Note on Integer Factoring Methods I

论无限整数的分解

On factoring of unlimited integers

本文在假设Dickson猜想成立的前提下，否定了A. Boudaoud提出的所有无限整数均可分解为有限整数加两无限整数乘积的猜想，修正了已发表版本的相关错误证明并解答了另一相关问题。

The South Caicos Factoring Algorithm

Let $N=UV$, where $U,V$ are integers, with $1< U,V <N$, and $\gcd(U,V)=1$. We describe a probabilistic algorithm for factoring $N$ using $O(\max(U,V)^{1/2+ε})$ bit operations.

Factoring integer using elliptic curves over rational number field $\mathbb{Q}$

For the integer $ D=pq$ of the product of two distinct odd primes, we construct an elliptic curve $E_{2rD}:y^2=x^3-2rDx$ over $\mathbb Q$, where $r$ is a parameter dependent on the classes of $p$ and $q$ modulo 8, and show, under the parity conjecture, that the elliptic curve has rank one and $v_p(x([k]Q))\not=v_q(x([k]Q))$ for odd $k$ and a generator $Q$ of the free part of $E_{2rD}(\mathbb Q)$. Thus we can recover $p$ and $q$ from the data $D$ and $ x([k]Q))$. Furthermore, under the Generalized Riemann hypothesis, we prove that one can take $r<c\log^4D$ such that the elliptic curve $E_{2rD}$…

恢复模形式的傅里叶系数与整数分解

Recovering Fourier coefficients of modular forms and factoring of integers

本文证明若定义在[-1,1]上的函数能被勒让德多项式展开的部分和充分好地逼近，则给定该函数在某个子集上的傅里叶系数c_n，可近似恢复所有n∈[n1,n2]的系数，并由此提出一种整数分解的新方法。

Factoring Numbers with elliptic curves

In the present paper we provide a probabilistic polynomial time algorithm that reduces the complete factorization of any squarefree integer $n$ to counting points on elliptic curves modulo $n$, succeeding with probability $1-\varepsilon$, for any $\varepsilon<0$ .

Algorithm for factoring some RSA and Rabin moduli

In this paper we present a new efficient algorithm for factoring the RSA and the Rabin moduli in the particular case when the difference between their two prime factors is bounded. As an extension, we also give some theoretical results on factoring integers.

Deterministic factoring with oracles

We revisit the problem of integer factorization with number-theoretic oracles, including a well-known problem: can we factor an integer $N$ unconditionally, in deterministic polynomial time, given the value of the Euler totient $$Φ$(N)$? We show that this can be done, under certain size conditions on the prime factors of N. The key technique is lattice basis reduction using the LLL algorithm. Among our results, we show for example that if $N$ is a squarefree integer with a prime factor $p > $\sqrt$ N$ , then we can recover p in deterministic polynomial time given $$Φ$(N)$. We also shed some li…

Shor算法的实现：特定约束下4096位整数的分解

Implementation of Shor Algorithm: Factoring a 4096-Bit Integer Under Specific Constraints

针对量子计算实用化需求，研究聚焦Shor算法实现，通过提升模运算效率，在特定约束下完成4096位整数分解，相比SOTA方法效率显著提升且支持更长整数分解。

Factoring polynomials in the ring of formal power series over Z

We consider polynomials with integer coefficients and discuss their factorization properties in Z[[x]], the ring of formal power series over Z. We treat polynomials of arbitrary degree and give sufficient conditions for their reducibility as power series. Moreover, if a polynomial is reducible over Z[[x]], we provide an explicit factorization algorithm. For polynomials whose constant term is a prime power, our study leads to the discussion of p-adic integers.

利用模$n$乘法关系的因式分解：一种受指标演算启发的次指数算法

Factoring using multiplicative relations modulo $n$: a subexponential algorithm inspired by the index calculus

该研究提出一种受指标演算启发的整数因式分解次指数算法，将因式分解归约为求模n因子基乘法关系的超定系统，虽慢于最优算法，但因简洁性和与指标演算的相似性具研究意义。

A deterministic algorithm for integer factorization

A deterministic algorithm for factoring $n$ using $n^{1/3+o(1)}$ bit operations is presented. The algorithm tests the divisibility of $n$ by all the integers in a short interval at once, rather than integer by integer as in trial division. The algorithm is implemented.

Approximately counting semismooth integers

An integer $n$ is $(y,z)$-semismooth if $n=pm$ where $m$ is an integer with all prime divisors $\le y$ and $p$ is 1 or a prime $\le z$. arge quantities of semismooth integers are utilized in modern integer factoring algorithms, such as the number field sieve, that incorporate the so-called large prime variant. Thus, it is useful for factoring practitioners to be able to estimate the value of $Ψ(x,y,z)$, the number of $(y,z)$-semismooth integers up to $x$, so that they can better set algorithm parameters and minimize running times, which could be weeks or months on a cluster supercomputer. In t…

A deterministic version of Pollard's p-1 algorithm

In this article we present applications of smooth numbers to the unconditional derandomization of some well-known integer factoring algorithms. We begin with Pollard's $p-1$ algorithm, which finds in random polynomial time the prime divisors $p$ of an integer $n$ such that $p-1$ is smooth. We show that these prime factors can be recovered in deterministic polynomial time. We further generalize this result to give a partial derandomization of the $k$-th cyclotomic method of factoring ($k\ge 2$) devised by Bach and Shallit. We also investigate reductions of factoring to computing Euler's totient…

Digit Polynomials and their application to integer factorization

This paper presents the concept of digit polynomials, which leads to a deterministic and unconditional integer factorization algorithm with the runtime complexity $\mathcal{O}(N^{1/4+ε})$. Strassen's well known factoring approach is a special case of our method. We will also consider a possibility to improve upon the complexity bound.

Notions for RSA integers

The key-generation algorithm for the RSA cryptosystem is specified in several standards, such as PKCS#1, IEEE 1363-2000, FIPS 186-3, ANSI X9.44, or ISO/IEC 18033-2. All of them substantially differ in their requirements. This indicates that for computing a "secure" RSA modulus it does not matter how exactly one generates RSA integers. In this work we show that this is indeed the case to a large extend: First, we give a theoretical framework that will enable us to easily compute the entropy of the output distribution of the considered standards and show that it is comparatively high. To do so, …

有理底降阶：一种确定性因数分解结构半素数的算法

Rational Base Descent: A Deterministic Algorithm for Factoring Structured Semiprimes

本文提出一种专门算法，用于分解形式为N=pq的半素数，其中一个素因子满足p≈c(a/b)^n，算法在特定参数下以O(log^3 N)时间隔离因子，并提供Python实现。

Non-unique factorization of polynomials over residue class rings of the integers

We investigate non-unique factorization of polynomials in Z_{p^n}[x] into irreducibles. As a Noetherian ring whose zero-divisors are contained in the Jacobson radical, Z_{p^n}[x] is atomic. We reduce the question of factoring arbitrary non-zero polynomials into irreducibles to the problem of factoring monic polynomials into monic irreducibles. The multiplicative monoid of monic polynomials of Z_{p^n}[x] is a direct sum of monoids corresponding to irreducible polynomials in Z_p[x], and we show that each of these monoids has infinite elasticity. Moreover, for every positive integer m, there exis…

因式分解时减少埃拉托斯特尼筛法使用的空间

Reducing the Space Used by the Sieve of Eratosthenes When Factoring

提出一种改进的埃拉托斯特尼筛法版本，能在 $O(x \log\log x)$ 运算内分解所有 $\le x$ 的整数，且将空间界限降至 $O(\sqrt{x}/\log\log x)$ 位，同时保持相同时间复杂度，且实际表现良好。

一种基于新双曲线的整数分解方法

A New Hyperbola based Approach to factoring Integers

本文针对半素数分解问题，提出基于双曲线代数集的新方法，通过引入双曲线 X-root 和 Y-root 概念，在奇异 Weierstrass 曲线上求解关键点以实现整数分解。

Sharp Transitions in Making Squares

In many integer factoring algorithms, one produces a sequence of integers (created in a pseudo-random way), and wishes to rapidly determine a subsequence whose product is a square (which we call a square product). In his lecture at the 1994 International Congress of Mathematicians, Pomerance observed that the following problem encapsulates all of the key issues: Select integers a_1, a_2, >... at random from the interval [1,x], until some (non-empty) subsequence has product equal to a square. Find good estimate for the expected stopping time of this process. A good solution to this problem shou…

Good integers : A note on results of Jitman and Prugsapitak

In this paper, we fix some errors made by Jitman [1] and by Prugsapitak and Jitman [3] while characterizing good integers and $2^β$-good integers.

A note on the alternating sums of powers of consecutive integers

Following an idea due to Euler, we evaluate the alternating sums of powers of consrcutive integers.

一个蕴含多项式因式分解与整数因式分解更快算法的数论猜想

A number-theoretic conjecture implying faster algorithms for polynomial factorization and integer factorization

本文针对有限域一元多项式因式分解的3/2指数壁垒，提出基于数论猜想的新策略，可优化多项式因式分解指数，还能降低确定性整数因式分解算法的指数。

An Improvement to the Number Field Sieve

We improve the "sieve" part of the number field sieve used in factoring integer and computing discrete logarithm. The runtime of our method is shorter than that of existing methods. Under some reasonable assumptions, we prove that it is less than two-thirds of the running time of the algorithm used before asymptotically with probability gr

A Note On The Spectral Norms of The Circulant Matrices Connected Integer Number Sequences

In this paper, we compute the spectral norms of the matrices related with integer squences and we give some example related with Fibonacci, Lucas, Pell and Perrin numbers.

利用类群进行整数的快速无平方因子分解

Fast square-free decomposition of integers using class groups

本文提出一种基于二元二次型类群的算法，用于快速求 $n=a^2b$ 的无平方因子分解，在 $a,b$ 为密码学规模素数时是目前最快的方法，对依赖此类整数分解困难性的密码系统具有重要意义。

整数的塔式分解

On the tower factorization of integers

研究整数塔式分解的高度，给出高度为k的整数集合密度公式，估计平均层数，并证明存在任意长且高度任意大的连续整数序列。

A note on the alternating sums of powers of consecutive q-integers

In this paper we construct a new q-Euler numbers and polynomials. By using these numbers and polynomials, we give the interesting formulae related to alternating sums of powers of consecutive q-integers following an idea due to Euler.

Note on the sums of powers of consecutive $q$-integers

In this paper we construct the $q$-analogue of Barnes's Bernoulli numbers and polynomials of degree 2, for positive even integers, which is an answer to a part of Schlosser's question. For positive odd integers, Schlosser's question is still open. Finally, we will treat the $q$-analogue of the sums of powers of consecutive integers.

A Note on Cyclotomic Integers

In this note, we present a new proof that the cyclotomic integers constitute the full ring of integers in the cyclotomic field.

Short Note: Every Large Set of Integers Contains a Three Term Arithmetic Progression

I show that a trivial modification of a standard proof of the Roth's Theorem on triples in arithmetic progression would lead to the following Theorem: If A is a "large set" that is its elements are monotone increasing integers and the sum of reciprocals of its elements diverges then the sequence contains an arithmetic progression of length three.

A note on the sums of powers of consecutive q-integers

In this paper we construct the q-analogue of Barnes' Bernoulli numbers and plynomials of degree 2, which is an answer to a part of Schlosser's question. Finally, we treat the q-analogue of the sums of powers of consecutive integrs.

A note about invariant polynomial transformations of integer sequences

We present an algorithm to find invariant poynomial transformations of integer sequences, using the classical invariant theory approach.

A note on the weighted q-Hardy-Littlewood-type maximal operator with respect to q-Volkenborn integral in the p-adic integer ring

The essential aim of this paper is to define weighted q-Hardy-littlewood-type maximal operator by means of p-adic q-invariant distribution on Zp. Moreover, we give some interesting properties concerning this type maximal operator.

A note on an irreducible class of polynomials over integers

In this note, we prove an irreducibility criterion for the polynomial of the form $f(x) = a_{n}x^{n} + a_{n-1}x^{n-1} + \cdots + a_{m}x^{m} + p^{u} \in \mathbb{Z}[x]$, where $p$ is a prime number, $u \geqslant 1$, $\gcd(u, m) = 1$, $p \nmid a_{m}$ and $p^{u} > |a_{n}| + |a_{n-1}| + \cdots + |a_{m}|$. In particular, we show that the conjecture of Koley and Reddy is true.

A note on polynomial sequences modulo integers

We study the uniform distribution of the polynomial sequence $λ(P)=(\lfloor P(k) \rfloor )_{k\geq 1}$ modulo integers, where $P(x)$ is a polynomial with real coefficients. In the nonlinear case, we show that $λ(P)$ is uniformly distributed in $\mathbb{Z}$ if and only if $P(x)$ has at least one irrational coefficient other than the constant term. In the case of even degree, we prove a stronger result: $λ(P)$ intersects every congruence class modulo every integer if and only if $P(x)$ has at least one irrational coefficient other than the constant term.

A Note on p-Adic Invariant Integral in the Rings of p-Adic Integers

In [2], I constructed the p-adic q-integral on Zp. In this paper, we consider the properties of the p-adic invariant p-adic q-integral in the ring of p-adic integers at q=-1. Finally we give the some applications of p-adic q-integration at q=-1. These properties are useful and worthwhile to study Euler numbers and polynomials.

关于Hildebrand关于可分解整数的一个猜想的注记

Note on a conjecture of Hildebrand regarding friable integers

本文针对Hildebrand关于可分解整数个数的光滑近似成立的猜想，在Gorodetsky复杂证明的基础上，提出了一个简短直接的证明。

A note on the first cuboid conjecture

Recently the problem of constructing a perfect Euler cuboid was related with three conjectures asserting the irreducibility of some certain three polynomials depending on integer parameters. In this paper a partial result toward proving the first cuboid conjecture is obtained. The polynomial which, according to this conjecture, should be irreducible over integers is proved to have no integer roots.

关于非主和非互质可允许 $C^{(1)}_2$-模块的模性质的注记

A note on modular properties of non-principal and non-coprincipal admissible $C^{(1)}_2$-modules of integer level

本文研究了 $C_2^{(1)}$ 代数整级非主和非互质可允许模块的模性质及其量子哈密顿约简的不变性。

A short note on Jacobsthal's function

The function g(n) represents the smallest number Q such that every sequence of Q consecutive integers contains an integer coprime to n. We give a new and explicit upper bound on this function.

Note on the diophantine equation X^t+Y^t=BZ^t

In this paper, we obtain new results on the integers solutions X, Y, Z of the diophantine equation X^t+Y^t=BZ^t for a rationnal integer B and a prime number t verifying some conditions explained in the paper.

Notes on the Quadratic Integers and Real Quadratic Number Fields

It is shown that when a real quadratic integer $ξ$ of fixed norm $μ$ is considered, the fundamental unit $\varepsilon_d$ of the field $\mathbb{Q}(ξ) = \mathbb{Q}(\sqrt{d})$ satisfies $\log \varepsilon_d \gg (\log d)^2$ almost always. An easy construction of a more general set containing all the radicands $d$ of such fields is given via quadratic sequences, and the efficiency of this substitution is estimated explicitly. When $μ= -1$, the construction gives all $d$'s for which the negative Pell's equation $X^2 - d Y^2 = -1$ (or more generally $X^2 - D Y^2 = -4$) is soluble. When $μ$ is a prime,…

A Note on Goldbach Partitions of Large Even Integers

Let $Σ_{2n}$ be the set of all partitions of the even integers from the interval $(4,2n], n>2,$ into two odd prime parts. We show that $\midΣ_{2n}\mid\sim 2n^2/\log^2{n}$ as $n\to\infty$. We also assume that a partition is selected uniformly at random from the set $Σ_{2n}$. Let $2X_n\in (4,2n]$ be the size of this partition. We prove a limit theorem which establishes that $X_n/n$ converges weakly to the maximum of two random variables which are independent copies of a uniformly distributed random variable in the interval $(0,1)$. Our method of proof is based on a classical Tauberian theorem du…

A note on the converse of Wolstenholme's Theorem

Given a prime p and a positive integer m satisfying a certain inequality, the converse of Wolstenholme's Theorem is shown to hold for the product mp^k where k is any positive integer, generalizing a result by Helou and Terjanian.

A Note on Additive Complements

Two infinite sequences A and B of non-negative integers are called additive complements, if their sum contains all sufficiently large integers. Let $A(x)$ and $B(x)$ be the counting functions of A and B. In this paper, we extend the results of Liu and Fang in 2016 and obtain some results on additive complements. For example, we prove that there exist additive complements $A$ and $B$ such that $\limsup_{x\to+\infty} A(x)B(x)/x= 2$ and $A(x)B(x) - x = 1$ for infinitely positive integers $x$.

A Note on the Quartic Diophantine Equation $A^4+hB^4=C^4+hD^4$

Integer solutions of the diophantine equation $A^4+hB^4=C^4+hD^4$ are known for all positive integer values of $h < 1000$. While a solution of the aforementioned diophantine equation for any arbitrary positive integer value of $h$ is not known, Gerardin and Piezas found solutions of this equation when $h$ is given by polynomials of degrees 5 and 2 respectively. In this paper, we present several new solutions of this equation when $h$ is given by polynomials of degrees $2,\;3$ and 4.

关于无限大素数模整数环的代数性质和非标准分析的注记

Notes on Algebraic Properties and Non-Standard Analysis of the Ring of Integers Modulo Infinitely Large Primes

本文探讨无限大素数模整数环的代数与模型论性质，结合非标准分析方法扩展超越数理论中的判别准则。