Galois Field Cryptographic System Using Extended Field Multiplication
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Solution Overview
Problem
Conventional methods for combining data streams in cryptographic systems are computationally inefficient and lack resistance to errors, with fixed generating algorithms tied to specific irreducible polynomials, making them vulnerable to sequence prediction and error detection challenges.
Innovation Solution
The method involves generating a data sequence and an encryption sequence using pseudo-random or pseudo-chaotic numbers, mapped through an Affine transformation from a Galois field GF[p^k] to GF[p^k+1], combined using Galois field multiplication, with an inverse decryption sequence generated for efficient error correction and security.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If conventional methods increase the length of the generating polynomial or use nonlinear combination techniques to increase security, then sequence security is improved, but implementation complexity significantly increases
Solution Approach 1:
The patent changes the parameter of the Galois field from GF[2^k] to GF[2^(k+1)], mapping sequences to an extended field. This parameter change enables the use of closed-form multiplication formulas that reduce computational complexity while maintaining security through the extended field's mathematical properties.
Solution Approach 2:
The patent replaces complex nonlinear combination techniques with a systematic Affine transformation followed by closed-form Galois field multiplication. This substitution uses algebraic structures (Affine transformations and field multiplication formulas) to achieve security without the implementation complexity of nonlinear methods.
2Ease of operation
If conventional methods use fixed generating algorithms tied to specific irreducible polynomials, then sequence generation is simplified, but resistance to sequence prediction decreases
Solution Approach 1:
The patent introduces dynamic elements by using Affine transformations with adjustable parameters (a, b) and extending the field to GF[2^(k+1)]. This allows the sequence generation to adapt to different security requirements while maintaining a systematic structure, preventing fixed polynomial vulnerabilities.
Solution Approach 2:
The patent moves the sequence generation from a fixed polynomial space to an extended Galois field GF[2^(k+1)], adding a dimensional aspect to the field extension. This dimensional change provides additional mathematical layers that enhance prediction resistance while maintaining operational simplicity through structured transformations.
3Reliability
If conventional methods increase the length of the sequence to improve security, then repetition period increases, but processing time increases
Solution Approach 1:
The patent performs preliminary Affine transformations to map input sequences to the extended Galois field before multiplication. This preliminary action organizes the data in a way that enables efficient closed-form multiplication, reducing overall processing time while achieving enhanced security through the field extension.
Solution Approach 2:
The patent uses closed-form multiplication formulas that replicate efficient computational patterns across the extended field. By copying and applying the same structured multiplication approach used in GF[2^k] to GF[2^(k+1)], the system maintains processing efficiency while achieving longer effective sequence periods through the extended field's properties.
4Device complexity
If conventional cryptographic systems lack error detection and correction capabilities, then system simplicity is maintained, but resistance to transmission and storage errors decreases
Solution Approach 1:
The patent makes the Galois field multiplication operation universal by defining closed-form formulas that work consistently across GF[2^k] and GF[2^(k+1)]. This universal operation can serve both encryption functions and error detection/correction functions, adding error resistance capability without requiring separate dedicated error correction mechanisms.
Data Source
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AI summary
A cryptographic system (CS) comprised of generators (502), (504), (510), an encryption device (ED), and a decryption device (DD). The generator (502) generates a data sequence (DS) including payload data. The generator (504) generates an encryption sequence (ES) including random numbers. The ED (506) is configured to perform a CGFC arithmetic process. As such, the ED is comprised of a mapping device (MD) and an encryptor. The MD is configured to map the DS and ES from Galois field GF[pk] to Galois extension field GF[pk+1]. The encryptor is configured to generate an encrypted data sequence (EDS) by combining the DS and ES utilizing a Galois field multiplication operation in Galois extension field GF[pk+1]. The generator (510) is configured to generate a decryption sequence (DS). The DD (508) is configured to generate a decrypted data sequence by performing an inverse of the CGFC arithmetic process utilizing the EDS and DS.