proof_system::UltraArithmeticRelationImpl< FF_ > Class Template Reference

Public Types
using	FF = FF_

Static Public Member Functions
template<typename ContainerOverSubrelations , typename AllEntities , typename Parameters >
static void	accumulate (ContainerOverSubrelations &evals, const AllEntities &in, const Parameters &, const FF &scaling_factor)
	Expression for the Ultra Arithmetic gate.

Static Public Attributes
static constexpr std::array< size_t, 2 >	SUBRELATION_PARTIAL_LENGTHS

Member Function Documentation

accumulate()

template<typename FF_ >

template<typename ContainerOverSubrelations , typename AllEntities , typename Parameters >

static void proof_system::UltraArithmeticRelationImpl< FF_ >::accumulate ( ContainerOverSubrelations &  evals, const AllEntities &  in, const Parameters &  , const FF &  scaling_factor  )

inlinestatic

Expression for the Ultra Arithmetic gate.

This relation encapsulates several idenitities, toggled by the value of q_arith in [0, 1, 2, 3, ...]. The following description is reproduced from the Plonk analog 'plookup_arithmetic_widget': The whole formula is:

q_arith * ( ( (-1/2) * (q_arith - 3) * q_m * w_1 * w_2 + q_1 * w_1 + q_2 * w_2 + q_3 * w_3 + q_4 * w_4 + q_c ) + (q_arith - 1)*( α * (q_arith - 2) * (w_1 + w_4 - w_1_omega + q_m) + w_4_omega) ) = 0

This formula results in several cases depending on q_arith:

1. q_arith == 0: Arithmetic gate is completely disabled
2. q_arith == 1: Everything in the minigate on the right is disabled. The equation is just a standard plonk equation with extra wires: q_m * w_1 * w_2 + q_1 * w_1 + q_2 * w_2 + q_3 * w_3 + q_4 * w_4 + q_c = 0
3. q_arith == 2: The (w_1 + w_4 - ...) term is disabled. THe equation is: (1/2) * q_m * w_1 * w_2 + q_1 * w_1 + q_2 * w_2 + q_3 * w_3 + q_4 * w_4 + q_c + w_4_omega = 0 It allows defining w_4 at next index (w_4_omega) in terms of current wire values
4. q_arith == 3: The product of w_1 and w_2 is disabled, but a mini addition gate is enabled. α² allows us to split the equation into two:

q_1 * w_1 + q_2 * w_2 + q_3 * w_3 + q_4 * w_4 + q_c + 2 * w_4_omega = 0

w_1 + w_4 - w_1_omega + q_m = 0 (we are reusing q_m here)

1. q_arith > 3: The product of w_1 and w_2 is scaled by (q_arith - 3), while the w_4_omega term is scaled by (q_arith

• 1). The equation can be split into two:

(q_arith - 3)* q_m * w_1 * w_ 2 + q_1 * w_1 + q_2 * w_2 + q_3 * w_3 + q_4 * w_4 + q_c + (q_arith - 1) * w_4_omega = 0

w_1 + w_4 - w_1_omega + q_m = 0

The problem that q_m is used both in both equations can be dealt with by appropriately changing selector values at the next gate. Then we can treat (q_arith - 1) as a simulated q_6 selector and scale q_m to handle (q_arith - 3) at product.

The The relation is defined as C(in(X)...) = q_arith * [ -1/2(q_arith - 3)(q_m * w_r * w_l) + (q_l * w_l) + (q_r * w_r) + (q_o * w_o) + (q_4 * w_4) + q_c + (q_arith - 1)w_4_shift ]

q_arith * (q_arith - 2) * (q_arith - 1) * (w_l + w_4 - w_l_shift + q_m)

Parameters

evals	transformed to evals + C(in(X)...)*scaling_factor
in	an std::array containing the fully extended Univariate edges.
parameters	contains beta, gamma, and public_input_delta, ....
scaling_factor	optional term to scale the evaluation before adding to evals.

Member Data Documentation

SUBRELATION_PARTIAL_LENGTHS

template<typename FF_ >

constexpr std::array<size_t, 2> proof_system::UltraArithmeticRelationImpl< FF_ >::SUBRELATION_PARTIAL_LENGTHS

staticconstexpr

Initial value:

{
        6, 
        5  
    }

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The documentation for this class was generated from the following file:

• src/barretenberg/relations/ultra_arithmetic_relation.hpp