Falcon Key Generation Using NTT to Cut Recursive Multiplication Cost
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Solution Overview
Problem
Existing Falcon key generation methods are inefficient, particularly in the last recursive calls where polynomials have fewer coefficients but larger sizes, leading to increased complexity and reduced efficiency.
Innovation Solution
Optimize Falcon key generation by employing techniques such as FFT, multi-modular NTT, Karatsuba, and Toom-Cook strategies during initial recursions, and using evaluation-interpolation techniques in an evaluated format for subsequent recursions to reduce polynomial multiplications and coefficient processing time.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Productivity
If traditional polynomial multiplication methods are used in the last recursive calls of Falcon key generation, then the algorithm completes the key generation process, but the complexity increases and efficiency decreases due to larger coefficient sizes and fewer coefficients
Solution Approach 1:
The patent applies segmentation by dividing the polynomial multiplication process into smaller modular components. Specifically, it uses the Number Theoretic Transform (NTT) to break down polynomial multiplication into point-wise multiplications in the transformed domain, and further segments the coefficients into smaller chunks that can be processed independently. This segmentation reduces the overall complexity of polynomial multiplication in the last recursive calls where coefficient sizes are large.
Solution Approach 2:
The patent replaces traditional mechanical polynomial multiplication methods with algebraic and number-theoretic techniques. Instead of directly multiplying polynomials coefficient-by-coefficient, it substitutes the mechanical process with NTT-based transformation, which converts polynomial multiplication into simpler point-wise operations in the frequency domain, significantly reducing computational complexity for large coefficients.
2Measurement precision
If intermediate conversions are performed during polynomial multiplication in recursive calls, then the algorithm maintains numerical precision, but the processing time increases due to repeated format conversions
Solution Approach 1:
The patent implements continuity of useful action by maintaining coefficients in a unified evaluated format (such as NTT domain representation) throughout the entire recursive key generation process. Instead of repeatedly converting between different formats (e.g., coefficient format, evaluated format, and back), the algorithm performs all polynomial operations continuously in the evaluated format, eliminating intermediate conversion steps and reducing processing time while maintaining numerical precision through careful modular arithmetic.
Solution Approach 2:
The patent introduces an intermediary evaluated format (NTT domain representation) that serves as a mediator between different computational requirements. By transforming polynomials into this intermediary format at the beginning of the recursive process and maintaining operations in this format throughout, the algorithm avoids repeated conversions between coefficient and evaluated formats, reducing time loss while preserving precision through the properties of the Number Theoretic Transform.
Applied Scientific Principles
This section explains which scientific principles are used to turn an abstract innovation direction into a practical engineering solution.
Function Achieved in This Case
Significantly enhances the speed and efficiency of Falcon key generation by reducing polynomial multiplication complexity and minimizing intermediate conversions, especially in the last recursive stages.
Implementation Method 1
The FFT converts an input, such as a large integer, from its domain of origin to a corresponding output in an FFT domain. The inverse FFT, invFFT, converts an input from an FFT domain to a corresponding output, in the domain of origin
Implementation Method 2
The Number Theoretic Transform (NTT) is analogous to the FFT, but in a prime field, where p is a prime such that p = 1 mod 2n. Under these conditions, φ has exactly n roots (x k) over, and any polynomial can be represented by the values f(x k). Conversion to and from NTT representation can be done efficiently in O(n log n) operations in.
Data Source
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Figure 5A
AI summary
The disclosure relates to a cryptographic device and to a method to increase the speed of the cryptographic device when it generates post-quantum key pairs of the Falcon type.