Integer Factorization via Polynomial Base Transformation
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Solution Overview
Problem
Current methods for factoring large integers into their prime factors are inefficient, particularly in resolving the factorization of odd integers represented as multiples of powers of an odd prime, which is crucial for cryptographic applications like decoding encrypted signals.
Innovation Solution
A method involving selecting a prime number of the form p=4k+1, calculating specific transformations to convert the factorization problem into a quadratic residue modulo p, and using representations of integers as sums of powers of p to efficiently decrypt encrypted signals.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Productivity
If traditional factoring methods are used for large integers, then the factorization can be completed, but the time required is excessively long and efficiency is poor
Solution Approach 1:
The patent transforms the factoring problem by changing the parameter representation of the integer N into a polynomial form in base p (where p is a prime factor of N-1). This parameter transformation allows the use of polynomial arithmetic operations instead of traditional integer factorization methods, significantly improving computational efficiency. The integer N is represented as N = a0 + a1*p + a2*p^2 + ... + ak*p^k, converting a difficult factoring problem into a more manageable polynomial arithmetic problem.
2Ease of manufacture
If the factorization problem is solved directly without transformation, then the solution is straightforward in concept, but the computational complexity is too high for practical application
Solution Approach 1:
The patent introduces an intermediary polynomial representation as a mediator between the original integer N and its prime factors. By representing N in base p and using polynomial arithmetic operations (multiplication, division, GCD computation) on these polynomial representations, the system avoids the computational complexity of direct integer factorization while maintaining the ability to extract prime factors through the polynomial arithmetic process.
3Reliability
If cryptographic security is maintained with larger key sizes, then security strength increases, but the time required for decryption and factorization operations increases
Solution Approach 1:
The patent applies parameter changes by representing large cryptographic integers in polynomial form based on a prime p where p-1 divides N-1. This transformation enables the use of efficient polynomial arithmetic operations for factorization and decryption, allowing larger key sizes to be used for enhanced security while maintaining reasonable decryption times through the efficiency of polynomial-based computations.
Data Source
AI summary
This patent describes a method, apparatus and computer program which factor a large integer N0 in a time of the order of p2·logp4 N0, where p denotes a prime.