Prime Modulus Research Articles

An Algorithm to Find the Irreducible Polynomials Over Galois Field GF(pm)

Irreducible Polynomials over GF(p) and the multiplicative inverses under it are important in cryptography. Presently the method of deriving irreducible polynomials of a particular prime modulus is very primitive and time consuming. In this paper, in order to find all irreducible polynomials, be it monic or non-monic, of all prime moduli p with all its order m, a fast deterministic computer algorithm based on an algebraic method producing a (m×m) matrix is proposed. The maximum number of terms in each column of the matrix is 2 where j is the column index. General Terms Algorithms, Irreducible polynomial.

Image Encryption Using Chaotic Map and Prime modulo Multiplicative LinearCongruential Generator

The chaos based cryptographic algorithms have suggested some new and efficient ways to develop secure image encryption techniques because of its exceptionally desirable properties of sensitivity to initial condition and parameters of chaotic map. In this study an image encryption algorithm is presented, in which chaos based encryption technique and prime modulo multiplicative linear Congruential generator (PMMLCG) has been used to improve the quality of image encryption and to achieve the goal of pixel scrambling. The proposed algorithm generates random numbers, which are used in row shuffling, column shuffling and pixel scrambling which results in encryption technique highly resistive to cryptanalytic attack. Results and effectiveness is measured in qualitative and quantitative metrics which shows that this encryption algorithm is securefor encryption of images.

Sign changes of Kloosterman sums with almost prime moduli

We prove that the Kloosterman sum $S(1,1;c)$ can change sign infinitely often as $c$ runs over squarefree moduli with at most 10 prime factors, which improves the previous results of E. Fouvry and Ph. Michel, J. Sivak-Fischler and K. Matom\"{a}ki, replacing 10 by 23, 18 and 15, respectively. The method combines the Selberg sieve, equidistribution of Kloosterman sums and spectral theory of automorphic forms.

Kloosterman sums in residue rings

We generalize some of our previous results on Kloosterman sums [Izv. Mat., to appear] for prime moduli to general moduli. This requires establishing the corresponding additive properties of the reciprocal-set $ I^{-1}=\{x^{-1}: x\in I\}, $ where $I$ is a

Equivalence classes of colorings

For any link and for any modulus $m$ we introduce an equivalence relation on the set of non-trivial m-colorings of the link (an m-coloring has values in Z/mZ). Given a diagram of the link, the equivalence class of a non-trivial m-coloring is formed by each assignment of colors to the arcs of the diagram that is obtained from the former coloring by a permutation of the colors in the arcs which preserves the coloring condition at each crossing. This requirement implies topological invariance of the equivalence classes. We show that for a prime modulus the number of equivalence classes depends on the modulus and on the rank of the coloring matrix (with respect to this modulus).

Differential characterization of Wilson primes for 𝔽q[t

We consider an analog, when ℤ is replaced by Fq[t], of Wilson primes, namely the primes satisfying Wilson’s congruence (p−1)!≡−1 to modulus p2 rather than the usual prime modulus p. We fully characterize these primes by connecting these or higher power congruences to other fundamental quantities such as higher derivatives and higher difference quotients as well as higher Fermat quotients. For example, in characteristic p>2, we show that a prime ℘ of Fq[t] is a Wilson prime if and only if its second derivative with respect to t is 0 and in this case, further, that the congruence holds automatically modulo ℘p−1. For p=2, the power p−1 is replaced by 4−1=3. For every q, we show that there are infinitely many such primes.

On the least prime primitive root

We study the uniform upper bound for the least prime that is a primitive root. Let g ⁎ ( q ) be the least prime primitive root (mod q ) where q is a prime power or twice a prime power of a prime p . The upper bound for g ⁎ ( q ) is studied by many authors who succeeded in establishing various conditional upper bounds. However, no uniform bounds were known other than Linnikʼs bound on the least prime in an arithmetic progression. In this paper, we prove that g ⁎ ( q ) ≪ p 3.1 . The exponent 3.1 is improved from the known exponent 4.5 from Linnikʼs bound for the prime modulus.

Finding the best portable congruential random number generators

Linear congruential random number generators must have large moduli to attain maximum periods, but this creates integer overflow during calculations. Several methods have been suggested to remedy this problem while obtaining portability. Approximate factoring is the most common method in portable implementations, but there is no systematic technique for finding appropriate multipliers and an exhaustive search is prohibitively expensive. We offer a very efficient method for finding all portable multipliers of any given modulus value. Letting M=AB+C, the multiplier A gives a portable result if B−C is positive. If it is negative, the portable multiplier can be defined as A=⌊M/B⌋. We also suggest a method for discovering the most fertile search region for spectral top-quality multipliers in a two-dimensional space. The method is extremely promising for best generator searches in very large moduli: 64-bit sizes and above. As an application to an important and challenging problem, we examined the prime modulus 263−25, suitable for 64-bit register size, and determined 12 high quality portable generators successfully passing stringent spectral and empirical tests.

Construction of filled phased antenna arrays from thinned arrays with position-coded elements on the basis of the properties of prime and composite number moduli

A method for construction of filled phased antenna arrays (PAAs) by placing thinned arrays with elements whose positions are coded in a lattice of nodes (PAA-Cs) at equidistant nodes of the same lattice with the use of the properties of prime and composite number moduli p is proposed. It is shown that, for moduli p satisfying certain conditions, it is possible to fill asymptotically one-half of the lattice nodes with elements of different PAA-Cs constructed from all primitive roots of a particular modulus p so that the elements of different PAA-Cs coincide at only two lattice nodes. Conditions for prime modulus p and the number of corresponding primitive roots as well as all values of modulus p up to 1000 satisfying these conditions are presented. For several values from this number set, examples of the arrangement of the elements of all PAA-Cs in a lattice of equidistant nodes are given. For these configurations, resulting patterns of receiving PAAs in which the pattern is formed after amplifiers (receiving active PAAs) are calculated.

Large-Order Multiple Recursive Generators with Modulus 231 − 1

The performance of a maximum-period multiple recursive generator (MRG) depends on the choices of the recurrence order k, the prime modulus p, and the multipliers used. For a maximum-period MRG, a large-order k not only means a large period length (i.e., pk − 1) but, more importantly, also guarantees the equidistribution property in high dimensions (i.e., up to k dimensions), a desirable feature for a good random-number generator. As to generating efficiency, in addition to the multipliers, some special choices of the prime modulus p can significantly speed up the generation of pseudo-random numbers by replacing the expensive modulo operation with efficient logical operations. To construct efficient maximum-period MRGs of a large order, we consider the prime modulus p = 231 − 1 and, via extensive computer search, find two large values of k, 7,499 and 20,897, for which pk − 1 can be completely factorized. The successful search is achieved with the help of some results in number theory as well as some modern factorization methods. A general class of MRGs is introduced, which includes several existing classes of efficient generators. With the factorization results, we are able to identify via computer search within this class many portable and efficient maximum-period MRGs of order 7,499 or 20,897 with prime modulus 231 − 1 and multipliers of powers-of-two decomposition. These MRGs all pass the stringent TestU01 test suite empirically.

ON SOME EXACT RECIPROCITY FORMULAS FOR TWISTED SECOND MOMENTS OF DIRICHLET L-FUNCTIONS

We derive the exact reciprocity formula connecting the twisted second moments at 1 for families of odd Dirichlet characters corresponding to two prime moduli, which parallels such formula discovered earlier for the special point 1/2.

오류주입 공격에 강건하며 병렬연산이 가능한 RSA-CRT

RSA-CRT는 RSA의 속도를 개선하기 위한 가장 대표적인 기법이다. RSA-CRT는 RSA에 사용되는 두 비밀소수의 법에서 각각 연산을 수행하기 때문에 RSA에 비해 약 4배가량 빠른 속도로 연산할 수 있다. 하지만 RSA에서 법 생성 후 바로 파기할 수 있었던 비밀 소수를 연산에 직접 사용함으로서 오류 주입공격 시 이를 노출하게 되는 문제가 있다. 이를 해결하기 위한 가장 대표적인 기법이 오류 확산에 기반을 둔 기법이다. 이 기법은 주입된 오류가 암호문 전체에 영향을 미치기 때문에 공격자가 비밀 소수를 얻기 힘들지만 독립적으로 진행되었던 연산을 순차적으로 해야 하며, 여전히 오류주입 공격에 취약하다는 문제점이 있다. 본 논문에서는 오류주입 공격에 강건하며 병렬처리가 가능하도록 공통법을 이용한 RSA-CRT 기법과 메시지를 각각의 법에서 연산한 RSA-CRT기법을 제안한다. 제안하는 기법은 최대 병렬연산을 통해 2회의 지수연산 시간밖에 소요되지 않기 때문에 빠른 연산속도를 제공하면서 오류주입 공격으로부터 비밀 소수의 노출을 보호할 수 있다. RSA-CRT is one of the commonly used techniques to speedup RSA operation. Since RSA-CRT performs its operations based on the modulus of two private primes, it is about four times faster than RSA. In RSA, the two primes are normally thrown away after generating the public key pair. However, in RSA-CRT, the two primes are directly used in RSA operations. This led to hardware fault attacks which can be used to factor the public modulus. The most common way to counter these attacks is based on error propagation. In these schemes, all the outputs of RSA are affected by the infected error which makes it difficult for an adversary to use the output to factor the public modulus. However, the error propagation has sequentialized the RSA operation. Moreover, these schemes have been found to be still vulnerable to hardware fault attacks. In this paper, we propose two new RSA-CRT schemes which are both resistant to hardware fault attack and support parallel execution: one uses common modulus and the other one perform operations in each prime modulus. Both proposed schemes takes about a time equal to two exponentiations to complete the RSA operation if parallel execution is fully used and can protect the two private primes from hardware fault attacks.

Splitting Fields and Periods of Fibonacci Sequences Modulo Primes

SummaryWe consider the period of a Fibonacci sequence modulo a prime and provide an accessible, motivated treatment of this classical topic using only ideas from linear and abstract algebra. Our methods extend to general recurrences with prime moduli and provide some new insights. And our treatment highlights a nice application of the use of splitting fields that might be suitable to present in an undergraduate course in abstract algebra or Galois theory.

Construction of thinned phased antenna arrays by coding the positions of array elements in a equidistant grid of nodes on the basis of the properties of prime and composite number moduli

A method for construction of thinned phased antenna arrays (PAAs) by coding the positions of array elements in an equidistant grid of nodes on the basis of the properties of prime and composite number moduli is proposed. (Such arrays are abbreviated as PAA-C.) An algorithm for the arrangement of array elements is described, examples of the arrangement of elements are presented, and the array patterns and their cross sections are calculated. A method for suppression of sidelobes of the PAA-C pattern is proposed. The method is based on the multiplication of the patterns of PAA-Cs differing in their codes corresponding to one modulus or in coding by twin prime numbers.

An exhaustive search for good 64-bit linear congruential random number generators with restricted multiplier

This paper explores that the different prime moduli can affect both the number of primitive root and the spectral test performance for 64-bit linear congruential generators (LCGs). Three forms of prime modulus and two types restriction on multiplier are considered in this paper. We perform computerized experiments that indicate significant differences exist among the number of primitive root of three forms of prime modulus. These differences can affect the performance of the spectral test. Two good 64-bit LCGs with significantly better spectral values and excellent empirical performance are presented. They are suitable for the requirements of todayʼs computer simulation studies.

Number theory: a lively introduction with proofs, applications, and stories

Preface. To the Student. To the Instructor. Acknowledgements. 0. Prologue. 1. Numbers, Rational and Irrational. (Historical figures: Pythagoras and Hypatia). 1.1 Numbers and the Greeks. 1.2 Numbers you know. 1.3 A First Look at Proofs. 1.4 Irrationality of he square root of 2. 1.5 Using Quantifiers. 2. Mathematical Induction. (Historical figure: Noether). 2.1.The Principle of Mathematical Induction. 2.2 Strong Induction and the Well Ordering Principle. 2.3 The Fibonacci Sequence and the Golden Ratio. 2.4 The Legend of the Golden Ratio. 3. Divisibility and Primes. (Historical figure: Eratosthenes). 3.1 Basic Properties of Divisibility. 3.2 Prime and Composite Numbers. 3.3 Patterns in the Primes. 3.4 Common Divisors and Common Multiples. 3.5 The Division Theorem. 3.6 Applications of gcd and lcm. 4.The Euclidean Algorithm. (Historical figure: Euclid). 4.1 The Euclidean Algorithm. 4.2 Finding the Greatest Common Divisor. 4.3 A Greeker Argument that the square root of 2 isIrrational. 5. Linear Diophantine Equations. (Historical figure: Diophantus). 5.1 The Equation aX + bY = 1. 5.2 Using the Euclidean Algorithm to Find a Solution. 5.3 The Diophantine Equation aX + bY = n. 5.4 Finding All Solutions to a Linear Diophantine Equation. 6. The Fundamental Theorem of Arithmetic. (Historical figure: Mersenne). 6.1 The Fundamental Theorem. 6.2 Consequences of the Fundamental Theorem. 7. Modular Arithmetic. (Historical figure: Gauss). 7.1 Congruence modulo n. 7.2 Arithmetic with Congruences. 7.3 Check Digit Schemes. 7.4 The Chinese Remainder Theorem. 7.5 The Gregorian Calendar. 7.6 The Mayan Calendar. 8. Modular Number Systems. (Historical figure: Turing). 8.1 The Number System Zn: an InformalView. 8.2 The Number System Zn: Definitionand Basic Properties. 8.3 Multiplicative Inverses in Zn. 8.4 Elementary Cryptography. 8.5 Encryption Using Modular Multiplication. 9. Exponents Modulo n. (Historical figure: Fermat). 9.1 Fermat's Little Theorem. 9.2 Reduced Residues and the Euler \phi-function. 9.3 Euler's Theorem. 9.4 Exponentiation Ciphers with a Prime modulus. 9.5 The RSA Encryption Algorithm. 10. Primitive Roots. (Historical figure: Lagrange). 10.1 Zn. 10.2 Solving Polynomial Equations inZn. 10.3 Primitive Roots. 10.4 Applications of Primitive Roots. 11. Quadratic Residues. (Historical figure: Eisenstein) 11.1 Squares Modulo n 11.2 Euler's Identity and the Quadratic Character of -1 11.3 The Law of Quadratic Reciprocity 11.4 Gauss's Lemma 11.5 Quadratic Residues and Lattice Points. 11.6 The Proof of Quadratic Reciprocity. 12. Primality Testing. (Historical figure: Erdos). 12.1 Primality testing. 12.2 Continued Consideration of Charmichael Numbers. 12.3 The Miller-Rabin Primality test. 12.4 Two Special Polynomial Equations inZp. 12.5 Proof that Millar-Rabin is Effective. 12.6 Prime Certificates. 12.7 The AKS Deterministic Primality Test. 13. Gaussian Integers. (Historical figure: Euler). 13.1 Definition of Gaussian Integers 13.2 Divisibility and Primes in Z[i]. 13.3 The Division Theorem for the Gaussian Integers. 13.4 Unique Factorization in Z[i]. 13.5 Gaussian Primes. 13.6 Fermat's Two Squares Theorem. 14. Continued Fractions. (Historical figure: Ramanujan). 14.1 Expressing Rational Numbers as Continued Fractions. 14.2 Expressing Irrational Numbers as Continued Fractions. 14.3 Approximating Irrational Numbers Using ContinuedFractions. 14.4 Proving that Convergents are Fantastic Approximations. 15. Some Nonlinear Diophantine Equations. (Historical figure: Germain). 15.1 Pell's Equation 15.2 Fermat's Last Theorem 15.3 Proof of Fermat's Last Theorem for n = 4. 15.4 Germain's Contributions to Fermat's Last Theorem 15.5 A Geometric look at the Equation x4 +y4 = z2. Appendix: Axioms of Number Theory. A.1 What is a Number System? A.2 Order Properties of the Integers. A.3 Building Results From Our Axioms. A.4 The Principle of Mathematical Induction.

The fourth moment of Dirichlet $L$-functions

We compute the fourth moment of Dirichlet L-functions at the central point for prime moduli, with a power savings in the error term.

Congruences for the Andrews spt function

Ramanujan-type congruences for the Andrews spt(n) partition function have been found for prime moduli 5 ≤ ℓ ≤ 37 in the work of Andrews [Andrews GE, (2008) J Reine Angew Math 624:133-142] and Garvan [Garvan F, (2010) Int J Number Theory 6:1-29]. We exhibit unexpectedly simple congruences for all ℓ≥5. Confirming a conjecture of Garvan, we show that if ℓ≥5 is prime and (-δ/ℓ) = 1, then spt[(ℓ2(ℓn+δ)+1)/24] ≡ 0 (mod ℓ). This congruence gives (ℓ - 1)/2 arithmetic progressions modulo ℓ(3) which support a mod ℓ congruence. This result follows from the surprising fact that the reduction of a certain mock theta function modulo ℓ, for every ℓ≥5, is an eigenform of the Hecke operator T(ℓ(2)).

A 32-bit linear congruential random number generator with prime modulus

This paper explores that the different prime moduli can affect both the number of primitive root and the spectral test performance for a 32-bit linear congruential generator (LCG). We consider five forms of prime modulus: the Mersenne prime modulus, the largest prime modulus, the Sophie-Germain prime modulus, the twin prime modulus and the factorial prime modulus. We perform a computerized experiment that indicates significant differences exist among the number of primitive root of the five forms of prime modulus, and demonstrate that these differences can affect the performance of spectral test.

Mapping the discrete logarithm

The discrete logarithm is a problem that surfaces frequently in the field of cryptography as a result of using the transformation g mod n. This paper focuses on a prime modulus, p, for which it is shown that the basic structure of the functional graph is largely dependent on an interaction between g and p − 1. In fact, there are precisely as many different functional graph structures as there are divisors of p− 1. This paper extracts two of these structures, permutations and binary functional graphs. Estimates exist for the shape of a random permutation, but similar estimates must be created for the binary functional graphs. Experimental data suggests that both the permutations and binary functional graphs correspond well to the theoretical data which provides motivation to extend this to larger divisors of p − 1 and study the impact this forced structure has on the many cryptographic algorithms that rely on the discrete logarithm for their security. This is especially applicable to those algorithms that require a “safe” prime (p = 2q + 1, where q is prime) modulus since all non-trivial functional graphs generated using a safe prime modulus can be analyzed by the framework presented here.