Euler's Totient Function, the Mangoldt Function, and a Sequence of Mertens Function Values

The Mobius function is commonly used to define Euler's totient function and the Mangoldt function. Similarly, the summatory Mobius function (the Mertens function) can be used to define the summatory totient function and the summatory Mangoldt function (the second Chebyshev function).

The Order of Euler’s Totient Function

The Mobius function is commonly used to define Euler’s totient function and the Mangoldt function. Similarly, the summatory Mobius function (the Mertens function) is used to define the summatory totient function and the summatory Mangoldt function (the second Chebyshev function). Analogues of the product formula for the totient function are introduced. An analogue of the summatory totient function with many additive properties is introduced.

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

Fórmulas explícitas para las funciones Phi de Euler y contador de números primos

An explicit formula which characterizes the pairs of integers that are relatively prime (that is, without common prime factors) is obtained here. It is deduced using sine functions and doesn’t require the knowledge of the prime factors of the arguments. From it, other explicit formulas are derived for the Euler’s Totient function and the Prime Counting function .

Prime number theorem for regular Toeplitz subshifts

We 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.

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

On the pseudorandom properties of $ k $-ary Sidel'nikov sequences

<p style='text-indent:20px;'>In 2002 Mauduit and Sárközy started to study finite sequences of <inline-formula><tex-math id="M2">\begin{document}$ k $\end{document}</tex-math></inline-formula> symbols</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ E_{N} = \left(e_{1},e_{2},\cdots,e_{N}\right)\in \mathcal{A}^{N}, $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M3">\begin{document}$ \mathcal{A} = \left\{a_{1},a_{2},\cdots,a_{k}\right\}(k\in \mathbb{N},k\geq 2) $\end{document}</tex-math></inline-formula> is a finite set of <inline-formula><tex-math id="M4">\begin{document}$ k $\end{document}</tex-math></inline-formula> symbols. Later many pseudorandom sequences of <inline-formula><tex-math id="M5">\begin{document}$ k $\end{document}</tex-math></inline-formula> symbols have been given and studied by using number theoretic methods. In this paper we study the pseudorandom properties of the <inline-formula><tex-math id="M6">\begin{document}$ k $\end{document}</tex-math></inline-formula>-ary Sidel'nikov sequences with length <inline-formula><tex-math id="M7">\begin{document}$ q-1 $\end{document}</tex-math></inline-formula> by using the estimates for certain character sums with exponential function, where <inline-formula><tex-math id="M8">\begin{document}$ q $\end{document}</tex-math></inline-formula> is a prime power. Our results show that Sidel'nikov sequences enjoy good well-distribution measure and correlation measure. Furthermore, we prove that the set of size <inline-formula><tex-math id="M9">\begin{document}$ \phi(q-1) $\end{document}</tex-math></inline-formula> of <inline-formula><tex-math id="M10">\begin{document}$ k $\end{document}</tex-math></inline-formula>-ary Sidel'nikov sequences is collision free and possesses the strict avalanche effect property provided that <inline-formula><tex-math id="M11">\begin{document}$ k = o(q^{\frac{1}{4}}) $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M12">\begin{document}$ \phi $\end{document}</tex-math></inline-formula> denotes Euler's totient function.</p>

Solutions of ϕ (n) = ϕ (n + k) and σ (n) = σ (n + k)

We show that for some even $k\leqslant 3570$ and all $k$ with $442720643463713815200|k$, the equation $\phi (n)=\phi (n+k)$ has infinitely many solutions $n$, where $\phi $ is Euler’s totient function. We also show that for a positive proportion of all $k$, the equation $\sigma (n)=\sigma (n+k)$ has infinitely many solutions $n$. The proofs rely on recent progress on the prime $k$-tuples conjecture by Zhang, Maynard, Tao, and PolyMath.

Integer factoring and compositeness witnesses

We describe a reduction of the problem of factorization of integers n ≤ x in polynomial-time (log x)M+O(1) to computing Euler’s totient function, with exceptions of at most xO(1/M) composite integers that cannot be factored at all, and at most x exp $\begin{array}{} \displaystyle \left(-\frac{c_M(\log\log x)^3}{(\log\log\log x)^2}\right) \end{array}$ integers that cannot be factored completely. The problem of factoring square-free integers n is similarly reduced to that of computing a multiple D of ϕ(n), where D ≪ exp((log x)O(1)), with the exception of at most xO(1/M) integers that cannot be factored at all, in particular O(x1/M) integers of the form n = pq that cannot be factored.