Glossary and notation

This page is a quick reference for terms and symbols used throughout the Akita Book. Individual chapters introduce each idea in more detail.

Glossary

Term	Meaning
Polynomial commitment scheme (PCS)	A protocol that commits to a polynomial and later proves a claimed evaluation without sending the whole polynomial.
Multilinear polynomial	A polynomial that has degree at most one in each variable. Its values on the Boolean cube determine it everywhere.
Evaluation table	The values of a multilinear polynomial at all Boolean inputs. A polynomial in $n$ variables has a table of length $2^{n}$ .
Commitment	A short public value that binds the polynomial for later opening checks. This does not promise hiding; see Akita's current privacy boundary.
Opening	A proof that a committed polynomial evaluates to a claimed value at a specified point.
Opening claim	The polynomial identity being proved: a commitment, an opening point, and the claimed value.
Commitment group	One or more related commitments that are opened together under one generated schedule.
Precommitted group	A commitment produced earlier and carried into a later grouped opening. Its profile is fixed when the commitment is created.
Fold	One step that replaces a large opening relation with a smaller one while preserving the claim the verifier must check.
Root fold	The first fold. It starts from the user's polynomial commitments and opening claims.
Recursive fold	An intermediate fold that consumes the relation produced by the preceding fold. A schedule may contain none.
Terminal fold	The final step, where the remaining witness is small enough to reveal and check directly.
Generated schedule	A precomputed choice of fold dimensions, bounds, opening methods, and proof layout for one supported request shape.
Catalog row	The generated record selected for a particular configuration and group shape. The prover and verifier read the same row.
Proof shape	The exact number, order, dimensions, and byte layout of every proof component required by a schedule.
Public setup	Public matrices and prepared data used to commit, prove, and verify. Akita does not require a secret trapdoor.
Transcript	The ordered record of public inputs and proof messages from which Fiat-Shamir challenges are derived.
Base field	The field that stores committed coefficients and supports the lattice relation.
Extension field	A larger field used for opening points, claimed evaluations, and transcript challenges in some configurations.
Cyclotomic ring	The quotient ring in which Akita's lattice commitments and fast polynomial arithmetic operate.
Dense polynomial	A polynomial whose table positions may each contain an arbitrary field element.
One-hot polynomial	A polynomial whose table has at most one `1` in each consecutive chunk of `onehot_k` entries; all other entries are zero. An all-zero chunk is allowed. Akita can use smaller schedules for this structure.
Gadget decomposition	Writing a field or ring value as a short vector of bounded signed digits in a power-of-two base.
Module-SIS	The lattice assumption used to bind Akita commitments. Informally, it says that finding a short nonzero relation among the public commitment columns is hard.
Sum-check	An interactive reduction that turns a sum over a Boolean cube into one polynomial evaluation at a random point.
Extension-opening reduction	The bridge from a base-field polynomial opened at an extension-field point to a packed extension-field polynomial with fewer variables.
Ring switching	Akita's lattice step that lifts a relation from a quotient ring to an integer polynomial relation with an explicit quotient. It is distinct from extension-opening reduction.
Setup offloading	Carrying a committed setup prefix into a later fold so that recursive verification does less setup work directly.

Scalar and grouped catalog rows

A trusted external catalog ships one row for each supported request shape.

A scalar row describes a polynomial group opened without precommitted groups. An independent commitment always uses this row, including a commitment that will later be supplied as a precommitted group.
A grouped row describes a final group together with an exact ordered prefix of precommitted group profiles. The complete batch is opened under this row.

Each precommitted descriptor in a grouped row must equal the scalar-row profile under which that commitment was produced. The external-catalog audit test every_grouped_artifact_precommit_has_a_shipped_scalar_producer enforces this invariant.

Notation

Symbol	Meaning
$F$	The base field.
$E$	The extension field used for public challenges and evaluations. It may equal $F$ .
$[E : F]$	The extension degree, or the number of base-field coordinates in one extension-field value.
$q$	The prime modulus of the field in the surrounding chapter.
$R_{q}$	A cyclotomic polynomial ring with coefficients reduced modulo $q$ .
$n$	The number of variables in a multilinear polynomial. Its evaluation table has $2^{n}$ entries.
$r$	A public opening point.
$v$	A claimed polynomial value at the opening point.
$b = 2^{ℓ}$	A power-of-two gadget-decomposition base.
$ℓ$	The base-two logarithm of the decomposition base, called the basis exponent in code.
$δ$	The number of signed digits in a gadget decomposition.
$d_{j}$	The signed digit at position $j$ .
$eq (r, x)$	The multilinear equality polynomial. On Boolean inputs, it is one at $x = r$ and zero elsewhere.
$ρ$	A random point produced by a sum-check transcript.

Some implementation chapters use short names such as A, B, and D for matrix roles. Those letters identify protocol roles, not universal mathematical symbols. Each chapter defines them before use.

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Glossary and notation

Glossary

Scalar and grouped catalog rows

Notation