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b1ef2682d6
* implement arbitrary precision modular exponentiation (prerequisite EIP-198) * [modexp] implement exponentiation modulo 2ᵏ * add inversion (mod 2ᵏ) * [modexp] High-level wrapper for powmod with odd modulus * [modexp] faster exponentiation (mod 2ᵏ) for even case and Euler's totient function odd case * [modexp] implement general fast modular exponentiation * Fix modular reduction with 64-bit modulus + fuzz powmod vs GMP * add benchmark * add EIP-198 support * fixups following self review * fix test paths
Common configuration
- Low-level logical and physical word definitions
- Elliptic curve declarations
- Cipher suites
Algorithms
Modular inverses mod 2ⁿ
We use "Dumas iterations" to precompute Montgomery magic number -1/n[0] (mod 2^Wordbitwidth)
Explanation p11 "Dumas iterations" based on Newton-Raphson:
- Cetin Kaya Koc (2017), https://eprint.iacr.org/2017/411
- Jean-Guillaume Dumas (2012), https://arxiv.org/pdf/1209.6626v2.pdf
- Colin Plumb (1994), http://groups.google.com/groups?selm=1994Apr6.093116.27805%40mnemosyne.cs.du.edu Other sources:
- https://crypto.stackexchange.com/questions/47493/how-to-determine-the-multiplicative-inverse-modulo-64-or-other-power-of-two
- https://mumble.net/~campbell/2015/01/21/inverse-mod-power-of-two
- http://marc-b-reynolds.github.io/math/2017/09/18/ModInverse.html