Table 1 Notations definition.

n

Number of all gates in the circuit

H

Multiplicative subgroups on a finite field \(\mathbb{F}\)

\(Poseidon\left( x \right)\)

Poseidon hash encryption

ω

The generating element of H

x

Polynomials over a finite field \(\mathbb{F}\)

\({Z_H}\left( x \right)\)

\({x^n} - 1\)

\({\uptheta},{\uptheta}{\upomega\:}\)

Finite domain \(\mathbb{F}\) randomly generated points

u, v, α, β, γ

Finite domain \(\mathbb{F}\) randomly generated points

\(\sigma\)

Permutation function

\(~{W_\theta }\left( x \right)\), \({W_{\theta \omega }}\left( x \right)\)

Polynomials on \({\uptheta},{\uptheta}{\upomega\:}\)

[X]₁

The operation of X·g1 on generating elements for the g1 group \(\mathbb{G}\)1

[X]₂

The operation of X·g2 on generating elements for the g2 group \(\mathbb{G}\)2

PP

Common parameters

\({\bar {z}_\omega }\)

The value of the \(z\left( x \right)\) function on \({{\uptheta}{\upomega}}\)

\({S_{{\sigma _1}}}\left( x \right),{S_{{\sigma _2}}}\left( x \right),{S_{{\sigma _3}}}\left( x \right)\)

Polynomials after permutation

\(\bar {a},\bar {b},\bar {c},{\bar {S}_{{\sigma _1}}},{\bar {S}_{{\sigma _2}}}\), \(\bar {t}\)

The value of the function on \({\uptheta}\)

\({l_i}\left( x \right)\)

Lagrangian basis functions on multiplicative subgroups

\({q_L}\left( x \right),{q_R}\left( x \right),{q_M}\left( x \right),{q_O}\left( x \right),{q_C}\left( x \right)\)

Gateway circuit polynomial

a(x), b(x), c(x)

Left Input, Right Input, Output

\(t\left( x \right)\)

Total constraint

\(r\left( x \right)\)

Intermediate function

\(PI\left( x \right)\)

Public parameter

\({k_1},\;{k_2},\;{k_3}\)

Mutually unequal quadratic nonresidual

X

Random elements in the finite field \(\mathbb{F}\)

\(srs\)

Structured reference strings