kuro.md

F₂ tower field arithmetic for binary proving. kuro is to F₂ what nebu is to Goldilocks.

the tower

F₂ → F₂² → F₂⁴ → F₂⁸ → F₂¹⁶ → F₂³² → F₂⁶⁴ → F₂¹²⁸
 1b    2b    4b    8b    16b     32b     64b     128b

each extension: $F_{2^{2k}} = F_{2^k}[x] / (x^2 + x + \alpha_k)$

at the top: F₂¹²⁸ = 128 F₂ elements packed in one u128 machine word. one XOR = 128 additions. one AND = 128 multiplications. SIMD-native.

why kuro exists

bitwise operations in Goldilocks (F_p) cost ~32 constraints each — bit decomposition is expensive in a prime field. in F₂, they cost 1 constraint. the 32× gap is the algebraic distance between F_p and F₂.

two workloads dominate the binary regime:

• quantized AI inference: BitNet 1-bit models. matrix-vector multiply = XOR + popcount
• tri-kernel SpMV: quantized axon weights for φ* iteration

kuro provides the field arithmetic. the binary PCS (zheng) provides the commitment. nox Bt instantiation (nox<F₂>) provides the execution. together: binary workloads at native cost.

dependency graph

nebu (Goldilocks)    kuro (F₂) ← this repo
       ↓                ↓
     hemera (hash, trust anchor)
       ↓
     nox (VM)
       ↓
     zheng (proofs, binary PCS uses kuro)
       ↓
     bbg (state, binary traces fold into Goldilocks accumulator)

discover all concepts