but also use them as control parameters for Brownian motion to obtain the permutation matrix. In addition, multiplication on the finite field GF(2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">8</sup> ) is added to further enhance the cryptography. Finally, the simulation results verify that the proposed image encryption scheme has better performance and higher security, which can effectively resist various attacks.

Metode Reversible Self-Dual untuk Konstruksi Kode DNA atas Lapangan Hingga GF(4)

The DNA molecule chain consists of two complementary strands composed of a sequence of four nucleotide bases, namely adenine (A), cytosine (C), guanine (G) and thymine (T). DNA code is a set of codewords with a fixed length of the alphabet {A, C, T, G}. DNA coding is one application of coding theory over a finite field. The set {A, C, T, G} is identified as finite field GF(4) = {0, 1, w, w2} with w2 + w + 1 = 0. The reversible self-dual (RSD) code over the finite field GF(4) is a code whose dual is itself and the reverse of each codeword contained in the code. This study aims to obtain an algorithm to construct a DNA code derived from the RSD C code on the field to GF(4) which is called the Reversible Self-Dual Method. The aspects studied include the characteristics that form the basis properties of the theory in compiling the DNA code algorithm over the RSD code over GF(4). The compiled algorithm is a DNA code construction method of codeword length even that conforms to the Hamming distance constraint, reverse-complement constraint, and GC-content constraint. The input of the algorithm is a generator matrix of RSD code C with a minimum distance of d and the output is a DNA code that satisfies these three constraints.

90 nm CMOS Implementation of Multiplicative Inverse of the S-Box for AES Algorithm Using Six Transistor XOR Gate

The Substitution-BOX is the most difficult architecture and is at the heart of any Advanced Encryption Standard algorithm implementation. It is the most complicated non-linear architecture using multiplicative inverse. It consumes maximum amount of the energy and power budget of the algorithm. This paper introduces a full-custom CMOS implementation of Multiplicative Inverse module for Substitution/ Inverse Substitution transformation in composite field arithmetic using Galois field GF (28). The multiplicative inversion module utilizes large number of XOR gates in its implementation. This paper introduces implementation of a novel XOR gate using six transistors. Using this six transistor XOR gate, the multiplicative inverse module is implemented in 90 nm CMOS technology. Simulation of the proposed design is achieved using Tanner EDA v.16 software. The area of the multiplicative inverse circuit is 39.92 µm2 requiring 776 transistors. With 0.6 Volts supply voltage, the design shows a power dissipation of 2.386 µWatts making it ideal for applications such as smart cards and RFID tags.

Structure of a finite non-commutative algebra set by a sparse multiplication table

Four-dimensional finite non-commutative associative algebras represent practical interest as algebraic support of post-quantum digital signature algorithms, especially algebras with two sided global unit, set by sparse basis vectors multiplication tables. A new algebra of the latter type, set over the field GF(p), is proposed and its structure is investigated. The studied algebra is described as a set of p2 + p + 1 commutative subalgebras of three different types. All subalgebras intersect strictly in the subset of scalar vectors. Formulas are derived for the number of subalgebras of each type.

Collective Signature Schemes Using Problem of Finding Roots Modulo

Digital signature schemes in general and representative collective digital signature schemes, in particular, are often built based on the difficulty of the discrete logarithm problem on the finite field, of the discrete logarithm problem of the elliptic curve, of the problem of factor analysis, of the problem of finding the roots modulo of large primes or a combination of the difficult problems mentioned above. In this paper, we use the new difficult problem, which is to find the with root in the finite ground field GF(p) to build representative collective signature schemes, but the chosen modulo p has a special structure distinct p=Nt_0 t_1 t_2+1, where N is an even number and t0, t1, t2 are prime numbers of equal magnitude, about 80 bits. The characteristics of the proposed scheme are: i) The private key of each signer consists of 2 components (K_1, K_2), randomly selected, but the public key has only one component (Y) calculated by the formula Y= K_1^ (w_1) K_2^ (w_2); w_1 = t_0 t_1 and w_2 = t_0 t_2; and ii) The generated signature consists of a set of 3 components (e, S1, S2). We use the technique of hiding the signer's public key Y, which is the coefficient λ generated by the group manager, in the process of forming the group signature and representative collective signature to enhance the privacy of all members of the signing collective.

Hardware Implementation of Efficient Elliptic Curve Scalar Multiplication using Vedic Multiplier

This paper presents an area efficient and high-speed FPGA implementation of scalar multiplication using a Vedic multiplier. Scalar multiplication is the most important operation in Elliptic Curve Cryptography(ECC), which used for public key generation and the performance of ECC greatly depends on it. The scalar multiplication is multiplying integer k with scalar P to compute Q=kP, where k is private key and P is a base point on the Elliptic curve. The Scalar multiplication underlying finite field arithmetic operation i.e. addition multiplication, squaring and inversion to compute Q. From these finite field operations, multiplication is the most time-consuming operation, occupy more device space and it dominates the speed of Scalar multiplication. This paper presents an efficient implementation of finite field multiplication using a Vedic multiplier. The scalar multiplier is designed over Galois Binary field GF(2233) for field size=233-bit which is secured curve according to NIST. The performances of the proposed design are evaluated by comparing it with Karatsuba based scalar multiplier for area and delay. The results show that the proposed scalar multiplication using Vedic multiplier has consumed 22% less area on FPGA and also has 12% less delay, than Karatsuba, based scalar multiplier. The scalar multiplier is coded in Verilog HDL, synthesize and simulated in Xilinx 13.2 ISE on Virtex6 FPGA.

AMBTC-based secret image sharing by simple modular arithmetic

A distortionless secret image sharing scheme using finite field GF(PG) (PG is the largest prime less than a given number) is investigated for sharing the absolute moment block truncation coding (AMBTC) images. Two adjusting operations are devised to modify the AMBTC trios (i.e., quantization pixels and bit maps) suitable for GF(PG) sharing. Polynomials under GF(PG) are constructed for encrypting the quantization pixels and bit maps. When sharing the trios under GF(PG), the AMBTC image is perfectly recovered. Moreover, the bit map sharing under GF(PG) can be extended to GF(PS) (PS is the smallest prime larger than a given number). Another scheme using GF(PS) is constituted by combining quantization pixel sharing under GF(PG) and bit map sharing under GF(PS). But the scheme using GF(PS) is not always lossless. Experimental results show that the proposed schemes are effective. Since GF(PG) and GF(PS) are simple modular arithmetic, improved computational efficiency is obtained.

A Novel Method for Developing Post-quantum Digital Signature Algorithms on Non-commutative Associative Algebras

Introduction: Development of practical post-quantum signature algorithms is a current challenge in the area of cryptography. Recently, several candidates on post-quantum signature schemes, in which the exponentiation operations in a hidden commutative group contained in a non-commutative algebra is used, were proposed. Search for new mechanisms of using a hidden group, while developing signature schemes resistant to quantum attacks, is of significant practical interest. Purpose: Development of a new method for designing post-quantum signature algorithms on finite non-commutative associative algebras. Results: A novel method for developing digital signature algorithms on non-commutative algebras. A new four-dimensional finite non-commutative associative algebra set over the ground field GF(p) have been proposed as algebraic support of the signature algorithms. To provide a higher performance of the algorithm, in the introduced algebra the vector multiplication is defined by a sparse basis vector multiplication table. Study of the algebra structure has shown that it can be represented as a set of commutative subalgebras of three different types, which intersect exactly in the set of scalar vectors. Using the proposed method and introduced algebra, a new post-quantum signature scheme has been designed. The introduced method is characterized in using one of the elements of the signature (e, S) in form of the four-dimensional vector S that is computed as a masked product of two exponentiated elements G and H of a hidden commutative group: S = B-1GnHmC-1, where non-permutable vectors B and C are masking multipliers; the natural numbers n and m are calculated depending on the signed document M and public key. The pair &lt;G, H&gt; composes a minimum generator systems of the hidden group. The signature verification equation has the form R = (Y1SZ1)e(Y2SZ2)e2, where pairwise non-permutable vectors Y1, Z1, Y2, and Z2 are element of the public key and natural number e that is computed depending on the value M and the vector R. Practical relevance: Due to sufficiently small size of public key and signature and high, the developed digital signature scheme represents interest as a practical post-quantum signature algorithm. The introduced method is very attractive to develop a post-quantum digital signature standard.

Cayley Graphs Defined by Systems of Equations

Let R be a finite ring. In this paper, we mainly explore the conditions to ensure the graph BΓn defined by a system of equations {fi|i=2,…,n} to be a Cayley graph or a Hamiltonian graph. More precisely, we prove that BΓn is a Cayley graph with G=⟨ϕ,A⟩ a group of dihedral type if and only if the system Fn={fi|i=2,…,n} is Cayley graphic of dihedral type in R. As an application, the well-known Lova´sz Conjecture, which states that any finite connected Cayley graph has a Hamilton cycle, holds for the connected BΓn defined by Cayley graphic system Fn of dihedral type in the field GF(pk).

Classification of Unit Groups of Five Radical Zero Completely Primary Finite Rings Whose First and Second Galois Ring Module Generators Are of the Order pk, k = 2, 3, 4

Let R0 = GR(pkr, pk) be a Galois maximal subring of R so that R = R0 ⊕ U ⊕ V ⊕ W ⊕ Y, where U, V, W, and Y are R0/pR0 spaces considered as R0‐modules, generated by the sets {u1, ⋯, ue}, {v1, ⋯, vf}, {w1, ⋯, wg}, and {y1, ⋯, yh}, respectively. Then, R is a completely primary finite ring with a Jacobson radical Z(R) such that (Z(R))5 = (0) and (Z(R))4 ≠ (0). The residue field R/Z(R) is a finite field GF(pr) for some prime p and positive integer r. The characteristic of R is pk, where k is an integer such that 1 ≤ k ≤ 5. In this paper, we study the structures of the unit groups of a commutative completely primary finite ring R with pψui = 0, ψ = 2, 3, 4; pζvj = 0, ζ = 2, 3; pwk = 0, and pyl = 0; 1 ≤ i ≤ e, 1 ≤ j ≤ f, 1 ≤ k ≤ g, and 1 ≤ l ≤ h.

Алгоритмы ЭЦП на конечных некоммутативных алгебрах над полями характеристики два

Abstract Purpose of work is the increase in performance and decrease in the hardware implementation cost of the of post-quantum algebraic signature algorithms based on the computational difficulty of solving systems of many quadratic equations with many unknowns. Research method is i) the development of post-quantum signature algorithms on finite non-commutative associative algebras defined over finite fields of characteristic two, which have high performance and small sizes of signature and public and secret keys; ii) using the concept of constructing algebraic signature algorithms with a hidden commutative group, characterized by the use of a power-type vector verification equation with multiple occurrences of the signature S as a factor; iii) the choice of the degree of extension z of the field GF(2z) in which the order of the hidden group is divisible only by prime divisors of at least 24 bits. Results of the study are the formulated main provisions for the implementation of post-quantum signature algorithms with a hidden group, the security of which is based on the computational difficulty of solving systems of many quadratic equations with many unknowns, when using the finite non-commutative algebras given over the GF(2z) fields as algebraic support. The values of the extension degree z are established for which the order of the hidden commutative group is divisible only by prime divisors of a sufficiently large size. A new post-quantum signature algorithm with relatively high performance and small sizes of the signature and public and secret keys have been developed. Using an informal security index in the form of a product of the binary logarithm of the order of the field and the number of unknowns, the developed and known post-quantum algorithms for a given level of security are compared. Practical relevance. The main provisions for constructing signature algorithms with a hidden group are formulated for the case of using finite non-commutative algebras with computationally efficient operations of multiplication and exponentiation, providing prerequisites for improving performance and reducing the hardware implementation cost of post-quantum signature algorithms.

STRUCTURE OF A 4-DIMENSIONAL ALGEBRA AND GENERATING PARAMETERS OF THE HIDDEN DISCRETE LOGARITHM PROBLEM

Structure of a 4-dimensional algebra and generating parameters of the hidden discrete logarithm problem the field GF'(p) is studied in connection with using it as algebraic support of the hidden discrete logarithm problem that is an attractive primitive of post-quantum signature schemes. It is shown that each invertible 4-dimensional vector that is not a scalar vector is included in a unique commutative group representing a subset of algebraic elements. Three types of commutative groups are contained in the algebra and formulas for computing the order and the number of groups are derived for each type. The obtained results are used to develop algorithms for generating parameters of digital signature schemes based on computational difficulty of the hidden logarithm problem.

An improved upper bound on self-dual codes over finite fields GF(11), GF(19), and GF(23)

This paper gives new methods of constructing symmetric self-dual codes over a finite field GF(q) where q is a power of an odd prime. These methods are motivated by the well-known Pless symmetry codes and quadratic double circulant codes. Using these methods, we construct an amount of symmetric self-dual codes over GF(11), GF(19), and GF(23) of every length less than 42. We also find 153 new self-dual codes up to equivalence: they are [32, 16, 12], [36, 18, 13], and [40, 20, 14] codes over GF(11), [36, 18, 14] and [40, 20, 15] codes over GF(19), and [32, 16, 12], [36, 18, 14], and [40, 20, 15] codes over GF(23). They all have new parameters with respect to self-dual codes. Consequently, we improve bounds on the highest minimum distance of self-dual codes, which have not been significantly updated for almost two decades.

Boolean analogues of the Pascal triangle with maximal possible number of ones

Abstract The aim of the paper is to find the maximal possible number ξ of units in Boolean triangular array Ts formed by $\begin{array}{} \displaystyle \frac{s(s+1)}{2} \end{array}$ elements of the field GF(2) defined by the top row of s elements. Each element of each row except the top one is the sum (as in the Pascal’s triangle) of two elements of the above row. It is proved that ξ ⩽ ⌈ $\begin{array}{} \displaystyle \frac{s(s+1)}{3} \end{array}$ ⌉ and this value is attained only on triangles having the upper row as the Fibonacci series mod 2.

POST-QUANTUM BLIND SIGNATURE PROTOCOL ON NON-COMMUTATIVE ALGEBRAS

A method for constructing a blind signature scheme based on a hidden discrete logarithm problem defined in finite non-commutative associative algebras is proposed. Blind signature protocols are constructed using four-dimensional and six-dimensional algebras defined over a ground finite field GF(p) and containing a global two-sided unit as an algebraic support. The basic properties of the used algebra, which determine the choice of protocol parameters, are described.

Information encoding in an autonomous downhole telemetry system

One of the most important tasks of managing the oil production is providing a noiseproof way of obtaining information about producing reservoirs. This paper studies problems concerning construction of generating polynomials for convolution codes over a Galois field. Two theorems on convolution codes have been formulated and proven. Code distance of the studied convolution code over the field GF(2) at a speed of R=1/m has been evaluated. The results of this research serve as a foundation for studies of correction capabilities of a computing system during a digital signal processing in an autonomous downhole telemetry system.

Low-latency area-efficient systolic bit-parallel GF(2[formula omitted]) multiplier for a narrow class of trinomials

Edge computing in Internet of Things (IoT) requires high-performance as well as cost-effective hardware implementations to accommodate large number of computations. Implementation of security in IoT edge devices is indispensable, and attaining security is possible using Elliptic curve cryptography (ECC) algorithms. Finite field GF(2m) multiplication is a performance-critical operation in these algorithms, hence, it requires efficient hardware implementations. Systolic structures for GF(2m) multiplier are suitable for high-performance applications as they allow high-throughput implementations. Many systolic GF(2m) multipliers have been proposed in the literature that can achieve high throughput rates, however, they require large area and high latency. In this paper, we propose a polynomial basis GF(2m) systolic multiplier applicable for a narrow class of trinomials which includes both the NIST (National Institute of Standards and Technology) recommended trinomials for m=233 and 409 fields. Analytical comparisons with the related multipliers for m=409 show that the multiplier proposed in this paper reduces the latency by 32% and area by 5%. The proposed multiplier is synthesized using synopsis tools targeting for an ASIC implementation and the comparisons are also presented. The proposed low-latency area-efficient multiplier can be used in the implementation of hardware security for low-cost IoT edge devices.

On (− 1)-differential uniformity of ternary APN power functions

Very recently, a new concept called multiplicative differential and the corresponding c-differential uniformity were introduced by Ellingsen et al. (IEEE Trans. Inform. Theory 66(9), 5781–5789 2020). A function F(x) over finite field GF(pn) to itself is said to have c-differential uniformity δ, or equivalent, F(x) is differentially (c,δ)-uniform, when the maximum number of solutions x ∈GF(pn) of F(x + a) − cF(x) = b, a,b,c ∈GF(pn), c≠ 1 if a = 0, is equal to δ. The objective of this paper is to study the (− 1)-differential uniformity of some ternary APN power functions F(x) = xd over GF(3n). We obtain ternary power functions with low (− 1)-differential uniformity, and some of them are almost perfect (− 1)-nonlinear.

Maximizing throughput in RLNC-based multi-source multi-relay with guaranteed decoding

In this paper, an efficient and practical solution to cope with random linear network coding (RLNC) drawbacks and maximize the system throughput is proposed. No header overheads and no retransmissions are required as a result of the introduction of a unique codebook at the nodes in charge of RLNC coding and decoding. The codebook contains the seeds that allow for the pseudo-random generation of full-rank matrices of network coding elements over the finite Galois field GF(q), where q is the field size, for N generations. Each relay generates the same matrix and uses its corresponding row of N coding GF(q) coefficients. The destination, whose seeds codebook is synchronized on the same index value, generates the same full-rank matrix to decode the source nodes packets. Results show that the number of successfully transmitted packets is not generally affected by the choice of the generation size or the underlying finite field, which demonstrates the outperformance of the proposed technique over the traditional RLNC procedure.