Primitive Polynomials Research Articles

A Gist of Analogues μ_pto Mobius function μ

If f (x) [x] p Z is an irreducible polynomial, the number of polynomials g(x) with deg(g(x))  deg( f (x))  (g(x), f (x)) =1 is the order of the multiplicative group of [x]/( f (x)) p Z. In this paper we introducing analogues p  to Mobius function  defined on [x], p Z the set of all primitive polynomials in [x] p Z .

A 0.7 pJ/bit, 1.5 Gbps Energy-Efficient Image-Based True Random Number Generator

ABSTRACT Random numbers cover a vast spectrum of applications. Hence generating it effectively with high performance is the need of the hour. This paper presents the novel design and implementation of high performance and energy-efficient true random number generator (TRNG) using images as a source. In the proposed work, the harvesting mechanism that comprises of hashing technique block (HTB) is used to reduce the intensity of pattern, an event counting circuit (ECC) is used for comparing the events, while a linear feedback shift register (LFSR), designed by considering a primitive polynomial function, is used to obtain the random numbers. The addition of 8 × 1 multiplexer (MUX) with meta-stable circuit feeder control lines had further increased the unpredictability in the proposed system. The implementation of the work has been done in the Xilinx Vivado simulation tool followed by the Cadence Virtuoso circuit simulation environment. The maximum speed of 1.5 Gbps with power dissipation of 1 mW and 0.7 pJ/bit energy efficiency with a layout area of 2218 μm2 has been achieved in this work. NIST 800.22 statistical test suite and uniformity test comprising of Kolmogorov–Smirnov and Chi-square test have also been performed for validation of generated random numbers. The obtained binary sequences have passed all tests successfully with calculated entropy up to 0.999999999. The autocorrelation factor (ACF) of the output bit streams has been obtained as approximately zero (∼0) within 96% confidence bounds of a Gaussian distribution (µ = 0, 3σ). The proposed design is thus suitable for true random number generation.

Performance Study of Spread Spectrum Systems with Hard Limiters

Use of spread spectrum systems in telecommunications is studied. It is shown that spread spectrum techniques can substantially enhance noise and interference immunity in the currently deployed information transmission networks. Primitive polynomials are proposed to obtain respective spreading codes. A spreading code consisting of 1023 chips is synthesized and its characteristics are studied. It is deduced that powerful interferences can exceed dynamic range of the receiving part of the system and, as a result, deteriorate information transmission. To overcome this problem, utilization of limiters is proposed, in which limitation level equals that of the internal noise of the receiving part. Computer simulation is employed to test the performance of the proposed solution. Performance of the spread spectrum system for information transfer is studied both without the limiter and with the limiter. Research results show that, for binary modulation, performance of the system with limitation and without limitation is nearly identical while limiters can substantially reduce requirements for the dynamic range. Compared to the existing approaches, it is proposed to use the synthesized spreading coding sequence with the limitation technique in practical implementations of those telecommunication networks, in which noise immunity and transmission concealment are required, such as in unmanned aerial vehicles. This can replace currently used approaches, such as frequency hopping, transmission power adjustment and antenna pattern changes.

ROBUST TELECOMMUNICATION CHANNEL WITH PARAMETERS CHANGING ON A FRAME-BY-FRAME BASIS

An approach is proposed to design of a robust data transfer channel for telecommunication networks that is based on the desired bits spectrum spreading and their additional scrambling by using two pseudo noise coding sequences derived from primitive polynomials of eighth and fifteenth degrees. Further information protection is offered to be achieved by deploying four 32768 chip-long pseudo noise coding sequences, instead of one, with 128 cyclic shifts each. It is proven that selection of a scrambling sequence and its cyclic shift according to a secret algorithm on a frame-by-frame basis helps substantially improve performance of the telecommunication channel in terms of protection against interception while preserving its useful properties. Research results permit to claim that the designed telecommunication channel could be used in noise immune and concealed telecommunication networks.

Pseudo-random sequences with non-maximal length realized according to the Galois scheme based on the primitive polynomial in degree

We consider homogeneous and nonhomogeneous pseudo-random sequences of non-maximal length formed by Galois generator based on characteristic polynomial in degree. We discovered periodic polynomial structures in degree in general form and on characteristic examples. We presented and applied the decimation method of periodic sequences, giving their decomposition on elementary M-sequences and constant components. We solved the tasks of analyzing of the diversity of generated sequences, which are organized and ordered from elements of direct and inverse M-sequences, and zero and one constants. We have obtained a general analytical relationship between decimation results and probabilistic properties. We discovered conditions for forming equal probability properties of the resulting sequences and determined the boundary relations of the generator parameters with the initial task of the minimum order of elementary M-sequences.

The Effect of the Primitive Irreducible Polynomial on the Quality of Cryptographic Properties of Block Ciphers

Substitution boxes are the only nonlinear component of the symmetric key cryptography and play a key role in the cryptosystem. In block ciphers, the S-boxes create confusion and add valuable strength. The majority of the substitution boxes algorithms focus on bijective Boolean functions and primitive irreducible polynomial that generates the Galois field. For binary field F2, there are exactly 16 primitive irreducible polynomials of degree 8 and it prompts us to construct 16 Galois field extensions of order 256. Conventionally, construction of affine power affine S-box is based on Galois field of order 256, depending on a single degree 8 primitive irreducible polynomial over ℤ2. In this manuscript, we study affine power affine S-boxes for all the 16 distinct degree 8 primitive irreducible polynomials over ℤ2 to propose 16 different 8×8 substitution boxes. To perform this idea, we introduce 16 affine power affine transformations and, for fixed parameters, we obtained 16 distinct S-boxes. Here, we thoroughly study S-boxes with all possible primitive irreducible polynomials and their algebraic properties. All of these boxes are evaluated with the help of nonlinearity test, strict avalanche criterion, bit independent criterion, and linear and differential approximation probability analyses to measure the algebraic and statistical strength of the proposed substitution boxes. Majority logic criterion results indicate that the proposed substitution boxes are well suited for the techniques of secure communication.

Construction of spread codes based on Abelian non-cyclic orbit codes

Let Fq be the finite field with q elements (where q is a prime power). Since any invertible matrix maps subspaces to subspaces of the same dimension we have a group action of the general linear group, GLn(Fq), on Gq(k,n) (Grassmann variety). The orbits of a subgroup of GLn(Fq) acting on the Grassmann variety are called (subspace) orbit codes. When the subgroup acting on Gq(k,n) is cyclic the associated codes are called cyclic orbit codes. We make a construction abelian non-cyclic orbit codes by making full use of the companion matrix of a primitive polynomial over finite fields, it is a partial spread code. Based on this code, an optimal partial spread code is obtained. Our results answer the first of two open problems presented by Climent et al.

An extension of a construction of covering arrays

AbstractBy Raaphorst et al, for a prime power , covering arrays (CAs) with strength 3 and index 1, defined over the alphabet , were constructed using the output of linear feedback shift registers defined by cubic primitive polynomials in . These arrays have rows and columns. We generalize this construction to apply to all polynomials; provide a new proof that CAs are indeed produced; and analyze the parameters of the generated arrays. Besides arrays that match the parameters of those of Raaphorst et al, we obtain arrays matching some constructions that use Chateauneuf‐Kreher doubling; in both cases these are some of the best arrays currently known for certain parameters.

A novel discrete image encryption algorithm based on finite algebraic structures

The arithmetic properties of a finite field have a remarkable impact on the security features of symmetric and asymmetric cryptosystems. Conventionally, in modern symmetric-key cryptosystems, a 256 elements Galois field depending on a single 8 degree primitive irreducible polynomial over ℤ2 is used for designing an S-box, the main nonlinear component in AES. The experience of S-box based image encryption schemes is not appreciable than chaotic systems based methods. In this study, we improve the S-box based image encryption algorithms by the usage of all 16 distinct degree 8 primitive irreducible polynomials over ℤ2 and by introducing a new role of the ring ℤn of integers modulo n in the permutation steps. The strength of this novel 2-algebraic structures based image encryption scheme is evaluated by some statistical analyses and a significant performance level is achieved. A comparison of encryption quality with some of the recent chaos-based image encryption algorithms is given and evidently the newly launched scheme spectacles high performance. Thus this new development in image encryption methods may provide an alternative to chaos dependent image encryption.

Pseudo Random Binary Sequence Based on Cyclic Difference Set

With the increasing reliance on technology, it has become crucial to secure every aspect of online information where pseudo random binary sequences (PRBS) can play an important role in today’s world of Internet. PRBS work in the fundamental mathematics behind the security of different protocols and cryptographic applications. This paper proposes a new PRBS namely MK (Mamun, Kumu) sequence for security applications. Proposed sequence is generated by primitive polynomial, cyclic difference set in elements of the field and binarized by quadratic residue (QR) and quadratic nonresidue (QNR). Introduction of cyclic difference set makes a special contribution to randomness of proposed sequence while QR/QNR-based binarization ensures uniformity of zeros and ones in sequence. Besides, proposed sequence has maximum cycle length and high linear complexity which are required properties for sequences to be used in security applications. Several experiments are conducted to verify randomness and results are presented in support of robustness of the proposed MK sequence. The randomness of proposed sequence is evaluated by popular statistical test suite, i.e., NIST STS 800-22 package. The test results confirmed that the proposed sequence is not affected by approximations of any kind and successfully passed all statistical tests defined in NIST STS 800-22 suite. Finally, the efficiency of proposed MK sequence is verified by comparing with some popular sequences in terms of uniformity in bit pattern distribution and linear complexity for sequences of different length. The experimental results validate that the proposed sequence has superior cryptographic properties than existing ones.

WAGE: An Authenticated Encryption with a Twist

This paper presents WAGE, a new lightweight sponge-based authenticated cipher whose underlying permutation is based on a 37-stage Galois NLFSR over F27. At its core, the round function of the permutation consists of the well-analyzed Welch-Gong permutation (WGP), primitive feedback polynomial, a newly designed 7-bit SB sbox and partial word-wise XORs. The construction of the permutation is carried out such that the design of individual components is highly coupled with cryptanalysis and hardware efficiency. As such, we analyze the security of WAGE against differential, linear, algebraic and meet/miss-in-the-middle attacks. For 128-bit authenticated encryption security, WAGE achieves a throughput of 535 Mbps with hardware area of 2540 GE in ASIC ST Micro 90 nm standard cell library. Additionally, WAGE is designed with a twist where its underlying permutation can be efficiently turned into a pseudorandom bit generator based on the WG transformation (WG-PRBG) whose output bits have theoretically proved randomness properties.

S-box Construction Based on Linear Fractional Transformation and Permutation Function

Substitution boxes (S-box) with strong and secure cryptographic properties are widely used for providing the key property of nonlinearity in block ciphers. This is critical to be resistant to a standard attack including linear and differential cryptanalysis. The ability to create a cryptographically strong S-box depends on its construction technique. This work aims to design and develop a cryptographically strong 8 × 8 S-box for block ciphers. In this work, the construction of the S-box is based on the linear fractional transformation and permutation function. Three steps involved in producing the S-box. In step one, an irreducible polynomial of degree eight is chosen, and all roots of the primitive irreducible polynomial are calculated. In step two, algebraic properties of linear fractional transformation are applied in Galois Field GF (28). Finally, the produced matrix is permuted to add randomness to the S-box. The strength of the S-box is measured by calculating its potency to create confusion. To analyze the security properties of the S-box, some well-known and commonly used algebraic attacks are used. The proposed S-box is analyzed by nonlinearity test, algebraic degree, differential uniformity, and strict avalanche criterion which are the avalanche effect test, completeness test, and strong S-box test. S-box analysis is done before and after the application of the permutation function and the analysis result shows that the S-box with permutation function has reached the optimal properties as a secure S-box.

Construction of primitive polynomials over finite fields

In this paper, using the polynomial composition methods some computationally simple and explicit ways for constructing higher degrees primitive polynomials from a given primitive polynomial over [Formula: see text] are given.

Method for Synthesis of Gordon–Mills–Welch Sequences for Discrete Information Transmission Systems

A method for synthesis of Gordon–Mills–Welch sequences (GMWSs) over extended field GF[(2m)n], based on the relationship of the roots of check polynomial hМS(x) for the basic M-sequence (MS) with the roots of polynomial factors hсi(x) of check polynomial hGMWS(x) for a GMWS is considered. It is shown that the synthesis includes 1) an algorithm for determining polynomials hсi(x), based upon the use of p-conjugate elements with odd powers βr of an element belonging to subfield GF(2m), 2) an algorithm for determining the initial states of the shift registers constructed from polynomials hсi(x), which is based on the decimation of the symbols of the basis MS, 3) and a method for determining the complete lists of check polynomials hGMWS(x) for all primitive polynomials of field GF[(2m)n], based on which the basis MSs are formed.

Conversion of Element Representations in Galois Rings

We are developing a symbolic calculator in order to computationally operate within Galois rings algebraic structure. In any Galois ring, whose characteristic is a power of a prime, each element has an additive representation, which is basically a remainder polynomial when dividing by a basic irreducible polynomial, and a p-adic representation given by Teichmuller elements, which are powers of roots of basic primitive polynomials. In this paper we introduce basic procedures to obtain Hensel’s lifts of primitive polynomials and the conversion between additive and p-adic representations.

Studying of the scrambling coding sequence performance based on the ninth order primitive polynomial in telecommunication networks of information systems

Scrambling coding sequences find wide application in telecommunication networks to improve noise immunity and information transmission concealment. There are many different scrambling coding sequences with different autocorrelation properties from which one can be chosen. In the previous paper the authors researched a possibility of using scrambling coding sequences built on the base of the primitive eighth degree polynomial in telecommunication networks of information systems to enhance noise immunity and concealment of information transmission. This sequence, consisting of two hundred and fifty five chips, showed some good performance in terms of information transmission quality indicators. Nevertheless, some of its limitations were also revealed, which are mostly linked to the number of used chips. This paper aims to overcome the problems by proposing to use scrambling coding sequences based on the ninth degree primitive polynomial. The required polynomial was selected and the needed scrambling sequence generated. This scrambling coding sequence includes five hundred and eleven chips. Computer simulation helped to establish that scrambling coding sequence synthesized using this polynomial permits to substantially improve information transmission quality indicators as compared to the case of the scrambling coding sequence based on the eighth degree polynomial. For example, the interfering signal and internal noise to desired signal ratio, for which reception of transmitted bit sequence is still possible, increased by three decibels. It is clearly linked to the doubling of the number of the used chips. To preserve the transmission rate, the occupied frequency spectrum needs to be doubled. An inference is made as to a possibility of using the synthesized coding sequence in practical realizations of telecommunication channels.

ВИКОРИСТАННЯ СКРЕМБЛЮЮЧИХ КОДОВИХ ПОСЛІДОВНОСТЕЙ НА ОСНОВІ ПРИМІТИВНИХ ПОЛІНОМІВ У ТЕЛЕКОМУНІКАЦІЙНИХ МЕРЕЖАХ ІНФОРМАЦІЙНИХ СИСТЕМ

ВИКОРИСТАННЯ СКРЕМБЛЮЮЧИХ КОДОВИХ ПОСЛІДОВНОСТЕЙ НА ОСНОВІ ПРИМІТИВНИХ ПОЛІНОМІВ У ТЕЛЕКОМУНІКАЦІЙНИХ МЕРЕЖАХ ІНФОРМАЦІЙНИХ СИСТЕМ

Study of scrambling coding sequence utilization in telecommunication networks of information systems

Problems of securing noise immunity and information transmission concealment in telecommunication networks of information systems are considered. In order to improve performance, it is proposed to use scrambling coding sequences, which spread the spectrum of the bits being transmitted. To synthesize new scrambling coding sequences, tenth degree primitive polynomial is used. Autocorrelation properties of the obtained scrambling coding sequence are studied. Limitations caused by those properties are deduced. In order to establish a possibility of synthesized scrambling coding sequence use in the telecommunication networks data transmission channels, it is carried out computer simulation of the transmission process of the bit set, whose spectrum is spread by the obtained sequence. In the process of computer simulation, a mixture of the desired signal in the form of a scrambled bit set, internal noise of information transmission channel and interfering signal was formed. While the powers of the internal noise and interfering signal remained constant, power of the bit sequence gradually assumed three discrete values. As a result of the research, it is established that utilization of the synthesized scrambling sequence permits to perform extraction of the bits being transmitted, under conditions in which the power of the noise interfering signal and the internal noise of the channel surpasses power of the desired channel by up to 18 decibel. The limitations set by the use of the synthesized coding sequence are described. An inference concerning a possibility of obtained coding sequence use in practical implementation of telecommunication channels of information networks is made.

ВИКОРИСТАННЯ ЦИКЛІЧНИХ ЗСУВІВ ПСЕВДОВИПАДКОВОЇ КОДОВОЇПОСЛІДОВНОСТІ ДЛЯ ПІДВИЩЕННЯ ХАРАКТЕРИСТИК ТЕЛЕКОМУНІКАЦІЙНОГО КАНАЛУ

An approach is proposed to design of noise immune and concealed data transfer channel for telecommunication networks. Attention is paid to securing hidden information transmission, as well as its protection from interception by rogue actors. The approach is based on the desired bits spectrum spreading and their additional scrambling by using pseudo noise coding sequences derived from primitive polynomials of eighth and fifteenth orders, which possess good auto and inter correlation properties. It is studied performance of the telecommunication channel that includes frames of 128 bit length, each of which is spectrally spread 256 times with the help of a synthesized pseudo noise coding sequence. The second 32768 chip-long pseudo noise coding sequence is used to mark the frame duration and perform additional information scrambling. Computer simulation is used to study performance of the designed telecommunication channel. The computer simulation helped to establish that the processing of the additive mixture of the desired signal and interfering ones, which surpass the desired signal two times in terms of power, by the matched filters permits to confidently reveal the information frame structure being transmitted by determining frame beginning pulse and establish the bit values of the desired information. Further improvement of information protection from interception is proposed to achieve by using cyclic shifts of 32768 chip-long pseudo noise coding sequence. Computer simulation helped to find out that ignorance of the cyclic shift leads to inability of information interception by the rogue elements. Research results, obtained in this paper, permit to claim that the designed telecommunication channel, with cyclic shifts according to a secret rule, could be successfully used in practical implementations of noise immune and concealed telecommunication networks.

Zero-knowledge identification scheme with companion matrices of primitive polynomials

This paper proposes the matrix power problem, that is, to find x given CxDx, where C and D are the companion matrices of primitive polynomials over finite field. And a new zero-knowledge identification scheme based on matrix power problem is proposed. It is perfect zero-knowledge for honest verifiers. Due to its simplicity, and low-memory and low-computation costs, the proposed scheme is suitable for using in computationally limited devices for identification such as smart cards.