binary field arithmetic
an encyclopedia of the mathematics behind kuro's F₂ tower — from first principles to applications. every concept is grounded in the tower we implement: F₂ → F₂² → F₂⁴ → F₂⁸ → F₂¹⁶ → F₂³² → F₂⁶⁴ → F₂¹²⁸.
foundations
binary-fields — field axioms in characteristic 2, GF(2) as the algebra of bits, GF(2^n) as vector spaces
tower-construction — quadratic extensions, why x² + x + α, the canonical generator, visual tower diagram
algorithms
karatsuba — Karatsuba multiplication through the tower, 3 multiplications instead of 4, recursive cost analysis
inversion — Fermat's little theorem in GF(2^n), tower-recursive inversion, the Frobenius endomorphism
data structures
packed-operations — Packed128: 128 F₂ elements in one u128, SIMD-native addition/multiplication, the inner product kernel
context
applications — BitNet AI inference, binary STARKs, coding theory, AES, quantized SpMV
f2-vs-fp — when to use binary fields (kuro) vs prime fields (nebu), cost model comparison, decision guide
Dimensions
explanation
rs/docs/explanation
explanation
zheng/docs/explanation
zheng explanations zheng is a polynomial proof system. these documents explain the concepts, design decisions, and historical context. for formal definitions, see reference/. for the hash primitive, see hemera. for the VM whose traces we prove, see nox. reading path pages vision | page | topic |…
trop/docs/explanation
tropical semiring arithmetic an encyclopedia of the mathematics behind trop's (min, +) algebra — from first principles to applications. every concept is grounded in the semiring we implement: tropical addition is min, tropical multiplication is ordinary addition, and shortest paths are matrix…
trident/docs/explanation
💡 Trident Explanation [← Documentation Index](/trident/docs/readme) Understanding-oriented. Deep dives into why Trident works the way it does, for readers who want the full picture. 🏗️ Core Architecture | Document | Description | |----------|-------------| |…
genies/docs/explanation
isogeny group action arithmetic an encyclopedia of the mathematics behind genies — supersingular isogenies, class group actions, and the CSIDH construction. every concept connects back to the central operation: the commutative group action of cl(O) on supersingular elliptic curves over F_q.…
nox/docs/explanation
nox explanations conceptual documentation — why nox works the way it does, the design choices behind it, and the consequences that emerge. these pages illuminate the architecture. for formal definitions, see reference/. for task-oriented instructions, see docs/guides/ (when available). pages | page…
bbg/docs/explanation
explanation
hemera/docs/explanation
why Hemera works the way it does design decisions behind the Hemera hash primitive. philosophy why-hemera — eight design principles: permanence, the tree, endofunction, self-reference, identity, unity, beauty, the name the-name — etymology: Hemera in the Protogenoi, genealogy of hash names…
jali/docs/explanation
polynomial ring arithmetic an encyclopedia of the mathematics behind jali's polynomial ring R_q = F_p[x]/(x^n+1) — from first principles to applications. every concept is grounded in the ring we implement: structured vectors of Goldilocks field elements with negacyclic convolution. foundations…
nebu/docs/explanation
finite field arithmetic an encyclopedia of the mathematics behind the Goldilocks field — from first principles to applications. every concept is grounded in the field we implement: p = 2⁶⁴ − 2³² + 1. foundations finite-fields — field axioms, existence and uniqueness, GF(p), characteristic, the…