crypto_research

crypto research stuffs from numeric arithmetics to ZK-applied protocols all written with dirty hands.

Code Structure

• docs

  • bls12_381 note and python versioned bls12-381 implementation [$\color{green}\checkmark$]

  • on_proving_pairings note and python versioned bn254 implementation [$\color{green}\checkmark$]

• ecc

  • integer_arithmetic

    • basic arithmetics on big integers

      • add/substruction/multiplication/division [$\color{green}\checkmark$]

      • euclid extended gcd/lehmer extended gcd [$\color{green}\checkmark$]

  • finite_field_arithmetic

    • basic arithmetics on base field $F_q$ and its instantiation

      • add/substruction/multiplication/division/inversion/modulo/exponentiation/sqrt/square [$\color{green}\checkmark$]
      • field implementation for pallas/vasta curves [$\color{green}\checkmark$]

    • basic arithmetics on extension field $F_{q^k}$ of $F_q$

      • quadratic extension $F_{q^2}/F_{q}$ [$\color{green}\checkmark$]

      • cubic extension $F_{q^3}/F_{q}$ [$\color{green}\checkmark$]

      • cyclotomic [$\color{green}\checkmark$]

      • twist/untwist $\Phi: F_{q^k} \mapsto F_{q^{k / d}}$ [$\color{green}\checkmark$]

      • frobenius map [$\color{green}\checkmark$]

      • field implementation for BLS12/MNT/BN pairing-family curves [$\color{green}\checkmark$]

  • elliptic_curve_arithmetic

    • neccessary arithmetics on elliptic curves over base field $F_q$

      • add/doubling/scalar_mul/... [$\color{green} \checkmark$]

    • neccessary arithmetics on pairing-friendly elliptic curves over extension field $F_{q^k}$ and its instantiation

      • add/doubling/scalar_mul/... [$\color{green} \checkmark$]

      • BLS12/MNT/BN pairing-friendly curves [$\color{green} \checkmark$]

  • hyperelliptic_curve_arithmetic [TODO]

  • special_curve_arithmetic [TODO]

  • pairings

    • Weil Pairing [$\color{green}\checkmark$]

    • Tate Pairing [$\color{green}\checkmark$]

    • Ate Pairing [Ongoing]

    • Optimal Ate Pairings [Ongoing]

  • ...

• hash

  • shake128(variable output length) [$\color{green}\checkmark$]
  • poseidon [TODO]

• pcs

  • sparse_polynomial [$\color{green}\checkmark$]
  • IPA [$\color{green} \checkmark$]
  • KZG [Ongoing]

• recursive snark ...

Notes

Small Fields

• Circle Stark from A New Starker
• Binius

Pairings

• Theory and Practical Implementation of BLS12-381
• On Proving Pairings

Proving System

• Plonk From Scratch
• Relaxed Plonk Step by Step
• Cross Term in Plonk

Credits

[1] Handbook of Elliptic and Hyperelliptic Curve Cryptography

[2] Guide to Elliptic Curve Cryptography

[3] Pairings For Beginners

[4] Algorithms for Modern Hardware

[5] IPA PCS

[6] Guide to Pairing-based Cryptography

[7] A note on twists for pairing friendly curves

[8] Pairing-Friendly Elliptic Curves of Prime Order

[9] Optimal Pairings

[10] On the Computation of the Optimal Ate Pairing at the 192-bit Security Level

[11] Efficient Final Exponentiation via Cyclotomic Structure for Pairings over Families of Elliptic Curves

[12] Faster Squaring in the Cyclotomic Subgroup of Sixth Degree Extensions

[13] A Guide to Plane Algebraic Curves

[14] Circle Stark

[15] Binius