polynomial ring arithmetic for cyber. R_q = F_p[x]/(x^n+1) over Goldilocks. jali is the encrypted computation engine — where nebu proves truth and genies proves privacy, jali proves that computation happened correctly on data nobody saw.
jali (जाली — lattice, mesh, screen) is the fifth execution algebra. n Goldilocks elements coupled by cyclotomic multiplication. one R_q multiply = 3n F_p multiplies via NTT. at n=1024: 3072× cost per operation over scalar nebu. the ring structure is what makes Ring-LWE hard — and what makes TFHE, lattice KEM, and structured noise possible.
workloads
| domain | mechanism |
|---|---|
| TFHE ciphertexts | encrypt/decrypt over R_q, programmable bootstrapping |
| lattice KEM (seal) | Module-RLWE key encapsulation over R_q |
| blind rotation | n polynomial multiplies during FHE bootstrapping |
| key switching | Galois automorphisms of R_q (slot permutation) |
| noise tracking | bound estimation through ring operations |
| convolution | native polynomial multiply = convolution |
nox integration
no new Layer 1 patterns. polynomial multiply decomposes to ntt + pointwise_mul + intt — existing nebu jets. ring-specific acceleration through Layer 3:
Layer 3 jets: jet_ntt_batch, jet_key_switch, jet_gadget_decomp, jet_noise_track, jet_blind_rotate.
verification: ring operations proved via PCS₃ (ring-aware Brakedown with NTT batching). zheng ring-aware CCS exploits R_q structure. HyperNova folds into Goldilocks accumulator.
dependency graph
nebu (F_p) — scalar arithmetic, NTT roots
↓
jali (R_q) ← this repo
↓
mudra (veil — FHE scheme over jali)
↓
nox (jets)
↓
zheng (PCS₃ — ring-aware proving)
↓
bbg (encrypted state)
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