Modular Arithmetic Encryption Using Non-Prime Numbers
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Solution Overview
Problem
Existing encryption algorithms using modular arithmetic are limited by their inability to utilize non-prime numbers, leading to potential encryption results of zero and non-unique mappings, which hampers secure and efficient data transmission.
Innovation Solution
A method employing modular arithmetic with random numbers P and Q, where a message M is encrypted as E = (MP^i) mod (PQ - 1), and decrypted as M = (EQ^i) mod (PQ - 1), using a tuple (E, i) for secure transmission, allowing non-prime numbers and reducing computational burden through precomputed tables.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Speed
If modular arithmetic is used in encryption algorithms, then encryption speed is improved, but the algorithm cannot use non-prime numbers which limits versatility
Solution Approach 1:
The patent changes the parameter constraints by allowing non-prime numbers to be used in modular arithmetic encryption. Specifically, it uses the property that for any integer a coprime to n, there exists an integer b such that (a*b) mod n = 1, extending the applicability of modular arithmetic from only prime moduli to any integer modulus, thereby improving versatility while maintaining encryption speed.
2Productivity
If modular arithmetic is used in encryption, then computational efficiency is improved, but encryption may result in zero or non-unique mappings which reduces reliability
Solution Approach 1:
The patent applies local quality by imposing specific conditions on the choice of parameters: the message M must be less than the modulus n, and the exponent a must be coprime to n. These localized constraints ensure that the encryption function E = (M^a) mod n produces unique, non-zero results while maintaining computational efficiency. The decryption uses the corresponding exponent b where (a*b) mod φ(n) = 1, ensuring unique mapping recovery.
3Reliability
If traditional encryption algorithms are used, then security is maintained, but data transmission length and computational effort increase
Solution Approach 1:
The patent extracts only the essential elements needed for encryption and decryption: the message M, modulus n, exponent a for encryption, and exponent b for decryption. By using the mathematical property that (M^a)^b mod n = M when (a*b) mod φ(n) = 1, it transmits only the encrypted value E = (M^a) mod n and the exponent a, rather than transmitting the entire message or complex key material, thereby reducing data transmission length while maintaining security.
Data Source
AI summary
The invention includes a method for encrypting and transmitting a message M from a first location to a second location. Random numbers P and Q are initially selected, each of them being either a prime or a non-prime number. P and Q are both furnished to a sender and a recipient at the first and second locations, respectively. A client at the first location generates an encrypted message E, which is the remainder left when a modulus operator divides a first quantity, equal to M multiplied by P to the i power, where i is a selected number, by a second quantity equal to (PQ−1). A tuple comprising E and i is transmitted from the first location to the second location, and a second data processing system at the second location to determines the message M from E, i and Q.