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Implement initial Montgomery multiplication framework in secp256k1 field operations

This commit introduces the foundational structure for Montgomery multiplication in `field.go`, including methods for converting to and from Montgomery form, as well as a multiplication function. The current implementation uses standard multiplication internally, with a placeholder for future optimizations. Additionally, a new markdown file, `MONTGOMERY_NOTES.md`, outlines the current status, issues, and next steps for completing the Montgomery multiplication implementation.

2025-11-02 15:30:17 +00:00

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Montgomery Multiplication Implementation Notes

Status

Montgomery multiplication has been partially implemented in field.go. The current implementation provides the API structure but uses standard multiplication internally.

Current Implementation

ToMontgomery(): Converts to Montgomery form using R² multiplication
FromMontgomery(): Converts from Montgomery form (currently uses standard multiplication)
MontgomeryMul(): Multiplies two Montgomery-form elements (currently uses standard multiplication)
montgomeryReduce(): REDC algorithm implementation (partially complete)

Issues

The FromMontgomery() implementation needs proper R⁻¹ computation
The MontgomeryMul() should use the REDC algorithm directly instead of standard multiplication
The R² constant may need verification
Tests are currently failing due to incomplete implementation

Next Steps

Compute R⁻¹ mod p correctly
Implement proper REDC algorithm in MontgomeryMul
Verify R² constant against reference implementation
Add comprehensive tests

References

Montgomery reduction: https://en.wikipedia.org/wiki/Montgomery_modular_multiplication
secp256k1 field implementation: src/field_5x52.h

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