Michael Rubinstein
We consider the uniform distribution of solutions $(x,y)$ to $xy=N \mod a$, and obtain a bound on the second moment of the number of solutions in squares of length approximately $a^{1/2}$. We use this to study a new factoring algorithm that factors $N=UV$ provably in $O(N^{1/3+ε})$ time, and discuss the potential for improving the runtime to sub-exponential.
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This note introduces a new class of integer factoring algorithms. Two versions of this method will be described, deterministic and probabilistic. These algorithms are practical, and can factor large classes of balanced integers N = pq, p < q < 2p in superpolynomial time. Further, an extension of the Fermat factoring method is proposed.
This note presents the basic mathematical structure of a new integer factorization method based on systems of linear Diophantine equations.
This note continues the theoretical development of deterministic integer factorization algorithms based on systems of polynomials equations. The main result establishes a new deterministic time complexity bench mark in integer factorization.
We present an oracle factorisation algorithm which finds a nontrivial factor of almost all positive integers $n$ based on the knowledge of the number of points on certain elliptic curves in residue rings modulo $n$.
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.
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 $E_{2rD}$…
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 best deterministic unconditionally proven integer factorization algorithms have exponential running time complexities of O(N^(1/4)) arithmetic operations, and conditional on the Riemann hypothesis, there is a deterministic algorithm of exponential running time complexity O(N^(1/5)). This note proposes a new deterministic integer factorization algorithm of deterministic exponential time complexity O(N^(1/6)). Furthermore, an algorithm for decomposing composite integers that have factor differences of the form q - p = (r - 1)N^(1/2) + u, where r > 1 is a fixed parameter, and | u | < N^(1/3),…
In this paper, we create a systematic and automatic procedure for transforming the integer factorization problem into the problem of solving a system of Boolean equations. Surprisingly, the resulting system of Boolean equations takes on a "life of its own" and becomes a new type of integer, which we call a generic integer. We then proceed to use the newly found algebraic structure of the ring of generic integers to create two new integer factoring algorithms, called respectively the Boolean factoring (BF) algorithm, and the multiplicative Boolean factoring (MBF) algorithm. Although these two a…
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。
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…
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。
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.
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.
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…
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。
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 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。
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, …
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…
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…
We consider the distribution of the largest prime divisor of the integers in the interval $[2,x]$, and investigate in particular the mode of this distribution, the prime number(s) which show up most often in this list. In addition to giving an asymptotic formula for this mode as $x$ tends to infinity, we look at the set of those prime numbers which, for some value of $x$, occur most frequently as the largest prime divisor of the integers in the interval $[2,x]$. We find that many prime numbers never have this property. We compare the set of "popular primes," those primes which are at some poin…
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 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。
On Factoring and Power Divisor Problems via Rank-3 Lattices and the Second Vector
本文提出基于秩3格和第二向量的算法，改进了因数分解、幂和差分解及r-幂除数问题的复杂度。
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