Carmichael Function Research Articles

Інформаційна технологія і програмне забезпечення для імітаційного моделювання, синтезу і досліджень методів криптографічного захисту даних

The described information technology for improving data protection (ITIDP) in information communication systems (ICS) is built on the basis of a system approach for the implementation of hardware and software solutions for encryption/decryption of data flows in a given continuum of hardware-software-space-time limitations. The formulation of the task for the implementation of ITIDP is substantiated, and a variant of its architecture is proposed. Examples of the development of possible hardware and software modules and resources for the creation of both ITIDP and ICS with increased protection of real-time data flows are given. The issues of choosing methods and means of data encryption in real technical systems and criteria for assessing the necessity and adequacy of encrypted protection of information flows depending on the usefulness and confidentiality of transmitted data are discussed. As a practical approbation of the application of the proposed technology for solving applied problems, examples of the synthesis and research of a special processor for a block cipher with sequential data processing and dynamic key correction, as well as the results of research and optimization of the RSA encryption model for its use in critical application mobile systems with limited hardware and software resources. It is shown that for systems with limited hardware resources in the RSA model of the cipher, it is more correct to use not the Euler function, but the Carmichael function. This approach, together with the use of a modified method of successive approximations according to the rules of modular algebra for calculating large powers of a large base with the subsequent determination of the remainder by a given modulus of a very large number, makes it possible to remove restrictions on the bit rate of data in low-power computers and speed up data decryption processes. The use of modular architecture in the proposed information technology ensures its scalability and quick reconfiguration for the study of various methods of cryptographic data protection.

Two problems on the distribution of Carmichael's lambda function

AbstractLet denote the exponent of the multiplicative group modulo n. We show that when q is odd, each coprime residue class modulo q is hit equally often by as n varies. Under the stronger assumption that , we prove that equidistribution persists throughout a Siegel–Walfisz‐type range of uniformity. By similar methods we show that obeys Benford's leading digit law with respect to natural density. Moreover, if we assume Generalized Riemann Hypothesis, then Benford's law holds for the order of a mod n, for any fixed integer .

An evaluation of the RSA private keys and the presence of weak keys

Numerous applications that rely on asymmetric cryptography use the RSA algorithm. It can be applied to digital signatures and the encryption of sensitive data. The secure storage of the private key is essential for the algorithm’s strength. Finding ways to use factorization or other heuristics to determine the value of the private key, was the goal of several academic efforts. Both the Euler totient and the Carmichael functions are used in this study to analyze the private key properties and demonstrate the existence of many private keys for the same public key. We further demonstrate that a universal private key that complies with the FIPS standard exists. Moreover, by taking advantage of a condition imposed by FIPS recommendations, we present a new method for attacking the RSA modulus (N).

Idempotent Factorizations of Square-Free Integers

We explore the class of positive integers n that admit idempotent factorizations n = p ¯ q ¯ such that λ ( n ) ∣ ( p ¯ − 1 ) ( q ¯ − 1 ) , where λ is the Carmichael lambda function. Idempotent factorizations with p ¯ and q ¯ prime have received the most attention due to their cryptographic advantages, but there are an infinite number of n with idempotent factorizations containing composite p ¯ and/or q ¯ . Idempotent factorizations are exactly those p ¯ and q ¯ that generate correctly functioning keys in the Rivest–Shamir–Adleman (RSA) 2-prime protocol with n as the modulus. While the resulting p ¯ and q ¯ have no cryptographic utility and therefore should never be employed in that capacity, idempotent factorizations warrant study in their own right as they live at the intersection of multiple hard problems in computer science and number theory. We present some analytical results here. We also demonstrate the existence of maximally idempotent integers, those n for which all bipartite factorizations are idempotent. We show how to construct them, and present preliminary results on their distribution.

The average number of divisors of the Euler function

The upper bound and the lower bound of average numbers of divisors of Euler Phi function and Carmichael Lambda function are obtained by Luca and Pomerance (see \cite{LP}). We improve the lower bound and provide a heuristic argument which suggests that the upper bound given by \cite{LP} is indeed close to the truth.

The arithmetic of Carmichael quotients

Carmichael quotients for an integer $m\ge 2$ are introduced analogous to Fermat quotients, by using Carmichael function $\lambda(m)$. Various properties of these new quotients are investigated, such as basic arithmetic properties, sequences derived from Carmichael quotients, Carmichael-Wieferich numbers, and so on. Finally, we link Carmichael quotients to perfect nonlinear functions.

What are the last digits of …?

We propose a class assignment where students are asked to construct and implement an efficient algorithm to calculate the last digits of a positive integral power of a positive integer. The mathematical prerequisites for this assignment are very limited: knowledge of remainder calculus and the binary representation of a positive integer. The periodicity of the last digits is studied by means of the Euler totient function and the Carmichael function.

On the range of Carmichael's universal-exponent function

Let $\lambda $ denote Carmichael's function, so $\lambda (n)$ is the universal exponent for the multiplicative group modulo $n$. It is closely related to Euler's $\varphi $-function, but we show here that the image of $\lambda $ is much denser than the im

On a problem of Arnold: The average multiplicative order of a given integer

For coprime integers g and n, let l(g) (n) denote the multiplicative order of g modulo n. Motivated by a conjecture of Arnold, we study the average of l(g) (n) as n <= x ranges over integers coprime to g, and x tending to infinity. Assuming the generalized Riemann Hypothesis, we show that this average is essentially as large as the average of the Carmichael lambda function. We also determine the asymptotics of the average of l(g) (p) as p <= x ranges over primes.

The Impossibility of Certain Types of Carmichael Numbers

Abstract.The proof of the existence of infinitely many Carmichael numbers depends heavily upon the study of the Carmichael lambda function

Values of the Euler and Carmichael Functions which are Sums of Three Squares

AbstractLet

FIXED AND PERIODIC POINTS OF POLYNOMIALS GENERATED BY MINIMAL POLYNOMIALS OF 2cos(2π/n)

In this paper certain identities are generated for minimal polynomials Θn(z) of numbers 2 cos (2π/n) and for Chebyshev polynomials of the first kind. Against the background of these identities, sets of cyclic points of some modified versions of these polynomials are derived. The discussion leads to the analysis of a certain new counting sequence of numbers connected with Carmichael function.

Values of the Carmichael Function Equal to a Sum of Two Squares

In this note, we determine the order of growth of the number of positive integers n 6 x such that λ(n) is a sum of two square numbers, where λ(n) is the Carmichael function.

Residue classes having tardy totients

We show, in an effective way, that there exists a sequence of congruence classes $a_k\pmod {m_k}$ such that the minimal solution $n=n_k$ of the congruence $\phi(n)\equiv a_k\pmod {m_k}$ exists and satisfies $\log n_k/\log m_k\to\infty $ as $k\to\infty$. Here, $\phi(n)$ is the Euler function. This answers a question raised in \cite{FS}. We also show that every congruence class containing an even integer contains infinitely many values of the Carmichael function $\lambda(n)$ and the least such $n$ satisfies $n\ll m^{13}$.

Divisors of the Euler and Carmichael functions

We study the distribution of divisors of Euler's totient function and Carmichael's function. In particular, we estimate how often the values of these functions have "dense" divisors.

On values taken by the largest prime factor of shifted primes

AbstractLet P denote the set of prime numbers, and let P(n) denote the largest prime factor of an integer n &gt; 1. We show that, for every real number , there exists a constant c(η) &gt; 1 such that for every integer a ≠ 0, the set has relative asymptotic density one in the set of all prime numbers. Moreover, in the range , one can take c(η) = 1+ε for any fixed ε &gt; 0. In particular, our results imply that for every real number 0.486 ≤ b.thetav; ≤ 0.531, the relation P(q − a) ≍ qθ holds for infinitely many primes q. We use this result to derive a lower bound on the number of distinct prime divisor of the value of the Carmichael function taken on a product of shifted primes. Finally, we study iterates of the map q ↦ P(q - a) for a &gt; 0, and show that for infinitely many primes q, this map can be iterated at least (log logq)1+o(1) times before it terminates.

Arithmetic properties of $φ(n)/λ(n)$ and the structure of the multiplicative group modulo $n$

For a positive integer n , we let \varphi(n) and \lambda(n) denote the Euler function and the Carmichael function, respectively. We define \xi(n) as the ratio \varphi(n)/\lambda(n) and study various arithmetic properties of \xi(n) .

Coincidences in the values of the Euler and Carmichael functions

The well-known Carmichael conjecture concerns the set of positive integers (presumed empty) which occur as a value of the Euler function at one positive integer but at no others. We study the analogous problem for the Carmichael function. We also study the (far more numerous) sets of integers which occur in the image of one of these functions but not of the other, as well as those which occur in the image of both.

Compositions with the euler and carmichael functions

Let ϕ and λ be the Euler and Carmichael functions, respectively. In this paper, we establish lower and upper bounds for the number of positive integers n ≤ x such that ϕ(λ(n)) = λ(ϕ(n)). We also study the normal order of the function ϕ(λ(n))/λ(ϕ(n)).

Values of the Euler Function in Various Sequences

Let ?(n) and ?(n) denote the Euler and Carmichael functions, respectively. In this paper, we investigate the equation ?(n)r = ?(n)s, where r = s = 1 are fixed positive integers. We also study those positive integers n, not equal to a prime or twice a prime, such that ?(n) = p - 1 holds with some prime p, as well as those positive integers n such that the equation ?(n) = f(m) holds with some integer m, where f is a fixed polynomial with integer coefficients and degree degf > 1.