RSA Factorization Using Linear Integer Programming on Decimal Digits
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Solution Overview
Problem
Current RSA factorization methods are computationally inefficient, heuristic-based, and require significant computational resources, failing to utilize decimal digit information effectively, and are limited by the need for prior knowledge of small prime factors or specific number structures.
Innovation Solution
A novel Linear-Integer-Programming (LIP) method that employs optimization-based integer programming techniques using the decimal digit information of semi-prime numbers to formulate subproblems as linear integer programs, allowing any commercially available solver to factorize semi-prime numbers into two primes without requiring prior knowledge of small primes or specific structures.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Productivity
If traditional RSA factorization methods (Trial Sieve, Pollard's Rho, GNFS) are used, then factorization can be performed on semi-prime numbers, but the computational complexity becomes infeasible for large semi-prime numbers with hundreds of digits
Solution Approach 1:
The patent segments the semi-prime number into individual decimal digits and formulates linear equations based on the relationship between digits and modular arithmetic. This transforms the complex factorization problem into a system of linear equations with binary variables, making it computationally tractable.
Solution Approach 2:
The patent replaces traditional mechanical/computational factorization methods (sieve methods, elliptic curve methods) with a mathematical programming approach using linear equations and binary variables. This substitution transforms the problem from a computational search into a structured mathematical optimization problem.
2Productivity
If heuristic techniques (Pollard's Rho, GNFS) are employed for factorization, then factorization can be attempted, but there is no guarantee of convergence to a feasible solution
Solution Approach 1:
The patent creates a self-contained mathematical system where the linear equations and binary variables inherently guarantee a solution. The system uses the decimal digit information of the semi-prime number itself to construct the equations, ensuring that a feasible solution exists and can be found through systematic solving rather than heuristic search.
3Productivity
If sieve methods (GNFS, Pollard's Rho) are used for factorization, then factorization can proceed, but prior knowledge of prime numbers smaller than √θ is required
Solution Approach 1:
The patent performs preliminary action by formulating the factorization problem in terms of decimal digit relationships and modular arithmetic before any factorization attempt. The linear equations are constructed using only the known decimal digits of the semi-prime number, eliminating the need for pre-computation of small primes or other preliminary data.
4Productivity
If special-purpose factorization algorithms (Trial Division, Wheel Factorization, Pollard's Rho) are applied, then small factors can be removed, but the running time depends on the size of the smallest prime factor
Solution Approach 1:
The patent creates a universal factorization method that works for any semi-prime number regardless of the size of its prime factors. The linear equation system with binary variables can handle both small and large factors uniformly, eliminating the time dependency on the smallest prime factor that plagues special-purpose algorithms.
Data Source
AI summary
A system for performing operations using linear integer programming for RSA factorization is provided, including an n/e extractor, a prime factorization calculator, a private key determiner, and a decryptor. The n/e extractor is configured to extract a modulus and a public key exponent from a public key. The prime factorization calculator is configured to: determine a semi-prime number of the modulus according to the modulus; use a tail digit and a head digit set of the semi-prime number of the modulus to perform decomposition and factorization with respect to the semi-prime number into two prime factors. The private key determiner is configured to determine a private key using the public key exponent and the two prime numbers. The decryptor is configured to decrypt an encrypted message using the private key so as to generate a decrypted message.