Scalar to Tau-Adic Conversion Algorithm for Cryptographic Circuits
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Solution Overview
Problem
Existing methods for converting scalars to τ-adic representation in elliptic curve cryptography are complex and time-consuming, especially when dealing with high scalar values that require multiple reduction operations.
Innovation Solution
A cryptographic method and device that perform scalar conversion by executing an algorithm applied to a scalar to generate an output vector, where each iteration updates the scalar value and sets digits of the output vector based on intermediate data values calculated modulo a complex number τ, with the number of iterations determined by a stop criterion.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Productivity
If traditional scalar conversion methods are used, then the conversion process is straightforward, but it becomes complex and time-consuming for high scalar values requiring multiple reduction operations
Solution Approach 1:
The patent segments the scalar conversion process into iterative steps that process the scalar in chunks rather than requiring complete reduction. The algorithm divides the conversion into manageable iterations where each iteration processes a portion of the scalar and produces intermediate results, avoiding the need for multiple complete reduction operations on high scalars.
Solution Approach 2:
The patent performs preliminary actions by pre-computing and storing reduction results for common cases. The algorithm prepares intermediate values and reduction constants in advance, so that during actual scalar conversion, these pre-computed values can be quickly applied without performing full reduction operations from scratch.
2Difficulty of detecting and measuring
If the scalar value is kept visible during conversion, then the conversion process is simple to implement, but the scalar value becomes easily detectable during cryptographic operations
Solution Approach 1:
The patent introduces intermediate representations as mediators between the original scalar and the final τ-adic form. Instead of directly converting and exposing the scalar, the algorithm uses intermediate values that obscure the original scalar while maintaining mathematical equivalence. These intermediates serve as protective layers that make it difficult to detect the original scalar value during conversion operations.
Solution Approach 2:
The patent changes parameters during conversion by transforming the scalar through multiple representation forms with different properties. The algorithm dynamically adjusts the representation parameters (such as the base and digit constraints) throughout the conversion process, making it difficult to trace back to the original scalar while maintaining the mathematical relationship needed for correct conversion.
Data Source
AI summary
The present disclosure relates to a cryptographic method including the execution, by a cryptographic circuit, of an algorithm applied to a scalar in order to generate an output vector, of length L+n, which digits are d0, . . . , dL+n−1, the algorithm comprising iterations i, each iteration i taking an input data value, initially equal to said scalar and an input vector of length c, which digits are d′i, . . . , d′i+c−1, where for each j∈{i, . . . , i+c−1}, the digit d′j is such that:dj′={dj if j<Ldj-m otherwise.