Asymmetric Cryptography Using Polynomial Systems in Extension Fields
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Solution Overview
Problem
Existing asymmetrical encryption and electronic signature algorithms face challenges such as high computational burdens and limitations in dual functionality for encryption and signature applications, with some being ineffective or vulnerable to cracking.
Innovation Solution
A method involving affine invertible transformations and systems of polynomials in finite fields to process data blocks, allowing for efficient decryption and encryption by transforming data into higher extension fields for secure processing, masking the trap-door from public access and requiring low computational power.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If asymmetrical encryption algorithms such as RSA or elliptic curve algorithms are used, then security and public-key functionality are improved, but computational burden and calculation complexity increase significantly
Solution Approach 1:
The patent segments the computational problem by processing data in blocks of (n·d) successive data items rather than handling entire messages at once. The data is divided into manageable blocks that can be processed efficiently through affine transformations and polynomial systems, reducing the overall computational burden while maintaining security.
Solution Approach 2:
The patent changes the mathematical parameters by working in extension fields E=GF(q^d) rather than base fields, and by using systems of n polynomials with specific degree constraints. This parameter transformation allows for more efficient computation compared to traditional RSA or elliptic curve methods while preserving cryptographic security.
2Adaptability or versatility
If traditional asymmetrical algorithms are used for both encryption and signature verification, then algorithm versatility is improved, but the same computational complexity and burden apply to both operations
Solution Approach 1:
The patent creates a universal algorithm that can perform both encryption and signature verification using the same mathematical framework. The system uses public polynomials for both encryption of messages and verification of signatures, while the secret key enables decryption or signature generation. This multi-functional approach eliminates the need for separate algorithms for different cryptographic operations.
3Productivity
If the unbalanced oil and vinegar scheme is used, then signature verification efficiency is improved, but the algorithm cannot serve as an encryption algorithm and requires adding many vinegar variables
Solution Approach 1:
The patent employs asymmetry in the polynomial system by specifying that degree(x_i) = 1 and degree(x_j) ≤ 2 with i < j, creating an unbalanced structure similar to oil-vinegar schemes. However, this asymmetry is combined with affine transformations and extension fields to enable both encryption and signature functions, unlike the original oil-vinegar scheme which was signature-only.
4Reliability
If key length is increased to at least 128 bits for symmetrical algorithms, then security is improved, but the precautions required for secure key communication become more severe constraints
Solution Approach 1:
The patent implements asymmetrical cryptography where public polynomials can be freely shared without security risks, while secret keys remain private. This eliminates the need for secure key communication channels required by symmetrical algorithms. The public key consists of polynomials that can be published openly, and the asymmetric mathematical structure ensures that deriving the secret key from public polynomials is computationally infeasible.
Data Source
AI summary
The invention proposes a method and a device for decrypting an encrypted message represented by a sequence C or applying an electronic signature to a sequence C, said sequence C consisting of data belonging to a finite field K=GF(q), where q>1, in which there are processed successive blocks each comprising (n·d) successive data items from the sequence C, where n and d are predetermined integers greater than 1, the processing of such a block including the following steps: applying a predetermined affine invertible transformation t−1 to said block; interpreting the resulting block as being formed of n successive elements (y1, y2, . . . , yn) of an extension E=GF(qd) of the field K; calculating an n-tuplet (x1, x2, . . . , xn) of elements of the field E by solving a system f of n predetermined polynomials of the formyk=∑1≤i≤j≤nak(ij)xiqαixjqβj+∑1≤i≤nbkixiqγi+ck(1≤k≤n)in which the coefficients ak(ij), bk(i), and ck belong to E and the exponents αi, βj, and γi are positive or zero integers; interpreting said n-tuplet (x1, x2, . . . , xn)as being a new block formed of (n·d) successive elements from the field K; and applying a predetermined affine invertible transformation s−1 to said new block.