Totient Function Research Articles

Revisited Carmichael’s Reduced Totient Function

The modified Totient function of Carmichael λ(.) is revisited, where important properties have been highlighted. Particularly, an iterative scheme is given for calculating the λ(.) function. A comparison between the Euler φ and the reduced totient λ(.) functions aiming to quantify the reduction between is given.

Search Heuristics and Constructive Algorithms for Maximally Idempotent Integers

Previous work established the set of square-free integers n with at least one factorization n=p¯q¯ for which p¯ and q¯ are valid RSA keys, whether they are prime or composite. These integers are exactly those with the property λ(n)∣(p¯−1)(q¯−1), where λ is the Carmichael totient function. We refer to these integers as idempotent, because ∀a∈Zn,ak(p¯−1)(q¯−1)+1≡na for any positive integer k. This set was initially known to contain only the semiprimes, and later expanded to include some of the Carmichael numbers. Recent work by the author gave the explicit formulation for the set, showing that the set includes numbers that are neither semiprimes nor Carmichael numbers. Numbers in this last category had not been previously analyzed in the literature. While only the semiprimes have useful cryptographic properties, idempotent integers are deserving of study in their own right as they lie at the border of hard problems in number theory and computer science. Some idempotent integers, the maximally idempotent integers, have the property that all their factorizations are idempotent. We discuss their structure here, heuristics to assist in finding them, and algorithms from graph theory that can be used to construct examples of arbitrary size.

Coprime partitions and Jordan totient functions

We show that while the number of coprime compositions of a positive integer n into k parts can be expressed as a Q-linear combination of the Jordan totient functions, this is never possible for the coprime partitions of n into k parts. We also show that the number pk′(n) of coprime partitions of n into k parts can be expressed as a C-linear combination of the Jordan totient functions, for n sufficiently large, if and only if k∈{2,3} and in a unique way. Finally we introduce some generalizations of the Jordan totient functions and we show that pk′(n) can be always expressed as a C-linear combination of them.

On a bound of Cocke and Venkataraman

Let G be a finite group with exactly k elements of largest possible order m. Let q(m) be the product of gcd (m,4) and the odd prime divisors of m. We show that |G|le q(m)k^2/varphi (m) where varphi denotes Euler’s totient function. This strengthens a recent result of Cocke and Venkataraman. As an application we classify all finite groups with k<36. This is motivated by a conjecture of Thompson and unifies several partial results in the literature.

A Menon-type identity concerning Dirichlet characters and a generalization of the gcd function

Menon’s identity is a classical identity involving gcd sums and the Euler totient function \(\phi \). In a recent paper, Zhao and Cao (Int. J. Number Theory 13(9) (2017) 2373–2379) derived the Menon-type identity \(\sum _{\begin{array}{c} k=1 \end{array}}^{n}(k-1,n)\chi (k) = \phi (n)\tau (\frac{n}{d})\), where \(\chi \) is a Dirichlet character mod n with conductor d. We derive an identity similar to this replacing gcd with a generalization it. We also show that some of the arguments used in the derivation of Zhao–Cao identity can be improved if one uses the method we employ here.

A New Methodology to Find Private Key of RSA Based on Euler Totient Function

The aim of this paper is to present a new methodology to find the private key of RSA. A new initial value which is generated from a new equation is selected to speed up the process. In fact, after this value is found, brute force attack is chosen to discover the private key. In addition, for a proposed equation, the multiplier of Euler totient function to find both of the public key and the private key is assigned as 1. Then, it implies that an equation that estimates a new initial value is suitable for the small multiplier. The experimental results show that if all prime factors of the modulus are assigned larger than 3 and the multiplier is 1, the distance between an initial value and the private key is decreased about 66%. On the other hand, the distance is decreased less than 1% when the multiplier is larger than 66. Therefore, to avoid attacking by using the proposed method, the multiplier which is larger than 66 should be chosen. Furthermore, it is shown that if the public key equals 3, the multiplier always equals 2.

On paranormed Ideal convergent sequence spaces defined by Jordan totient function

The study of sequence spaces and summability theory has been an important aspect in defining new notions of convergence for the sequences that do not converge in the usual sense. Paving the way into the applications of law of large numbers and theory of functions, it has proved to be an essential tool. In this paper we generalise the classical Maddox sequence spaces c_{0}(p), c(p), ell (p) and ell _{infty }(p) and define new ideal paranormed sequence spaces c^{I}_{0}(Upsilon ^{r}, p), c^{I}(Upsilon ^{r}, p), ell ^{I}_{ infty }(Upsilon ^{r}, p) and ell _{infty }(Upsilon ^{r}, p) defined with the aid of Jordan’s totient function and a bounded sequence of positive real numbers. We develop isomorphism between certain maps and also find their α-, β- and γ-duals. We examine algebraic and topological properties of these corresponding spaces. Further we study some standard inclusion relations and prove the decomposition theorem.

English

Menon’s identity is a classical identity involving gcd sums and the Euler totient function φ. A natural generalization of φ is the Klee’s function Φs. We derive a Menon-type identity using Klee’s function and a generalization of the gcd function. This identity generalizes an identity given by Y. Li and D. Kim (2017).

A novel synthesis method for reliable feedback shift registers via Boolean networks

A random fault or a malicious attack can compromise the security of decryption systems. Using a stable and monotonous feedback shift register (FSR) as the main building block in a convolutional decoder can limit some error propagation. This work foucus on the synthesis of reliable (i.e., globally stable and monotonous) FSRs using the Boolean networks (BNs) method. First, we obtain an algebraic expression of the FSRs, based on which one necessary and sufficient condition for the monotonicity of the FSRs is given. Then, we obtain the upper bound of the number of cyclic attractors for monotonous FSRs. Furthermore, we propose one method of constructing n-stage reliable FSRs, and figure out that the number of reliable FSRs is $${{{2^{{2^{n - 4}}}}} \over {\phi (n)}}$$ (n > 5) times of that constructed by the existing method, where ϕ denotes the Euler’s totient function. Finally, the proposed method and the obtained results are verified by some examples.

Some Pythagorean type equations concerning arithmetic functions

We investigate some equations involving the number of divisors d(n); the sum of divisors σ(n); Euler's totient function ϕ(n); the number of distinct prime factors ω(n); and the number of all prime factors (counted with multiplicity) Ω(n). The first part deals with equation f(xy) + f(xz) = f(yz). In the second part, as an analogy to x2 + y2 = z2, we study equation f(x2) + f(y2) = f(z2) and its generalization to higher degrees and more terms. We use just elementary methods and basic facts about the above functions and indicate why and how to discuss this topic in group study sessions or special maths classes of secondary schools in the framework of inquiry based learning.&#x0D; Subject Classification: 97F60, 11A25

Prime number theorem for regular Toeplitz subshifts

AbstractWe prove that neither a prime nor an l-almost prime number theorem holds in the class of regular Toeplitz subshifts. But when a quantitative strengthening of the regularity with respect to the periodic structure involving Euler’s totient function is assumed, then the two theorems hold.

On a generalization of the Euler totient function

In a recent paper, J. Kaczorowski defined the generalized Euler totient function $\varphi(n, F)$ corresponding to a polynomial Euler product $F$. Let $E(x, F)$ denote the error term in the asymptotic formula of the summatory function of $\varphi(n, F).$ J.~Kaczorowski proved that the mean square of $E(x, F)$ has an asymptotic formula if $F$ satisfies GRH. In this paper we shall prove a sharper asymptotic formula under GRH.

Totient quotient and small gaps between primes

For each $$m\ge 1$$ , there exists $$H=H(m)>0$$ such that the set $$\begin{aligned} \bigg \{\frac{\phi (p+1)}{\phi (p-1)}:\,p\text { is prime and }[p+1,p+H]\text { contains at least }m\text { primes}\bigg \} \end{aligned}$$ is dense in $$[0,+\infty )$$ , where $$\phi $$ denotes the Euler totient function. This gives a unconditional weak form of a recent result of Garcia, Luca, Shi and Udell, which was proved under the assumption of a conjecture of Dickson.

On the local structure of the set of values of Euler’s $\varphi $ function

Assuming the validity of Dickson’s conjecture, we show that the set $\mathcal {V}$ of values of Euler’s totient function $\varphi $ contains arbitrarily large arithmetic progressions with common difference 4. This leads to the question of proving uncondit

The Number of Almost Perfect Nonlinear Functions Grows Exponentially

Almost perfect nonlinear (APN) functions play an important role in the design of block ciphers as they offer the strongest resistance against differential cryptanalysis. Despite more than 25 years of research, only a limited number of APN functions are known. In this paper, we show that a recent construction by Taniguchi provides at least $$\frac{\varphi (m)}{2}\left\lceil \frac{2^m+1}{3m} \right\rceil $$ inequivalent APN functions on the finite field with $${2^{2m}}$$ elements, where $$\varphi $$ denotes Euler’s totient function. This is a great improvement of previous results: for even m, the best known lower bound has been $$\frac{\varphi (m)}{2}\left( \lfloor \frac{m}{4}\rfloor +1\right) $$ ; for odd m, there has been no such lower bound at all. Moreover, we determine the automorphism group of Taniguchi’s APN functions.

Ciphertext-Only Attack on RSA Using Lattice Basis Reduction

We use lattice basis reduction for ciphertext-only attack on RSA. Our attack is applicable in the conditions when known attacks are not applicable, and, contrary to known attacks, it does not require prior knowledge of a part of a message or key, small encryption key, e, or message broadcasting. Our attack is successful when a vector, comprised of a message and its exponent, is likely to be the shortest in the lattice, and meets Minkowski's Second Theorem bound. We have conducted experiments for message, keys, and encryption/decryption keys with sizes from 40 to 8193 bits, with dozens of thousands of successful RSA cracks. It took about 45 seconds for cracking 2001 messages of 2050 bits and for large public key values related with Euler’s totient function, and the same order private keys. Based on our findings, for RSA not to be susceptible to the proposed attack, it is recommended avoiding RSA public key form used in our experiments

Conjecturing with some Conjectures

This world has been seeing so many conjectures from the formation of this world. Particularly, mathematics world has been seeing so many conjectures from the civilization of this world, also venturing through on it. So many Indians, Chinese and Arabian scholars provided their participation in mathematics world. Since before Euclid so many mathematicians ventured on numbers, geometry, astronomy, etc... in number theory, there are so many conjectures like applications of GCD, Fermat Last theorem, Euler’s totient function, Gold Bach conjecture, ABC conjecture, etc... some of this has been proved but so many of that still has not been proved. In this paper, I try to prove Gold Bach conjecture, stating abc conjecture for composite numbers, and try to deliver some conjectures.

Complete regular dessins and skew-morphisms of cyclic groups

A dessin is a 2 -cell embedding of a connected 2 -coloured bipartite graph into an orientable closed surface. A dessin is regular if its group of orientation- and colour-preserving automorphisms acts regularly on the edges. In this paper we study regular dessins whose underlying graph is a complete bipartite graph K m , n , called ( m , n ) -complete regular dessins. The purpose is to establish a rather surprising correspondence between ( m , n ) -complete regular dessins and pairs of skew-morphisms of cyclic groups. A skew-morphism of a finite group A is a bijection φ : A → A that satisfies the identity φ ( x y ) = φ ( x ) φ π ( x ) ( y ) for some function π : A → ℤ and fixes the neutral element of A . We show that every ( m , n ) -complete regular dessin D determines a pair of reciprocal skew-morphisms of the cyclic groups ℤ n and ℤ m . Conversely, D can be reconstructed from such a reciprocal pair. As a consequence, we prove that complete regular dessins, exact bicyclic groups with a distinguished pair of generators, and pairs of reciprocal skew-morphisms of cyclic groups are all in a one-to-one correspondence. Finally, we apply the main result to determining all pairs of integers m and n for which there exists, up to interchange of colours, exactly one isomorphism class of ( m , n ) -complete regular dessins. We show that the latter occurs precisely when every group expressible as a product of cyclic groups of order m and n is abelian, which eventually comes down to the condition gcd ( m , ϕ ( n )) = gcd ( ϕ ( m ), n ) = 1 , where ϕ is Euler’s totient function.

Each Positive Rational Number Has the Form

In this note, we show that each positive rational number can be written as , where is Euler’s totient function and m and n are positive integers.

On Fixed Points of Iterations Between the Order of Appearance and the Euler Totient Function

Let Fn be the nth Fibonacci number. The order of appearance z(n) of a natural number n is defined as the smallest positive integer k such that Fk≡0(modn). In this paper, we shall find all positive solutions of the Diophantine equation z(φ(n))=n, where φ is the Euler totient function.