Polynomial Evaluation for Finite Field Cryptography
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Solution Overview
Problem
Existing methods for evaluating functions over finite fields, particularly in cryptographic data processing, are inefficient due to the high number of non-linear operations required, which can be compromised by malicious observation and data deduction.
Innovation Solution
A method that optimizes polynomial evaluation by determining monomials of integer powers of the characteristic p through successive raisings and multiplications, using a recursive process to minimize non-linear operations and maintain masking, allowing for efficient evaluation of polynomials while preserving data security through masked representations.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If conventional polynomial evaluation methods are used, then the function can be evaluated, but the number of non-linear operations is high which compromises security
Solution Approach 1:
The polynomial is segmented into two parts: terms with degrees that are integer powers of the characteristic p, and other terms. This segmentation allows the first part to be evaluated using only linear operations (successive raisings to power p), while the second part requires fewer multiplications, thereby reducing the total number of non-linear operations and improving security against side-channel attacks.
Solution Approach 2:
The method changes the parameter representation by expressing the polynomial in terms of the characteristic p of the finite field. By evaluating terms with degrees that are integer powers of p through successive linear operations rather than general multiplications, the method reduces non-linear operations while maintaining correctness in fields of characteristic p.
2Reliability
If the number of non-linear operations is reduced, then data security is enhanced, but the evaluation speed may be affected
Solution Approach 1:
The polynomial evaluation is segmented into two distinct computational paths: one for terms with degrees that are integer powers of the characteristic p (evaluated via successive linear raisings), and another for remaining terms (evaluated via fewer multiplications). This segmentation optimizes both security by reducing non-linear operations and speed by using efficient linear operations where applicable.
Solution Approach 2:
The method substitutes general multiplication operations with more efficient operations specific to fields of characteristic p. By using successive raisings to power p (which are linear operations in such fields) instead of general multiplications, the method replaces a slower, more complex operation with a faster, simpler one, thereby improving evaluation speed while enhancing security.
3Reliability
If successive raisings to power p are used, then linear operations replace non-linear operations, but additional computational steps are required
Solution Approach 1:
The polynomial terms are segmented based on whether their degrees are integer powers of the characteristic p. This segmentation ensures that successive raisings to power p are applied only where mathematically valid and efficient, avoiding unnecessary computational steps while maximizing the security benefits of reduced non-linear operations.
Solution Approach 2:
The method applies successive raisings to power p partially - only to the extent that the polynomial degree allows (i.e., only for terms where the degree is an integer power of p). This partial application avoids excessive computational steps while still achieving the security benefit of replacing non-linear operations with linear ones for the applicable portion of the polynomial.
Data Source
AI summary
A method for evaluating a function of a finite field of characteristic p into itself, for an element x of the field, uses an evaluation, for the element x, of a polynomial formed by a plurality of monomials. The evaluation of the polynomial includes the following steps: determining monomials the degree of which is an integer power of the characteristic p by successive raisings of the element x to the power p; and determining monomials the degree of which is different from an integer power of the characteristic p on the basis of the determined monomials, the degree of which is an integer power of the characteristic p, and by at least one multiplication. An evaluating device is also provided.