Asymmetric Encryption Method Using Iterative Factorization
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Solution Overview
Problem
Existing asymmetrical encryption methods, such as Rabin and Williams, face complexity restrictions and increased computing power demands due to limited data bit transmission and security vulnerabilities from message part dependencies, while RSA methods are inefficient in data transmission and decryption complexity.
Innovation Solution
An asymmetrical encryption method based on the factorization problem using a composite number n as the public key and its prime factors as the private key, with an encryption function iterated L times to encrypt messages m=(m1, m2) into c=(c1, c2), where decryption is facilitated by solving quadratic equations modulo n, and multivaluedness is resolved using additional bits and the Jacobi symbol.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Quantity of substance
If Rabin or Williams encryption methods are used, then data bit transmission is possible, but complexity restrictions and increased computing power demands occur
Solution Approach 1:
The message is divided into two independent parts m1 and m2, which are processed separately through the encryption function. This segmentation allows more data bits to be transmitted while the computational complexity remains manageable because each part can be processed independently modulo n.
Solution Approach 2:
The encryption method changes the parameter space by using a composite modulus n and iterating the encryption function L times. This allows the system to transmit more data bits efficiently while controlling computing power demands through the iterative structure and modular arithmetic properties.
2Reliability
If message parts are made dependent for security, then security vulnerability increases, but data transmission efficiency decreases
Solution Approach 1:
The encryption function uses asymmetric operations on the two message parts: m1 is added to m2 modulo n, while m1 is multiplied by m2 modulo n. This asymmetric treatment provides security through the difficulty of reversing these operations without the private key, while maintaining independence between message parts for efficient transmission.
Solution Approach 2:
The encryption transforms the message from a single-dimensional value to a two-dimensional tuple (m1, m2) that is processed in the modular arithmetic space. This dimensional change allows independent processing of message parts, improving transmission efficiency while security is maintained through the mathematical properties of the modular operations.
3Reliability
If RSA method is used for encryption, then data can be encrypted, but decryption complexity and transmission efficiency are reduced
Solution Approach 1:
The method extracts the core security requirement from RSA (factorization difficulty) while removing the inefficient aspects. By using a composite modulus and iterative encryption with quadratic equation solving, the patent achieves decryption complexity based on factorization without the overhead of RSA's exponentiation operations, improving decryption efficiency.
Solution Approach 2:
The encryption method creates a simplified copy of RSA's security model by using composite numbers and factorization difficulty, but implements it through iterative quadratic operations rather than RSA's modular exponentiation. This copying approach maintains encryption capability while reducing decryption complexity.
4Measurement precision
If quadratic equation multivaluedness is not resolved, then decryption ambiguity occurs, but additional bits increase transmission overhead
Solution Approach 1:
The parity bit and Jacobi symbol act as feedback mechanisms that provide additional information about the correct root selection. These minimal additional bits enable the decryptor to uniquely identify the correct square root without requiring extensive transmission overhead, resolving the multivaluedness issue efficiently.
Solution Approach 2:
The Jacobi symbol provides a distinctive identifier (similar to a color code) that helps distinguish between different square roots. By transmitting this symbolic indicator along with the encrypted data, the system enables precise root selection with minimal overhead, maintaining decryption accuracy while limiting transmission increase.
Data Source
AI summary
The present invention relates to an asymmetrical encryption method. The public key is made up of a large composite number n; the private key is made up of the factors of the composite number. The encryption is made up of a number of iterations of individual encryption steps that are successively reversed during the decryption. In this context, the reversal of an individual encryption step requires the solving of a quadratic equation modulo m [sic]. The private key is preferably made up of the large prime numbers p and q. The public key is the product n of these two prime numbers, as well as a comparatively small integer L which is greater than one. The message m is made up of two integral values m1 and m2, thusm=(m1, m2),both values being in the set Zn={0, 1, 2, . . . , n−1}.The encryption is accomplished via the equationc=fL(m).