ArmadilloEcdlpVerify

Python verification scripts for Armadillo ECDLP, operating over the elliptic curve E(GF(2¹¹³)): y² + xy = x³ + x² + 1.

Files

• verify_final.py Final cryptographic verification script.

  • Verifies that the base point G and public key Q precisely reside on the elliptic curve.
  • Performs Subgroup Order Verification: Checks that n × G = O and n × Q = O (where O is the Point at Infinity) to ensure point membership in the prime-order subgroup.
  • Computes the private key scalar multiplication Q = k × G to validate the solution over Polynomial Basis.

• basis_convert.py Conversion tool for bidirectional mapping between Optimal Normal Basis type 2 (ONB2) and Polynomial Basis (PB) representations over GF(2¹¹³).

• gen_basepoint.py Simulation of the Armadillo C++ base point generation logic.

  • Faithfully reproduces the custom 32-bit PRNG and mult() logic from keygen_random.cpp.
  • Reconstructs the ECC_RandomPoint and ECC_Embed logic, including the iterative X-coordinate incrementation when quadratic roots are absent.
  • Validates the base point G = 2 × P derivation for a given BasePointInit seed (e.g., 0xC1F7F755).

• derive_matrix.py Script to algebraically derive the rigorous 113×113 bit conversion matrices (M and M⁻¹) directly from the known coordinates of Q.

  • Generates isomorphic states using the Frobenius endomorphism (squaring) acting as a completely linear transformation constraint in GF(2¹¹³).
  • Dynamically forms linearly independent base projections internally leveraging cyclic shifts (ONB2's fundamental squaring property).
  • Completes GF(2) matrix reconstruction effortlessly using precise Gaussian elimination in O(n³) operations, avoiding complex period tracing.

Usage

You can run each of these tools independently via Python:

# To perform final proof validity tests:
python verify_final.py

# To test manual PB <-> ONB2 coordination points checks:
python basis_convert.py

# To deduce and regenerate the exact GF(2) conversion matrices analytically:
python derive_matrix.py

# To simulate C++ base point generation from a 32-bit seed:
python gen_basepoint.py