Integer Factorization for Encrypted Signal Decoding
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Solution Overview
Problem
The challenge lies in efficiently factoring a large odd integer N0 into its prime factors, particularly in decoding encrypted electromagnetic signals encoded with a public key N0=r×s, where N0, r, and s are integers, and determining if the decoded signal indicates illegal activities that may violate laws.
Innovation Solution
A method involving the selection of integers a and b, applying a semigroup law in a projective space to find shared factors of N0, and using these factors to decrypt the signal, allowing for the identification of prime factors of N0 and subsequent decryption of the electromagnetic signal to detect potential law violations.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Productivity
If conventional factorization methods are used to factor large integer N0, then the factorization can be performed, but the computational time becomes excessively long and infeasible for practical applications
Solution Approach 1:
The patent transforms the factorization problem by changing parameters: it factors N0-1 instead of N0 directly, and uses a specialized algorithm that operates on the transformed problem. This parameter transformation enables the factorization to be completed in O(log^6 N0) time, which is significantly faster than conventional methods would require for large integers.
Solution Approach 2:
The patent replaces conventional mechanical factorization algorithms with a mathematical transformation approach. Instead of using traditional trial division or simple sieve methods, the invention uses algebraic transformations and properties of quadratic residues to reduce the factorization problem to a more efficient computational form.
2Productivity
If the factorization problem is solved efficiently, then encrypted signals can be decrypted quickly, but the complexity of the decoding process increases
Solution Approach 1:
The patent segments the decoding process into distinct phases: (1) transforming the encrypted signal into a factorization problem, (2) factoring N0-1 using the specialized algorithm, (3) recovering the prime factors of N0, and (4) decrypting the signal. This segmentation allows each phase to be optimized independently, achieving efficient overall decoding while managing computational complexity through structured processing.
3Reliability
If the public key N0 is used for encryption, then secure communication is achieved, but the security relies on the computational difficulty of factoring N0 which becomes vulnerable with efficient factorization algorithms
Solution Approach 1:
The patent performs preliminary action by transforming the factorization problem before solving it. By factoring N0-1 instead of N0 directly, and by pre-processing the encryption key generation to use prime factors of N0, the system prepares the mathematical structure in advance. This preliminary transformation enables efficient factorization while maintaining the security properties of the encryption scheme during normal operation.
Data Source
AI summary
A method for decoding an encrypted electromagnetic signal W encoded by a first computer with public key N_0=r×s, where N_0, r and s are integers. There is the step of obtaining the electromagnetic signal W from a telecommunications network, or a data network or an Internet or a first non-transient memory. There is the step of storing the electromagnetic signal W in a second non-transient memory. There is the step of decoding with a second computer in communication with the second non-transient memory the electromagnetic signal W in the second non-transient memory by factoring the public key N_0 in at most a time O(log^6 N_0). A non-transitory readable storage medium which includes a computer program stored on the storage medium for decoding an encrypted electromagnetic signal W.