How Does Spektre's C Implementation Optimize Modular Arithmetic and NTT for Enhanced Performance?

Optimized Modular Arithmetics and NTT (Finite Field DFT) Implementation in C

The provided code by Spektre implements modular arithmetics and Number Theoretic Transform (NTT) optimizations in C . Here's an explanation of the code along with answers to the questions raised:

Main Function Flow:

1. The fourier_NTT class encapsulates functions for NTT, inverse NTT (INTT), modular arithmetic, and helper functions.
2. To perform NTT on an array of DWORDs src (unsigned 32-bit integers), call NTT() with src and the desired number of elements n (default is 0, indicating the length of src).
3. Similarly, INTT() performs the inverse NTT.
4. The class provides modular arithmetic functions: mod(), modadd(), modsub(), modmul(), and modpow().

Addressing Optimization Questions:

1. Optimizing NTT Performance:

To optimize NTT performance, the code employs several techniques:

• Precomputed Powers Tables: It precomputes powers of W and iW up to a certain threshold (NN) for faster access during recursion.
• Removed Safety Mods: Some unnecessary safety mods are removed, resulting in a 2.5% speedup.
• Improved Modmul Function: The modmul() function is optimized using inline assembly, providing a 34.9% speedup.

2. Safety of Modular Arithmetic Optimizations:

The optimizations in modular arithmetic make use of the specific properties of the modulo prime p being 0xC0000001. However, it's important to note that these optimizations may not be suitable for different values of p.

Additional Optimizations:

1. NTT Fast Loop Rearrangement:

The main NTT loop has been rearranged for better performance.

2. Reduced Branching in Modular Arithmetic:

Bitwise tricks have been used to eliminate branching in modadd(), resulting in faster execution.

3. Removed Unnecessary If Statements:

Unneeded if statements and bitwise functions have been removed, further streamlining the code.

4. New Modmul Inline Assembly:

The modmul() function has been enhanced with a new inline assembly implementation, offering additional speed improvements.

Conclusion:

The optimized code provided by Spektre improves the performance of NTT and modular arithmetics significantly. The optimizations include algorithm improvements, precomputed power tables, and efficient inline assembly.

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