Polynomial Congruences Research Articles

Curious congruences for cyclotomic polynomials II

Curious congruences for cyclotomic polynomials II

A digital signature scheme based on problem of solving polynomial congruence in residue class modulo n

Many digital signature schemes have been proposed based on different mathematical problems. Some of them are based on factoring into primes, discrete logarithm problem, elliptic curve discrete logarithm problem, lattice problem, multivariate quadratic problem etc. In this work a signature scheme is proposed which security is based on problem of solving polynomial congruence modulo some positive integer. In proposed scheme degree of polynomial is not fixed, so degree can be changed to increase security. It is also efficient because in all the phases modulo addition and multiplication are used and no need to calculate higher exponent

Curious congruences for cyclotomic polynomials

Let \(\Phi _n^{(k)}(x)\) be the kth derivative of the nth cyclotomic polynomial. We are interested in the values \(\Phi _n^{(k)}(1)\) for fixed positive integers n. D. H. Lehmer proved that \(\Phi _n^{(k)}(1)/ \Phi _n(1)\) is a polynomial of the Euler totient function \(\phi (n)\) and the Jordan totient functions and gave its explicit formula. In this paper, we give a quick proof that \(\Phi _n^{(k)}(1)/\Phi _n(1)\) is a polynomial of them without giving the explicit form. In the final section, we deduce some curious congruences: \(2\Phi ^{(3)}_n(1)\) is divisible by \(\phi (n)-2\). Moreover, if k is greater than 1, then \(\Phi ^{(2k+1)}_n(1)\) is divisible by \(\phi (n)-2k\). The proof depends on a new combinatorial identity for general self-reciprocal polynomials over \({{\mathbb {Z}}}\), which gives rise to a formula that expresses the value \(\Phi _n^{(k)}(1)\) as a \({{\mathbb {Z}}}\)-linear combination of the coefficients in the minimal polynomial of \(2\cos (2\pi /n)-2\). As a supplement, we show the monotonic increasing property of \(\Phi _n(x)\) on \([1,\infty )\) in two ways.

Parametrizing roots of polynomial congruences

We use the arithmetic of ideals in orders to parameterize the roots $\mu \pmod m$ of the polynomial congruence $F(\mu) \equiv 0 \pmod m$, $F(X) \in \mathbb{Z}[X]$ monic, irreducible and degree $d$. Our parameterization generalizes Gauss's classic parameterization of the roots of quadratic congruences using binary quadratic forms, which had previously only been extended to the cubic polynomial $F(X) = X^3 - 2$. We show that only a special class of ideals are needed to parameterize the roots $\mu \pmod m$, and that in the cubic setting, $d = 3$, general ideals correspond to pairs of roots $\mu_1 \pmod{m_1}$, $\mu_2 \pmod {m_2}$ satisfying $\gcd(m_1, m_2, \mu_1 - \mu_2) = 1$. At the end we illustrate our parameterization and this correspondence between roots and ideals with a few applications, including finding approximations to $\frac{\mu}{m} \in \mathbb{R}/ \mathbb{Z}$, finding an explicit Euler product for the co-type zeta function of $\mathbb{Z}[2^{\frac{1}{3}}]$, and computing the composition of cubic ideals in terms of the roots $\mu_1 \pmod {m_1}$ and $\mu_2 \pmod {m_2}$.

A Secured Storage and Communication System for Cloud Using ECC, Polynomial Congruence and DSA

Security is necessary for performing the effective data communications in internet and cloud environment. The data security in cloud is becoming very challenging task due to the presence of huge volume of cloud users. Even though, the various secured cloud data storage mechanisms are available and tried to fulfill the current requirement in terms of security level and efficiency. However, the available works are not obtained the enough security level and the efficiency while storing and sharing the data in cloud. For this purpose, this work proposes a new secured storage method for providing data security to store the data securely in cloud. The proposed system consists of three different algorithms for performing prime number generation, digital signature creation for generating keys, encryption, decryption and authorization. Here, a new technique to find a co-prime number which is used to generate keys in key generation process and also useful for performing encryption and decryption process. Moreover, a new key generation technique is also introduced for elliptic curve cryptography and digital signature algorithm using polynomial congruence for performing key generation process securely. In addition, a new elliptic curve and polynomial congruence based encryption/decryption algorithms for performing data encryption and decryption in the process of data storage and communication. Finally, the user authenticity is verified by the proposed digital signature algorithm in this work. The experiments have been conducted for evaluating the proposed secured storage and communication system and proved as better than others in terms of efficiency and security level.

Recurrences and Congruences for Higher-Order Geometric Polynomials and Related Numbers

We obtain new recurrence relations, an explicit formula, and convolution identities for higher-order geometric polynomials. These relations generalize known results for geometric polynomials and lead to congruences for higher-order geometric polynomials and, in particular, for p-Bernoulli numbers.

Recurrences and congruences for higher order geometric polynomials and related numbers

UDC 517.5We obtain new recurrence relations, an explicit formula, and convolution identities for higher order geometric polynomials. These relations generalize known results for geometric polynomials, and lead to congruences for higher order geometric polynomials, particularly for -Bernoulli numbers.

Chinese remainder theorem of modules over multivariate polynomial rings

The Chinese remainder theorem plays an important role in the areas of number theory and algebra. In the multivariate polynomial rings, the Chinese remainder theorem can be used to solve systems of multivariate polynomial congruence equations. With the help of algebra theory, polynomials that satisfy the congruence equations can be found. In this paper, we investigate the Chinese remainder theorem of modules over multivariate polynomial rings. Using the theory and method of Gröbner bases for modules, we find the polynomial vector that satisfies the Chinese remainder theorem of modules. Furthermore, we obtain the solutions of systems of congruence equations for modules over polynomial rings.

Sequences and Polynomial Congruence

Sequences and Polynomial Congruence

Correction to: Identities behind some congruences for r-Bell and derangement polynomials

In the original article there were capitalization errors in two of the references. The corrected references follow:

Identities behind some congruences for r-Bell and derangement polynomials

We derive new congruences bounding r-Bell and derangement polynomials, which generalize the existing ones, while the presented approach is significantly simpler and, at the same time, more informative. Namely, we provide precise identities that imply the congruences and explain somehow their nature.

P-MEC: Polynomial Congruence-Based Multimedia Encryption Technique Over Cloud

In this multimedia era, the security of multimedia content over the Cloud becomes a crucial research area. There are several modes of communication available for transmitting the information over the Cloud from one place to another place around the globe. Such communication networks are open to all where anyone can access the data very easily. Therefore, the security of multimedia data over the Cloud becomes a major challenge. This study highlights better protection of multimedia content over Cloud in comparison to the existing approaches, where the information cannot be unwrapped in a reasonable time. The Strength of RSA, ElGamal, Diffie Hellman, and many more Cryptographic algorithms depend on the concept of congruence. As of now, the solution for linear and quadratic congruence had made its use limited, but this study had changed it by introducing the solution for Cubic and Polynomial congruence. The polynomial congruence concept has been used to encrypt the multimedia content. Based on the qualitative and quantitative experimental analysis, it can be concluded that the P-MEC model stands firm compared with the state-of-the-art models. The security analysis and experimental results display that the proposed scheme has strong robustness against statistical attack, vulnerability attack, and other comprehensive attacks.

On a Conjecture of Igusa in Two Dimensions

We extend work of Denef and Sperber and also Cluckers regarding a conjecture of Igusa in the two dimensional setting by no longer requiring the polynomial to be nondegenerate with respect to its Newton diagram. More precisely we establish sharp, uniform bounds for complete exponential sums and the number of polynomial congruences for general quasi-homogeneous polynomials in two variables.

The smallest root of a polynomial congruence

The smallest root of a polynomial congruence

A q-microscope for supercongruences

By examining asymptotic behavior of certain infinite basic (q-) hypergeometric sums at roots of unity (that is, at a ‘q-microscopic’ level) we prove polynomial congruences for their truncations. The latter reduce to non-trivial (super)congruences for truncated ordinary hypergeometric sums, which have been observed numerically and proven rarely. A typical example includes derivation, from a q-analogue of Ramanujan's formula∑n=0∞(4n2n)(2nn)228n32n(8n+1)=23π, of the two supercongruencesS(p−1)≡p(−3p)(modp3)andS(p−12)≡p(−3p)(modp3), valid for all primes p>3, where S(N) denotes the truncation of the infinite sum at the N-th place and (−3⋅) stands for the quadratic character modulo 3.

Counting factorisations of monomials over rings of integers modulo N

A sharp bound is obtained for the number of ways to express the monomial X n as a product of linear factors over ℤ/p α ℤ. The proof relies on an induction-on-scale procedure which is used to estimate the number of solutions to a certain system of polynomial congruences. The method also applies to more general systems of polynomial congruences that satisfy a non-degeneracy hypothesis.

From generating series to polynomial congruences

Consider an ordinary generating function ∑k=0∞ckxk, of an integer sequence of some combinatorial relevance, and assume that it admits a closed form C(x). Various instances are known where the corresponding truncated sum ∑k=0q−1ckxk, with q a power of a prime p, also admits a closed form representation when viewed modulo p. Such a representation for the truncated sum modulo p frequently bears a resemblance with the shape of C(x), despite being typically proved through independent arguments. One of the simplest examples is the congruence ∑k=0q−1(2kk)xk≡(1−4x)(q−1)/2(modp) being a finite match for the well-known generating function ∑k=0∞(2kk)xk=1/1−4x.We develop a method which allows one to directly infer the closed-form representation of the truncated sum from the closed form of the series for a significant class of series involving central binomial coefficients. In particular, we collect various known such series whose closed-form representation involves polylogarithms Lid(x)=∑k=1∞xk/kd, and after supplementing them with some new ones we obtain closed-forms modulo p for the corresponding truncated sums, in terms of finite polylogarithms £d(x)=∑k=1p−1xk/kd.

On polynomial congruences

We deal with functions which fulfil the condition \({\Delta_h^{n+1} \varphi(x)\in\mathbb{Z}}\) for all x, h taken from some linear space V. We derive necessary and sufficient conditions for such a function to be decent in the following sense: there exist functions \({f\colon V\rightarrow \mathbb{R}, g\colon V \rightarrow \mathbb{Z}}\) such that \({\varphi = f + g}\) and \({\Delta_h^{n+1}f(x)=0}\) for all \({x, h\in V}\).

The Lazard formal group, universal congruences and special values of zeta functions

A connection between the theory of formal groups and arithmetic number theory is established. In particular, it is shown how to construct general Almkvist–Meurman–type congruences for the universal Bernoulli polynomials that are related with the Lazard universal formal group (based on earlier works of the author). Their role in the theory of L L –genera for multiplicative sequences is illustrated. As an application, sequences of integer numbers are constructed. New congruences are also obtained, useful to compute special values of a new class of Riemann–Hurwitz–type zeta functions.

Recursive Elucidation of Polynomial Congruences Using Root-Finding Numerical Techniques

In this paper we put forward a family of algorithms for lifting solutions of a polynomial congruencemod pto polynomial congruencemod pk. For this purpose, root-finding iterative methods are employed for solving polynomial congruences of the formaxn≡b(mod pk),k≥1,wherea,b,andn>0are integers which are not divisible by an odd primep. It is shown that the algorithms suggested in this paper drastically reduce the complexity for such computations to a logarithmic scale. The efficacy of the proposed technique for solving negative exponent equations of the formax-n≡b(mod pk)has also been addressed.