Pseudorandom Sequence Generator Using Quadratic Equations
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Solution Overview
Problem
Current pseudorandom sequence generators face challenges in balancing strong security, which relies on difficult mathematical problems, with efficiency in calculation resources, as existing secure algorithms are inefficient, while fast generators lack strong security foundations.
Innovation Solution
A method generating pseudorandom sequences by iteratively calculating n-tuples using m-tuples with at least half of their components represented by second-degree polynomial functions, leveraging the difficulty of solving quadratic equations on a finite body, ensuring security through NP-hard problems, while maintaining efficiency for small parameter values.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If pseudorandom sequence generators use algorithms based on difficult mathematical problems to ensure strong security, then security strength is improved, but calculation efficiency deteriorates
Solution Approach 1:
The patent changes the mathematical parameters from traditional hard problems (discrete logarithm, factorization) to quadratic equation systems over finite fields. By adjusting parameters like the size of the finite field and the number of quadratic equations, the system achieves both strong security (NP-hard problem) and acceptable calculation efficiency for cryptographic applications
Solution Approach 2:
The patent substitutes traditional cryptographic mathematical mechanisms (linear feedback shift registers, elliptic curves) with a new mechanism based on solving systems of quadratic equations. This substitution maintains security while improving efficiency by leveraging the computational properties of quadratic systems over finite fields
2Productivity
If pseudorandom sequence generators use fast algorithms to improve calculation efficiency, then calculation efficiency is improved, but security strength deteriorates
Solution Approach 1:
The patent selects specific parameters for the quadratic equation system (field size, number of equations, equation structure) that optimize the balance between calculation speed and security. The parameters are chosen so that while the problem remains NP-hard (ensuring security), the system can be implemented with relatively simple algebraic operations (improving efficiency)
3Reliability
If pseudorandom sequence generators use complex algorithms to ensure strong security, then security strength is improved, but device complexity increases
Solution Approach 1:
The patent replaces complex cryptographic primitives (hash functions, block ciphers, elliptic curve operations) with a simpler algebraic system based on quadratic equations. The complexity is reduced by using basic field operations (addition, multiplication, inversion) rather than complex algorithmic structures, while maintaining NP-hard security through the mathematical properties of quadratic systems
Data Source
AI summary
A method of generating a pseudorandom sequence of terms belonging to a finite body K of cardinal q≧2 intended to be used in a cryptographic procedure, said method comprising the iterative calculation, from an initialization n-tuple X(0)=(X(0)1, X(0)2, . . . , X(0)n), where n≧2, of elements of K, of n-tuples X(i)=(X(i)1, X(i)2, . . . , X(i)n) of elements of K (where i=1, 2, . . . ), each n-tuple X(i) being obtained on iteration number i in a predetermined manner at least from certain components Y(i)k of an m-tuple Y(i)=(Y(i)1, Y(i)2, . . . , Y(i)m), where m≧n, of elements of K and the terms of said pseudorandom sequence being extracted in a predetermined manner from the n-tuples X and/or the m-tuples Y. For at least one value of i, among said components Y(i)k of the m-tuple Y(i) that are used to obtain the multiplet X(i), at least E(n/2) of them are each represented by a predetermined second degree polynomial function, with coefficients in K, of the components of the n-tuple X(i−1).