Algorithms for Integer Factorization Based on Counting Solutions of Various Modular Equations

This paper is a logical continuation of my recently-published paper. Security of modern communication based on RSA cryptographic protocols and their analogues is as crypto-immune as integer factorization (iFac) is difficult. In this paper are considered enhanced algorithms for the iFac that are faster than the algorithm proposed in the previous paper. Among these enhanced algorithms is the one that is based on the ability to count the number of integer solutions on quadratic and bi-quadratic modular equations. Therefore, the iFac complexity is at most as difficult as the problem of counting. Properties of various modular equations are provided and confirmed in numerous computer experiments. These properties are instrumental in the proposed factorization algorithms, which are numerically illustrated in several examples.


Introduction and Problem Statement
Security of modern communication based on RSA or Rabin cryptographic protocols and their analogues is as crypto-immune as difficult is the integer factorization (iFac) [1][2][3].This paper is a continuation of the paper [4].In that paper is considered a factorization algorithm of semi-prime n=pq for two cases: where either both factors p and q are non-Blum primes i.e., p=q=1(mod4), (1.1) or at least one factor is a non-Blum prime.In this paper an iFac algorithm is provided, which also works if both factors p and q are Blum primes, i.e., p=q=3(mod4). (1. 2) The SQUAR-algorithm discussed in [4] is based on several properties (formulated as propositions and conjectures) of dual modular elliptic curves, where b is a positive integer:   If n is a prime and (1.5) holds, then for every b also holds Remark 1.1: Conjecture1.1 plays an important role in the design of the iFac described in [4]; further details are provided in the Appendix.
Proposition 1.2:If the factors p and q are congruent to 1 modulo n=pq, then the following identities hold for non-negative integers m and s: (1.10) Proposition 1.3:If the factors p and q are congruent to 1 modulo n=pq, , and The proposed iFac2 algorithm described below is less restrictive than the integer factorization SQUAR-algorithm described and analyzed in [4], because it is also applicable if both p and q satisfy (1.1 S 4 ={b=43,53,•••; B=6519205}.Therefore, the SQUAR-algorithm provided in [4] requires at least fifteen basic steps, because 43 is the fourteenth prime (2.5).Yet, since , , , , , , in fifteen elliptic curves, we determine both factors of n after three distinct counts.

iFac1 Validation Definition 3.1:
), then we need to compute the 3 rd distinct value {see Example 2.1}.However, if w>1, then we compute the 1 st factor, say, p, and then q:=n/p.The following proposition and examples provide explanations.

 
Proposition 3.1: If primes p and q are selected randomly, then with probability greater than 2/3 we can determine factors of semi-prime n if we know only two distinct counts and .

i
Proof: It is demonstrated in the paper [4] that if p=q=1(mod4), then there exist two positive integers c<p and d<q, and four sets 1   Since n is a semi-prime, then in each of these cases we compute a factor of n.For instance, if For more details see Table A2.
Although the iFac1 algorithm is computationally simpler than the SQUAR algorithm, we can further simplify the iFac algorithm via application of other modular equations.

Modular Quadratic and Bi-quadratic Equations
In this section are considered properties of quadratic, bi-quadratic modular equations and equations with , where the moduli are prime or semi-prime.and let G(n, m, b) denote the number of points on (4.5); if both factors p and q are primes, and if (4.5) is either a quadratic or bi-quadratic equation (i.e., if m=1 or m=2), then for every b>0 if an odd prime m is co-prime with , then for every b and m each co-prime with   Here is called the Euler totient function.{see also Table 6.1 and 6.2 below}.
The iFac algorithm described below is based on Proposition 4.1.This algorithm is computationally efficient if there exists an efficient procedure (an oracle) that counts the points on either the MQE (m=1) or bi-quadratic equation (m=2) (4.5).

iFac2 Algorithm
Conjecture 4.3 can be applied to design an iFac2 algorithm.As it implied from the following discussion, this algorithm is more efficient than the SQUAR-algorithm proposed in [4].Yet, for the seemingly simple iFac2 algorithm we need to know how to efficiently count the number of points G(n, m, b) on modular Equation (4.5) for m=1 or m=2.The algorithm  Therefore, by the Vieta theorem, p and q are the roots of quadratic equation Hence, : =9967 and :

Properties of Modular Equations for m>1: Computer Experiments
Q.E.D

Conjecture 1 . 1 :
Let us reiterate some of these properties and then consider their generalizations.Let p=q=1(mod4); n=pq; let P(n,b) and M(n,b) denote the number of points on elliptic curves (EC) (1.3) and (1.4) respectively.For the sake of brevity, we call P(n,b) and M(n,b) the counts.Consider n=pq, and let primes p and q satisfy (1.1); if

Definition 4 . 1 : 1 .
{equivalence}: Problem 1 A is equivalent to problem 2 A if their time complexities satisfy the inequality     Tables 6.1 and 6.2 illustrate Conjecture 4.2 and Conjecture 4.3.
t<d and gcd(d,r)=m, then for every positive integer b both HECs have equal number of points.Proof: 10) Now let us find such integers w and z, for which the following equation holds

Table 3 .
Since w=1, it means that we cannot find the factors of n because the combination {A, B} is not a resolventa {see 1}.Yet, after we find the third distinct value =L=9342205; the factorization is accomplished: Since c<p, therefore, p must divide d.then we can find factors p and q after considerations of only two distinct counts 1 and i .Although this case is possible {see Example 3.2}, for large primes p and q it is highly improbable.P PAnalogously, we proceed with an analysis of gcd(A+B, n).w:=gcd(A+B, n)=1.

Table 4 .3. Values of G(pq, m, b); m=1,2,3,5.
Remark 5.1: It is well-known that, if n is a semi-prime and if we know the value of Euler totient function   (4.7), then we can find the factors of n.The Conjecture 4.3 is the framework that allows us to compute   Therefore, the iFac2 problem is equivalent to the problem of counting points on the MQE (4.1).and n 

Table 8 .1. Algorithms & residues modulo 4. 7. iFac2 Algorithm Validation
There are several algorithms that count points on elliptic and hyper-elliptic curves.If some of these algorithms can be applied for counting points on quadratic or bi-quadratic modular equations with the same time complexities, then the Schoof-Elkies-Atkin (SEA) algorithm is currently the best known algorithm that counts points on a modular cubic curve with expected running time