jali reference

canonical specification for polynomial ring arithmetic R_q = F_p[x]/(x^n+1) over Goldilocks.

what jali is

jali (जाली — lattice/mesh) is the fifth execution algebra for cyber. polynomial ring elements are structured vectors of n Goldilocks field elements with multiplication defined by convolution modulo x^n+1.

one R_q multiply = 3n F_p multiplies (via NTT). the 3072x gap (at n=1024) over scalar nebu arithmetic justifies a dedicated algebra — same criterion that separates kuro (32x gap for binary).

spec pages

algebra

R_q = F_p[x] / (x^n + 1)

  p = 2^64 - 2^32 + 1           Goldilocks prime (nebu)
  n = power of 2 (1024, 2048, 4096)
  x^n + 1 = cyclotomic polynomial Φ_{2n}

  element: vector of n Goldilocks field elements (coefficients)
  addition: coefficient-wise (n parallel nebu adds)
  multiplication: NTT → pointwise → INTT (3n nebu muls)
  automorphisms: x → x^{5^k} (Galois group of Φ_{2n})

via NTT, R_q ≅ F_p^n (n independent copies of Goldilocks). the ring structure (cyclotomic wrapping) is what makes Ring-LWE hard. the NTT decomposition is what makes computation fast.

consumers

dependency

nebu (F_p scalar arithmetic + NTT roots of unity)
  ↓
jali (R_q polynomial ring arithmetic)
  ↓
mudra (seal, veil — protocols over jali)
zheng (PCS₃ — ring-aware proving)
nox (jets — accelerated execution)

jali depends only on nebu. no hemera dependency (hashing is the consumer's job). pure arithmetic.

see also

• nebu — Goldilocks field arithmetic (the scalar field)
• jali is to R_q what nebu is to F_p