Primitive Polynomials Research Articles

An enhanced key expansion module based on 2D hyper chaotic map and Galois field

Key expansion is an essential component in block cryptography, which serves the round function. By analyzing key expansion module of AES, it is found that the round-keys are highly correlated, and current round-key can be derived from its previous or the next round-key. To address these weaknesses, first, a 2D exponential chaotic map (2D-ECM) that exhibits ergodicity and superior randomness was constructed. Then, the Lyapunov exponents (LEs) were calculated based on the singular value decomposition (SVD) method. In addition, TestU01 test results showed that the sequences generated by 2D-ECM have better randomness. Further, an enhanced key expansion module was designed utilizing 2D-ECM and primitive polynomial over GF(2n), which has irreversibility and parallelism, and the round-keys are independent of each other. Simulation results and performance analysis demonstrated the effectiveness of the proposed enhanced key expansion module.

Synchronization of M-sequences based on fast Нadamard transform

Options have been developed for constructing circulant matrices of any M-sequence (MS) based on automorphic multiplicative groups of the extended Galois field, constructed using an irreducible primitive polynomial, on the basis of which the original MS is formed. The result of this approach is the identification of new methods for transforming MS circulant matrices to a matrix of Walsh functions, ordered by the powers of the antiderivative element of the field. It is shown for the first time that, depending on the initial conditions of the transformation, a set of any number of any cyclic shifts of the MP, shifted relative to each other by one symbol, can be transformed to any rows of the ordered matrix of Walsh functions, following one another. This circumstance makes it possible to simplify the MS synchronization algorithm for a known range of its cyclic shifts, especially in the case of large periods of its repetition, and also to reduce the computational complexity of the processing algorithm when working in a truncated basis of Walsh–Hadamard functions.

Rate-compatible LDPC Codes based on Primitive Polynomials and Golomb Rulers

Rate-compatible LDPC Codes based on Primitive Polynomials and Golomb Rulers

Filter generators with increased resistance against algebraic attacks

Filter generators form one of the best known and most studied classes of key stream generators used in synchronous stream ciphers. Each such generator consists of a binary linear shift register with a primitive feedback polynomial and a nonlinear Boolean function, which has a number of requirements related to the condition of generator security against known attacks. One such requirement is the high algebraic immunity of the generator's filter function; this parameter characterises security filter generator against modern algebraic attacks. In certain cases, this requirement is too restrictive in sense of practicality, as it increases the computational or circuit complexity of the key stream generation algorithm. This makes the actuality of increasing the resistance of filter generators with fixed filter functions, that have limited (low) algebraic immunity. The paper proposes to solve this problem by modifying the feedback function of the linear shift register. the security of the proposed generators against algebraic attacks is investigated and is shown that (under certain natural conditions) such generators are more secure at the same initial state length as compared to traditional filter generators. The proposed solution seems to be useful for practical application in advanced hardware-oriented stream ciphers design.

A Strong Key Expansion Algorithm Based on Nondegenerate 2D Chaotic Map Over GF(2n)

The strength of a cryptosystem relies on the security of its key expansion algorithm, which is an important component of a block cipher. However, numerous block ciphers exhibit the vulnerability of reversibility and serialization. Therefore, it is necessary to design an irreversible parallel key expansion algorithm to generate independent round keys. First, a 2D nondegenerate exponential chaotic map (2D-NECM) is constructed, and the results of the dynamic analysis show that the 2D-NECM possesses ergodicity and superior randomness within a large range of parameters. Then, an irreversible parallel key expansion algorithm is designed based on 2D-NECM and primitive polynomial over GF([Formula: see text]). By injecting random perturbation into the initial key, the algorithm can generate different round keys even if the same initial key is used. Simulation results indicate that the algorithm has high security performance. It effectively satisfies the requirements of irreversibility and parallelism, while ensuring the mutual independence of round keys.

Decimation of M Sequences As a Way of Obtaining Primitive Polynomials

Decimation of M Sequences As a Way of Obtaining Primitive Polynomials

Test Compression for Launch-on-Capture Transition Fault Testing

A new low-power test compression scheme, called Dcompress , is proposed for launch-on-capture transition fault testing by using a new seed encoding scheme, a new design for testability architecture, and a new low-power test application procedure. The new seed encoding scheme generates seeds for all tests by selecting a primitive polynomial that encodes all tests of a compact test set. A software-defined linear feedback shift register architecture, called SLFSR , is proposed to make the new method conform to the current flow of design and test. Experimental results on benchmark circuits show that test data volume can be compressed up to 6300X with the well-compacted baseline test set for a design with 11.8M gates and more than 1.1M scan flip-flops.

Quantum‐Accelerated Algorithms for Generating Random Primitive Polynomials Over Finite Fields

AbstractPrimitive polynomials over finite fields are crucial resources with broad applications across various domains in computer science, including classical pseudo‐random number generation, coding theory, and post‐quantum cryptography. Nevertheless, the pursuit of an efficient classical algorithm for generating random primitive polynomials over finite fields remains an ongoing challenge. In this work, it shows how this problem can be solved efficiently with the help of quantum computers. Moreover, the designs of specific quantum circuits to implement them are also presented. The research paves the way for the rapid and real‐time generation of random primitive polynomials in diverse quantum communication and computation applications.

On binary linear codes and binary pseudo-random sequences

In this paper, we study the relation between the linear subspace of the pseudo-noise (PN)-sequences generated by a primitive polynomial and the simplex code. This family of sequences can be also seen as an Maximum Distance Separable (MDS) [Formula: see text]-linear code over [Formula: see text]. Furthermore, we see how to compute the family of generalized sequences produced by a primitive polynomial by means of a first-order Reed–Muller code.

Hardware architecture of Dillon’s APN permutation for different primitive polynomials

Cryptographically strong functions used as S-boxes in block cyphers are fundamental for the cypher’s security. Their representation as lookup tables is possible for functions of small dimension. For larger dimensions, this is infeasible, especially if resources are limited. An alternative is representing the function as a polynomial over a finite field, and constructing a circuit evaluating this polynomial. We study how the choice of primitive polynomial affects the efficiency of hardware implementations. We take Dillon’s permutation on 6 bits (the only known permutation in even dimension from the cryptographically optimal Almost Perfect Nonlinear functions) as a relevant example, and present hardware architectures, polynomial representations and hardware theoretical complexities for all primitive polynomials of degree six. To compare the efficiency, we report on results obtained from FPGA (Field Programmable Gate Array) implementations. To the best of our knowledge, no similar study has been given in the literature. We observe that using the primitive trinomial y6+y+1 reduces the number of 2-input XOR gates up to 11.7% and the number of XOR gates × Delay metrics up to 13.2% with respect to the worst-case implementation. Therefore, the choice of primitive polynomial can profoundly impact the efficiency of such an implementation, and should be carefully considered.

Quasi-Cyclic Perfect Codes in Doob Graphs and Special Partitions of Galois Rings

The Galois ring GR$(4^\Delta)$ is the residue ring $Z_4[x]/(h(x))$, where $h(x)$ is a basic primitive polynomial of degree $\Delta$ over $Z_4$. For any odd $\Delta$ larger than $1$, we construct a partition of GR$(4^\Delta) \backslash \{0\}$ into $6$-subsets of type $\{a,b,-a-b,-a,-b,a+b\}$ and $3$-subsets of type $\{c,-c,2c\}$ such that the partition is invariant under the multiplication by a nonzero element of the Teichmuller set in GR$(4^\Delta)$ and, if $\Delta$ is not a multiple of $3$, under the action of the automorphism group of GR$(4^\Delta)$. As a corollary, this implies the existence of quasi-cyclic additive $1$-perfect codes of index $(2^\Delta-1)$ in $D((2^\Delta-1)(2^\Delta-2)/{6}, 2^\Delta-1 )$ where $D(m,n)$ is the Doob metric scheme on $Z^{2m+n}$.

The Toda flow on Hessenberg elements of real, split simple Lie algebras

In this work, we consider the Toda flow associated with compact/Borel decompositions of real, split simple Lie algebras. Using the primitive invariant polynomials of Chevalley, we show how to construct integrals in involution which are invariants of the maximal compact subgroup, and moreover, we show that the number of such integrals is given by a formula involving only Lie-theoretic data. We then introduce the space of Hessenberg elements, characterize the generic Hessenberg coadjoint orbits, and show that the dimension of such orbits is precisely twice the number of nontrivial invariants which appeared earlier. For the class of classical, real split simple Lie algebras, we construct angle-type variables which in particular shows that the Toda flow is Liouville integrable on generic Hessenberg coadjoint orbits.

Constructions of optimum distance full flag codes

The full flag codes of maximum distance and size on vector space Fq2ν are studied in this paper. We start to construct the subspace codes of maximum distance by making uses of the companion matrix of a primitive polynomial and the cosets of a subgroup in the general linear group over the finite field Fq. And a spread code is given. Then we focus on flag codes having maximum distance (optimum distance flag codes). Based on the constructed spread code, an optimum distance full flag code attaining the largest size is given. Then we consider the structure of flag codes on the symplectic space Fq2ν. We show that for any symmetric matrix A in GLν(Fq), there is a totally isotropic flag orbit code with optimum distance on the symplectic space Fq2ν. An optimum distance flag orbit code with larger size is presented on symplectic space Fq4k. And based on a partial spread code that are constructed, another optimum distance flag orbit code on the symplectic space Fq2ν is given.

Linear Gain Controller Aided Iterative Soft Sequential Acquisition for Primitive Polynomials

Linear Gain Controller Aided Iterative Soft Sequential Acquisition for Primitive Polynomials

The Self-Shrinking Conflation Generator: A Proposed Improvement to the Self-Shrinking Generator

The backbone of many cybersecurity applications and algorithms require random numbers. One of the most commonly used pseudo-random number generators is the Linear Feedback Shift Register (LFSR), which is fast, computationally inexpensive, and has excellent statistical properties. Unfortunately LFSRs have a number of weaknesses, some of which were addressed by decimation-based sequence generators such as the self-shrinking generator (SSG). Regrettably, the SSG was also found to be vulnerable to attack. In this paper, we propose an improvement to the SSG called the self-shrinking conflation generator (SSCG). Our approach is based on the observation that what is discarded during the self-shrinking process of the SSG, is from a cryptographic perspective, just as good as that which is kept. By combining the bits the SSG would normally discard with those it retains, using the exclusive OR (XOR) operation, we create a modified SSG bitstream with several improved characteristics. To highlight these improvements, we provide some mathematical security analysis associated with this approach, apply the NIST statistical test suite to several different bitstreams created using LFSRs driven by different degree primitive polynomials, and compare our results to that of the SSG.

Abelian lifts of polynomials

Let p be a prime, and let f∈Z[x] be a non-constant polynomial whose leading coefficient is prime to p and that f has no repeated root modulo p. Suppose the mod-p irreducible factors of f all have the same degree. Then there are infinitely many Abelian extensions of number fields L/K such that OL/pOL≃(Z/p)[x]/f, where OL denotes the ring of integers of L and p⊂OK is a maximal ideal with OK/pOK≃Z/p. Moreover, under the generalized Riemann hypothesis for the Dedekind zeta functions of number fields, for fixed deg⁡(f) we can construct Abelian lifts in polynomial time in log⁡p. This confirms in a stronger form a question raised by Mihăilescu and Vuletescu concerning polynomial cyclic algebras. The proofs make use of a weak form of Artin's conjecture on primitive roots and polynomial time polynomial factoring algorithms over number fields.

A fast method for blindly identifying Reed‐Solomon codes based on matrix analysis

For the identification of Reed-Solomon (RS) codes, the existing methods need to test the possible codeword length and the primitive polynomials exhaustively. Exhaustive search leads to high computational complexity. To overcome this limitation and fulfill the requirement in practice, a fast blind identification method is proposed. Different with most methods, our identification method is discussed in Galois Field of 2. First, by using Gaussian elimination, the bit matrix is transformed to the simplest upper triangular form. If the assumed codeword length is right, the matrix is of deficient rank, and the parameters of an RS code can be estimated according to the rank. The null space of the matrix, defined as the equivalent binary parity check matrix, can also be obtained from the simplification result. Then, a candidate set of primitive polynomials is constructed according to the estimated codeword length. By using the matrix null space, the correctness of a candidate primitive polynomial is tested through matrix analysis. Finally, by combining the estimated parameters, the generator polynomial of an RS code is identified. To decrease the bit errors in the matrix, soft-decision data is used to select reliable bits. Experimental results show the effectiveness and robustness of our method.

Fitting height of finite groups admitting a fixed-point-free automorphism satisfying an additional polynomial identity

Let f(x) be a non-zero polynomial with integer coefficients. An automorphism φ of a group G is said to satisfy the elementary abelian identity f(x) if the linear transformation induced by φ on every characteristic elementary abelian section of G is annihilated by f(x). We prove that if a finite (soluble) group G admits a fixed-point-free automorphism φ satisfying an elementary abelian identity f(x), where f(x) is a primitive polynomial, then the Fitting height of G is bounded in terms of deg⁡(f(x)). We also prove that if f(x) is any non-zero polynomial and G is a σ′-group for a finite set of primes σ=σ(f(x)) depending only on f(x), then the Fitting height of G is bounded in terms of the number irr(f(x)) of different irreducible factors in the decomposition of f(x). These bounds for the Fitting height are stronger than the well-known bounds in terms of the composition length α(|φ|) of 〈φ〉 when deg⁡f(x) or irr(f(x)) is small in comparison with α(|φ|).

Reconstruction of a Synchronous Scrambler Based on Average Check Conformity

Channel coding technology is indispensable in digital communication systems. In noncooperative contexts, the identification of the channel codes of the synchronous scrambler is essential. In this paper, a new algorithm that directly uses a soft decision sequence for blind reconstruction of the synchronous scrambler is proposed. First, considering imbalanced signal sources and the principle of scrambling and descrambling the synchronous scrambler, the error-containing equation of the synchronous scrambler is established. Second, the average check conformity is introduced to complete the check relationship detection. Then, based on the statistical characteristics of the average check conformity, the corresponding discrimination threshold is established, and the reconstruction of the feedback polynomial of the synchronous scrambler is completed by traversing the possible primitive polynomials. Finally, the verification equation is determined by the method of subsection optimization, which greatly reduces the number of initial states that need to be traversed; this is critical for scramblers with high-order feedback polynomials (i.e., when the order of the feedback polynomial is greater than 15). Simulations show that the algorithm can effectively reconstruct the synchronous scrambler under imbalanced signal sources. Moreover, the proposed algorithm offers improved performance with about 1–2 dB gain at low signal-to-noise ratios compared with existing methods, and its computational complexity is reasonable.

Security and Efficiency of Linear Feedback Shift Registers in GF(2n) Using n-Bit Grouped Operations

Many stream ciphers employ linear feedback shift registers (LFSRs) to generate pseudorandom sequences. Many recent LFSRs are defined in GF(2n) to take advantage of the n-bit processors, instead of using the classic binary field. In this way, the bit generation rate increases at the expense of a higher complexity in computations. For this reason, only certain primitive polynomials in GF(2n) are used as feedback polynomials in real ciphers. In this article, we present an efficient implementation of the LFSRs defined in GF(2n). The efficiency is achieved by using equivalent binary LFSRs in combination with binary n-bit grouped operations, n being the processor word’s length. This improvement affects the general considerations about the security of cryptographic systems that uses LFSR. The model also allows the development of a faster method to test the primitiveness of polynomials in GF(2n).