#include <bigfield.hpp>

Classes
struct	Basis

struct	Limb
	Represents a single limb of a bigfield element, with its value and maximum value. More...

Public Types
using	View = bigfield

using	CoefficientAccumulator = bigfield

using	TParams = T

using	native = bb::field< T >

using	field_ct = field_t< Builder >

Public Member Functions
	bigfield (const field_t< Builder > &low_bits, const field_t< Builder > &high_bits, const bool can_overflow=false, const size_t maximum_bitlength=0)
	Constructs a new bigfield object from two field elements representing the low and high bits.

	bigfield (Builder *parent_context=nullptr)
	Constructs a zero bigfield object: all limbs are set to zero.

	bigfield (Builder *parent_context, const uint256_t &value)
	Constructs a new bigfield object from a uint256_t value.

	bigfield (const uint256_t &value)

	bigfield (const int value)
	Constructs a new bigfield object from an int value. We first need to to construct a field element from the value to avoid bugs that have to do with the value being negative.

	bigfield (const unsigned long value)

	bigfield (const unsigned long long value)

	bigfield (const native value)
	Construct a new bigfield object from bb::fq. We first convert to uint256_t as field elements are in Montgomery form internally.

	bigfield (const byte_array< Builder > &bytes)
	Constructs a bigfield element from a 256-bit byte array.

	bigfield (const bigfield &other)

	bigfield (bigfield &&other) noexcept

	~bigfield ()=default

bigfield &	operator= (const bigfield &other)

bigfield &	operator= (bigfield &&other) noexcept

byte_array< Builder >	to_byte_array () const
	Convert the bigfield element to a byte array. Concatenates byte arrays of the high (2L bits) and low (2L bits) parts of the bigfield element.

uint512_t	get_value () const

uint512_t	get_maximum_value () const

bigfield	add_to_lower_limb (const field_t< Builder > &other, const uint256_t &other_maximum_value) const
	Add a field element to the lower limb. CAUTION (the element has to be constrained before using this function)

bigfield	operator+ (const bigfield &other) const
	Adds two bigfield elements. Inputs are reduced to the modulus if necessary. Requires 4 gates if both elements are witnesses.

bigfield	add_two (const bigfield &add_a, const bigfield &add_b) const
	Create constraints for summing three bigfield elements efficiently.

bigfield	operator- (const bigfield &other) const
	Subtraction operator.

bigfield	operator* (const bigfield &other) const
	Evaluate a non-native field multiplication: (a * b = c mod p) where p == target_basis.modulus.

bigfield	operator/ (const bigfield &other) const

bigfield	operator- () const
	Negation operator, works by subtracting `this` from zero.

bigfield	operator+= (const bigfield &other)

bigfield	operator-= (const bigfield &other)

bigfield	operator*= (const bigfield &other)

bigfield	operator/= (const bigfield &other)

bigfield	sqr () const
	Square operator, computes a * a = c mod p.

bigfield	sqradd (const std::vector< bigfield > &to_add) const
	Square and add operator, computes a * a + ...to_add = c mod p.

bigfield	pow (const uint32_t exponent) const
	Raise the bigfield element to the power of (out-of-circuit) exponent.

bigfield	madd (const bigfield &to_mul, const std::vector< bigfield > &to_add) const
	Compute a * b + ...to_add = c mod p.

bigfield	div_without_denominator_check (const bigfield &denominator)

bigfield	conditional_negate (const bool_t< Builder > &predicate) const

bigfield	conditional_select (const bigfield &other, const bool_t< Builder > &predicate) const
	Create an element which is equal to either this or other based on the predicate.

bool_t< Builder >	operator== (const bigfield &other) const
	Validate whether two bigfield elements are equal to each other.

void	assert_is_in_field () const

void	assert_less_than (const uint256_t &upper_limit) const

void	assert_equal (const bigfield &other) const

void	assert_is_not_equal (const bigfield &other) const

void	self_reduce () const

bool	is_constant () const
	Check if the bigfield is constant, i.e. its prime limb is constant.

bigfield	invert () const
	Inverting function with the assumption that the bigfield element we are calling invert on is not zero.

void	convert_constant_to_fixed_witness (Builder *builder)

void	fix_witness ()

Builder *	get_context () const

void	set_origin_tag (const bb::OriginTag &tag) const

bb::OriginTag	get_origin_tag () const

void	set_free_witness_tag ()
	Set the free witness flag for the bigfield.

void	unset_free_witness_tag ()
	Unset the free witness flag for the bigfield.

uint32_t	set_public () const
	Set the witness indices of the binary basis limbs to public.

Static Public Member Functions
static bigfield	unsafe_construct_from_limbs (const field_t< Builder > &a, const field_t< Builder > &b, const field_t< Builder > &c, const field_t< Builder > &d, const bool can_overflow=false)
	Construct a bigfield element from binary limbs that are already reduced.

static bigfield	construct_from_limbs (const field_t< Builder > &a, const field_t< Builder > &b, const field_t< Builder > &c, const field_t< Builder > &d, const bool can_overflow=false)
	Construct a bigfield element from binary limbs that are already reduced and ensure they are range constrained.

static bigfield	unsafe_construct_from_limbs (const field_t< Builder > &a, const field_t< Builder > &b, const field_t< Builder > &c, const field_t< Builder > &d, const field_t< Builder > &prime_limb, const bool can_overflow=false)
	Construct a bigfield element from binary limbs and a prime basis limb that are already reduced.

static bigfield	create_from_u512_as_witness (Builder *ctx, const uint512_t &value, const bool can_overflow=false, const size_t maximum_bitlength=0)
	Creates a bigfield element from a uint512_t. Bigfield element is constructed as a witness and not a circuit constant.

static bigfield	from_witness (Builder *ctx, const bb::field< T > &input)

static void	perform_reductions_for_mult_madd (std::vector< bigfield > &mul_left, std::vector< bigfield > &mul_right, const std::vector< bigfield > &to_add)
	Performs individual reductions on the supplied elements as well as more complex reductions to prevent CRT modulus overflow and to fit the quotient inside the range proof.

static bigfield	mult_madd (const std::vector< bigfield > &mul_left, const std::vector< bigfield > &mul_right, const std::vector< bigfield > &to_add, bool fix_remainder_to_zero=false)

static bigfield	dual_madd (const bigfield &left_a, const bigfield &right_a, const bigfield &left_b, const bigfield &right_b, const std::vector< bigfield > &to_add)

static bigfield	msub_div (const std::vector< bigfield > &mul_left, const std::vector< bigfield > &mul_right, const bigfield &divisor, const std::vector< bigfield > &to_sub, bool enable_divisor_nz_check=false)

static bigfield	sum (const std::vector< bigfield > &terms)
	Create constraints for summing these terms.

static bigfield	internal_div (const std::vector< bigfield > &numerators, const bigfield &denominator, bool check_for_zero)

static bigfield	div_without_denominator_check (const std::vector< bigfield > &numerators, const bigfield &denominator)

static bigfield	div_check_denominator_nonzero (const std::vector< bigfield > &numerators, const bigfield &denominator)

static bigfield	conditional_assign (const bool_t< Builder > &predicate, const bigfield &lhs, const bigfield &rhs)

static bigfield	one ()

static bigfield	zero ()

static constexpr bigfield	unreduced_zero ()
	Create an unreduced 0 ~ pk, where pk is the minimal multiple of modulus that should be reduced.

static bigfield	reconstruct_from_public (const std::span< const field_ct, PUBLIC_INPUTS_SIZE > &limbs)
	Reconstruct a bigfield from limbs (generally stored in the public inputs)

static constexpr uint512_t	get_maximum_unreduced_value ()

static constexpr uint512_t	get_prohibited_value ()

static constexpr uint1024_t	get_maximum_crt_product ()
	Compute the maximum product of two bigfield elements in CRT: M = 2^t * n.

static size_t	get_quotient_max_bits (const std::vector< uint1024_t > &remainders_max)
	Compute the maximum number of bits for quotient range proof to protect against CRT underflow.

static bool	mul_product_overflows_crt_modulus (const uint1024_t &a_max, const uint1024_t &b_max, const std::vector< bigfield > &to_add)

static bool	mul_product_overflows_crt_modulus (const std::vector< uint512_t > &as_max, const std::vector< uint512_t > &bs_max, const std::vector< bigfield > &to_add)

static constexpr uint256_t	get_maximum_unreduced_limb_value ()

static constexpr uint256_t	get_prohibited_limb_value ()

Public Attributes
Builder *	context

std::array< Limb, NUM_LIMBS >	binary_basis_limbs
	Represents a bigfield element in the binary basis. A bigfield element is represented as a combination of 4 binary basis limbs: a = a[0] + a[1] * 2^L + a[2] * 2^2L + a[3] * 2^3L.

field_t< Builder >	prime_basis_limb
	Represents a bigfield element in the prime basis: (a mod n) where n is the native modulus.

Static Public Attributes
static constexpr size_t	PUBLIC_INPUTS_SIZE = BIGFIELD_PUBLIC_INPUTS_SIZE

static constexpr size_t	NUM_LIMBS = 4

static constexpr uint256_t	modulus = (uint256_t(T::modulus_0, T::modulus_1, T::modulus_2, T::modulus_3))

static constexpr uint512_t	modulus_u512 = static_cast<uint512_t>(modulus)

static constexpr uint64_t	NUM_LIMB_BITS = NUM_LIMB_BITS_IN_FIELD_SIMULATION

static constexpr uint64_t	NUM_LAST_LIMB_BITS = modulus_u512.get_msb() + 1 - (NUM_LIMB_BITS * 3)

static const uint1024_t	DEFAULT_MAXIMUM_REMAINDER

static constexpr uint256_t	DEFAULT_MAXIMUM_LIMB = (uint256_t(1) << NUM_LIMB_BITS) - uint256_t(1)

static constexpr uint256_t	DEFAULT_MAXIMUM_MOST_SIGNIFICANT_LIMB

static constexpr uint64_t	LOG2_BINARY_MODULUS = NUM_LIMB_BITS * NUM_LIMBS

static constexpr bool	is_composite = true

static constexpr size_t	MAXIMUM_SUMMAND_COUNT_LOG = 4

static constexpr size_t	MAXIMUM_SUMMAND_COUNT = 1 << MAXIMUM_SUMMAND_COUNT_LOG

static constexpr uint256_t	prime_basis_maximum_limb

static constexpr Basis	prime_basis { uint512_t(bb::fr::modulus), bb::fr::modulus.get_msb() + 1 }

static constexpr Basis	binary_basis { uint512_t(1) << LOG2_BINARY_MODULUS, LOG2_BINARY_MODULUS }

static constexpr Basis	target_basis { modulus_u512, static_cast<size_t>(modulus_u512.get_msb() + 1) }

static constexpr bb::fr	shift_1 = bb::fr(uint256_t(1) << NUM_LIMB_BITS)

static constexpr bb::fr	shift_2 = bb::fr(uint256_t(1) << (NUM_LIMB_BITS * 2))

static constexpr bb::fr	shift_3 = bb::fr(uint256_t(1) << (NUM_LIMB_BITS * 3))

static constexpr bb::fr	shift_right_1 = bb::fr(1) / shift_1

static constexpr bb::fr	shift_right_2 = bb::fr(1) / shift_2

static constexpr bb::fr	negative_prime_modulus_mod_binary_basis = -bb::fr(uint256_t(modulus_u512))

static constexpr uint512_t	negative_prime_modulus = binary_basis.modulus - target_basis.modulus

static constexpr std::array< uint256_t, NUM_LIMBS >	neg_modulus_limbs_u256

static constexpr std::array< bb::fr, NUM_LIMBS >	neg_modulus_limbs

static constexpr uint64_t	MAX_ADDITION_LOG = 10

static constexpr uint64_t	MAXIMUM_LIMB_SIZE_THAT_WOULDNT_OVERFLOW

static constexpr uint64_t	MAX_UNREDUCED_LIMB_BITS = NUM_LIMB_BITS + 10

static constexpr uint64_t	PROHIBITED_LIMB_BITS = MAX_UNREDUCED_LIMB_BITS + 5

Private Member Functions
std::array< uint32_t, NUM_LIMBS >	get_binary_basis_limb_witness_indices () const
	Get the witness indices of the (normalized) binary basis limbs.

void	reduction_check () const
	Check if the bigfield element needs to be reduced.

void	sanity_check () const
	Perform a sanity check on a value that is about to interact with another value.

std::array< uint256_t, NUM_LIMBS >	get_binary_basis_limb_maximums ()
	Get the maximum values of the binary basis limbs.

Static Private Member Functions
static std::pair< uint512_t, uint512_t >	compute_quotient_remainder_values (const bigfield &a, const bigfield &b, const std::vector< bigfield > &to_add)
	Compute the quotient and remainder values for dividing (a * b + (to_add[0] + ... + to_add[-1])) with p.

static uint512_t	compute_maximum_quotient_value (const std::vector< uint512_t > &as, const std::vector< uint512_t > &bs, const std::vector< uint512_t > &to_add)
	Compute the maximum possible value of quotient of a*b+\sum(to_add)

static std::pair< bool, size_t >	get_quotient_reduction_info (const std::vector< uint512_t > &as_max, const std::vector< uint512_t > &bs_max, const std::vector< bigfield > &to_add, const std::vector< uint1024_t > &remainders_max={ DEFAULT_MAXIMUM_REMAINDER })
	Check for 2 conditions (CRT modulus is overflown or the maximum quotient doesn't fit into range proof). Also returns the length of quotient's range proof if there is no need to reduce.

static void	unsafe_evaluate_multiply_add (const bigfield &input_left, const bigfield &input_to_mul, const std::vector< bigfield > &to_add, const bigfield &input_quotient, const std::vector< bigfield > &input_remainders)
	Evaluate a multiply add identity with several added elements and several remainders.

static void	unsafe_evaluate_multiple_multiply_add (const std::vector< bigfield > &input_left, const std::vector< bigfield > &input_right, const std::vector< bigfield > &to_add, const bigfield &input_quotient, const std::vector< bigfield > &input_remainders)
	Evaluate a relation involving multiple multiplications and additions.

static void	unsafe_evaluate_square_add (const bigfield &left, const std::vector< bigfield > &to_add, const bigfield &quotient, const bigfield &remainder)
	Evaluate a square with several additions.

static std::pair< uint512_t, uint512_t >	compute_partial_schoolbook_multiplication (const std::array< uint256_t, NUM_LIMBS > &a_limbs, const std::array< uint256_t, NUM_LIMBS > &b_limbs)
	Compute the partial multiplication of two uint256_t arrays using schoolbook multiplication.

Detailed Description

template<typename Builder, typename T>
class bb::stdlib::bigfield< Builder, T >

Definition at line 21 of file bigfield.hpp.

Member Typedef Documentation

◆ CoefficientAccumulator

template<typename Builder , typename T >

using bb::stdlib::bigfield< Builder, T >::CoefficientAccumulator = bigfield

Definition at line 25 of file bigfield.hpp.

◆ field_ct

template<typename Builder , typename T >

using bb::stdlib::bigfield< Builder, T >::field_ct = field_t<Builder>

Definition at line 28 of file bigfield.hpp.

◆ native

template<typename Builder , typename T >

using bb::stdlib::bigfield< Builder, T >::native = bb::field<T>

Definition at line 27 of file bigfield.hpp.

◆ TParams

template<typename Builder , typename T >

using bb::stdlib::bigfield< Builder, T >::TParams = T

Definition at line 26 of file bigfield.hpp.

◆ View

template<typename Builder , typename T >

using bb::stdlib::bigfield< Builder, T >::View = bigfield

Definition at line 24 of file bigfield.hpp.

Constructor & Destructor Documentation

◆ bigfield() [1/11]

template<typename Builder , typename T >

bb::stdlib::bigfield< Builder, T >::bigfield	(	const field_t< Builder > &	low_bits,
		const field_t< Builder > &	high_bits,
		const bool	can_overflow = `false`,
		const size_t	maximum_bitlength = `0`
	)

Constructs a new bigfield object from two field elements representing the low and high bits.

Parameters

low_bits	The field element representing the low 2L bits of the bigfield.
high_bits	The field element representing the high 2L bits of the bigfield.
can_overflow	Whether the bigfield can overflow the modulus.
maximum_bitlength	The maximum bitlength of the bigfield. If 0, it defaults to \|p\| (target modulus bitlength).

Definition at line 44 of file bigfield_impl.hpp.

◆ bigfield() [2/11]

template<typename Builder , typename T >

bb::stdlib::bigfield< Builder, T >::bigfield ( Builder * parent_context = nullptr )

Constructs a zero bigfield object: all limbs are set to zero.

This constructor is used to create a bigfield element without any context. It is useful for creating bigfield elements that will be later assigned to a context.

Definition at line 25 of file bigfield_impl.hpp.

◆ bigfield() [3/11]

template<typename Builder , typename T >

bb::stdlib::bigfield< Builder, T >::bigfield	(	Builder *	parent_context,
		const uint256_t &	value
	)

Constructs a new bigfield object from a uint256_t value.

Parameters

parent_context	The circuit context in which the bigfield element will be created.
value	The uint256_t value to be converted to a bigfield element.

Definition at line 32 of file bigfield_impl.hpp.

◆ bigfield() [4/11]

template<typename Builder , typename T >

bb::stdlib::bigfield< Builder, T >::bigfield ( const uint256_t & value )

inlineexplicit

Definition at line 117 of file bigfield.hpp.

◆ bigfield() [5/11]

template<typename Builder , typename T >

bb::stdlib::bigfield< Builder, T >::bigfield ( const int value )

inline

Constructs a new bigfield object from an int value. We first need to to construct a field element from the value to avoid bugs that have to do with the value being negative.

Definition at line 126 of file bigfield.hpp.

◆ bigfield() [6/11]

template<typename Builder , typename T >

bb::stdlib::bigfield< Builder, T >::bigfield ( const unsigned long value )

inline

Definition at line 131 of file bigfield.hpp.

◆ bigfield() [7/11]

template<typename Builder , typename T >

bb::stdlib::bigfield< Builder, T >::bigfield ( const unsigned long long value )

inline

Definition at line 136 of file bigfield.hpp.

◆ bigfield() [8/11]

template<typename Builder , typename T >

bb::stdlib::bigfield< Builder, T >::bigfield ( const native value )

inline

Construct a new bigfield object from bb::fq. We first convert to uint256_t as field elements are in Montgomery form internally.

Parameters

value

Definition at line 146 of file bigfield.hpp.

◆ bigfield() [9/11]

template<typename Builder , typename T >

bb::stdlib::bigfield< Builder, T >::bigfield ( const byte_array< Builder > & bytes )

Constructs a bigfield element from a 256-bit byte array.

This constructor interprets the input byte array as a 256-bit little-endian integer, and decomposes it into 4 limbs as follows:

Limb 0: 68 bits — bytes b24 to b31 and the high 4 bits of b23
Limb 1: 68 bits — bytes b14 to b22, and the low 4 bits of b23
Limb 2: 68 bits — bytes b7 to b14 and the high 4 bits of b6
Limb 3: 52 bits — bytes b0 to b5 and the low 4 bits of b6

Parameters

bytes The 32-byte input representing the 256-bit integer (little-endian).

Definition at line 210 of file bigfield_impl.hpp.

◆ bigfield() [10/11]

template<typename Builder , typename T >

bb::stdlib::bigfield< Builder, T >::bigfield ( const bigfield< Builder, T > & other )

Definition at line 117 of file bigfield_impl.hpp.

◆ bigfield() [11/11]

template<typename Builder , typename T >

bb::stdlib::bigfield< Builder, T >::bigfield ( bigfield< Builder, T > && other )

noexcept

Definition at line 127 of file bigfield_impl.hpp.

◆ ~bigfield()

template<typename Builder , typename T >

bb::stdlib::bigfield< Builder, T >::~bigfield ( )

default

Member Function Documentation

◆ add_to_lower_limb()

template<typename Builder , typename T >

bigfield< Builder, T > bb::stdlib::bigfield< Builder, T >::add_to_lower_limb	(	const field_t< Builder > &	other,
		const uint256_t &	other_maximum_value
	)		const

Add a field element to the lower limb. CAUTION (the element has to be constrained before using this function)

Sometimes we need to add a small constrained value to a bigfield element (for example, a boolean value), but we don't want to construct a full bigfield element for that as it would take too many gates. If the maximum value of the field element being added is small enough, we can simply add it to the lowest limb and increase its maximum value. That will create 2 additional constraints instead of 5/3 needed to add 2 bigfield elements and several needed to construct a bigfield element.

Template Parameters

Builder	Builder
T	Field Parameters

Parameters

other	Field element that will be added to the lower
other_maximum_value	The maximum value of other

Returns: bigfield<Builder, T> Result

Definition at line 315 of file bigfield_impl.hpp.

◆ add_two()

template<typename Builder , typename T >

bigfield< Builder, T > bb::stdlib::bigfield< Builder, T >::add_two	(	const bigfield< Builder, T > &	add_a,
		const bigfield< Builder, T > &	add_b
	)		const

Create constraints for summing three bigfield elements efficiently.

Template Parameters

Builder
T

Parameters

add_a
add_b

Returns: The sum of three terms

Uses five gates (one for each limb) to add three bigfield elements.

Definition at line 459 of file bigfield_impl.hpp.

◆ assert_equal()

template<typename Builder , typename T >

void bb::stdlib::bigfield< Builder, T >::assert_equal ( const bigfield< Builder, T > & other ) const

Definition at line 1888 of file bigfield_impl.hpp.

◆ assert_is_in_field()

template<typename Builder , typename T >

void bb::stdlib::bigfield< Builder, T >::assert_is_in_field ( ) const

Definition at line 1819 of file bigfield_impl.hpp.

◆ assert_is_not_equal()

template<typename Builder , typename T >

void bb::stdlib::bigfield< Builder, T >::assert_is_not_equal ( const bigfield< Builder, T > & other ) const

Definition at line 1948 of file bigfield_impl.hpp.

◆ assert_less_than()

template<typename Builder , typename T >

void bb::stdlib::bigfield< Builder, T >::assert_less_than ( const uint256_t & upper_limit ) const

Definition at line 1824 of file bigfield_impl.hpp.

◆ compute_maximum_quotient_value()

template<typename Builder , typename T >

uint512_t bb::stdlib::bigfield< Builder, T >::compute_maximum_quotient_value	(	const std::vector< uint512_t > &	as,
		const std::vector< uint512_t > &	bs,
		const std::vector< uint512_t > &	to_add
	)

staticprivate

Compute the maximum possible value of quotient of a*b+\sum(to_add)

Parameters

as	Multiplicands
bs	Multipliers
to_add	Added elements

Returns: uint512_t The maximum value of quotient

Definition at line 2590 of file bigfield_impl.hpp.

◆ compute_partial_schoolbook_multiplication()

template<typename Builder , typename T >

std::pair< uint512_t, uint512_t > bb::stdlib::bigfield< Builder, T >::compute_partial_schoolbook_multiplication	(	const std::array< uint256_t, NUM_LIMBS > &	a_limbs,
		const std::array< uint256_t, NUM_LIMBS > &	b_limbs
	)

staticprivate

Compute the partial multiplication of two uint256_t arrays using schoolbook multiplication.

Parameters

a_limbs
b_limbs

Returns: std::pair<uint512_t, uint512_t>

Regular schoolbook multiplication of two arrays each with L = 4 limbs will produce a result of size 2 * L - 1 = 7. In this context, we can ignore the last three limbs as those terms have multiplicands: (2^4L, 2^5L, 2^6L) and since we are working modulo 2^t = 2^4L, those terms will always be zero. This is why we call this helper function "partial schoolbook multiplication".

Definition at line 2644 of file bigfield_impl.hpp.

◆ compute_quotient_remainder_values()

template<typename Builder , typename T >

std::pair< uint512_t, uint512_t > bb::stdlib::bigfield< Builder, T >::compute_quotient_remainder_values	(	const bigfield< Builder, T > &	a,
		const bigfield< Builder, T > &	b,
		const std::vector< bigfield< Builder, T > > &	to_add
	)

staticprivate

Compute the quotient and remainder values for dividing (a * b + (to_add[0] + ... + to_add[-1])) with p.

Parameters

a	Left multiplicand
b	Right multiplier
to_add	Vector of elements to add

Returns: std::pair<uint512_t, uint512_t> Quotient and remainder values

Definition at line 2568 of file bigfield_impl.hpp.

◆ conditional_assign()

template<typename Builder , typename T >

static bigfield bb::stdlib::bigfield< Builder, T >::conditional_assign	(	const bool_t< Builder > &	predicate,
		const bigfield< Builder, T > &	lhs,
		const bigfield< Builder, T > &	rhs
	)

inlinestatic

Definition at line 594 of file bigfield.hpp.

◆ conditional_negate()

template<typename Builder , typename T >

bigfield< Builder, T > bb::stdlib::bigfield< Builder, T >::conditional_negate ( const bool_t< Builder > & predicate ) const

Definition at line 1594 of file bigfield_impl.hpp.

◆ conditional_select()

template<typename Builder , typename T >

bigfield< Builder, T > bb::stdlib::bigfield< Builder, T >::conditional_select	(	const bigfield< Builder, T > &	other,
		const bool_t< Builder > &	predicate
	)		const

Create an element which is equal to either this or other based on the predicate.

Template Parameters

Builder
T

Parameters

other	The other bigfield element
predicate	Predicate controlling the result (0 for this, 1 for the other)

Returns: Resulting element

Definition at line 1624 of file bigfield_impl.hpp.

◆ construct_from_limbs()

template<typename Builder , typename T >

static bigfield bb::stdlib::bigfield< Builder, T >::construct_from_limbs	(	const field_t< Builder > &	a,
		const field_t< Builder > &	b,
		const field_t< Builder > &	c,
		const field_t< Builder > &	d,
		const bool	can_overflow = `false`
	)

inlinestatic

Construct a bigfield element from binary limbs that are already reduced and ensure they are range constrained.

Definition at line 185 of file bigfield.hpp.

◆ convert_constant_to_fixed_witness()

template<typename Builder , typename T >

void bb::stdlib::bigfield< Builder, T >::convert_constant_to_fixed_witness ( Builder * builder )

inline

Create a witness form a constant. This way the value of the witness is fixed and public.

Definition at line 677 of file bigfield.hpp.

◆ create_from_u512_as_witness()

template<typename Builder , typename T >

bigfield< Builder, T > bb::stdlib::bigfield< Builder, T >::create_from_u512_as_witness	(	Builder *	ctx,
		const uint512_t &	value,
		const bool	can_overflow = `false`,
		const size_t	maximum_bitlength = `0`
	)

static

Creates a bigfield element from a uint512_t. Bigfield element is constructed as a witness and not a circuit constant.

Parameters

ctx
value
can_overflow	Can the input value have more than log2(modulus) bits?
maximum_bitlength	Provide the explicit maximum bitlength if known. Otherwise bigfield max value will be either log2(modulus) bits iff can_overflow = false, or (4 * NUM_LIMB_BITS) iff can_overflow = true

Returns: bigfield<Builder, T>

This method is 1 gate more efficient than constructing from 2 field_ct elements.

Definition at line 137 of file bigfield_impl.hpp.

◆ div_check_denominator_nonzero()

template<typename Builder , typename T >

bigfield< Builder, T > bb::stdlib::bigfield< Builder, T >::div_check_denominator_nonzero	(	const std::vector< bigfield< Builder, T > > &	numerators,
		const bigfield< Builder, T > &	denominator
	)

static

Div method with constraints for denominator!=0.

Similar to operator/ but numerator can be linear sum of multiple elements

Definition at line 897 of file bigfield_impl.hpp.

◆ div_without_denominator_check() [1/2]

template<typename Builder , typename T >

bigfield< Builder, T > bb::stdlib::bigfield< Builder, T >::div_without_denominator_check ( const bigfield< Builder, T > & denominator )

Definition at line 886 of file bigfield_impl.hpp.

◆ div_without_denominator_check() [2/2]

template<typename Builder , typename T >

bigfield< Builder, T > bb::stdlib::bigfield< Builder, T >::div_without_denominator_check	(	const std::vector< bigfield< Builder, T > > &	numerators,
		const bigfield< Builder, T > &	denominator
	)

static

Div method without constraining denominator!=0.

Similar to operator/ but numerator can be linear sum of multiple elements

Definition at line 879 of file bigfield_impl.hpp.

◆ dual_madd()

template<typename Builder , typename T >

bigfield< Builder, T > bb::stdlib::bigfield< Builder, T >::dual_madd	(	const bigfield< Builder, T > &	left_a,
		const bigfield< Builder, T > &	right_a,
		const bigfield< Builder, T > &	left_b,
		const bigfield< Builder, T > &	right_b,
		const std::vector< bigfield< Builder, T > > &	to_add
	)

static

Compute (left_a * right_a) + (left_b * right_b) + ...to_add = c mod p

This is cheaper than two multiplication operations, as the above only requires one quotient/remainder

Definition at line 1453 of file bigfield_impl.hpp.

◆ fix_witness()

template<typename Builder , typename T >

void bb::stdlib::bigfield< Builder, T >::fix_witness ( )

inline

Fix a witness. The value of the witness is constrained with a selector

Definition at line 689 of file bigfield.hpp.

◆ from_witness()

template<typename Builder , typename T >

static bigfield bb::stdlib::bigfield< Builder, T >::from_witness	(	Builder *	ctx,
		const bb::field< T > &	input
	)

inlinestatic

Definition at line 294 of file bigfield.hpp.

◆ get_binary_basis_limb_maximums()

template<typename Builder , typename T >

std::array< uint256_t, NUM_LIMBS > bb::stdlib::bigfield< Builder, T >::get_binary_basis_limb_maximums ( )

inlineprivate

Get the maximum values of the binary basis limbs.

Returns: std::array<field_t<Builder>, NUM_LIMBS> An array containing the maximum values of the binary basis limbs.

Definition at line 1100 of file bigfield.hpp.

◆ get_binary_basis_limb_witness_indices()

template<typename Builder , typename T >

std::array< uint32_t, NUM_LIMBS > bb::stdlib::bigfield< Builder, T >::get_binary_basis_limb_witness_indices ( ) const

inlineprivate

Get the witness indices of the (normalized) binary basis limbs.

Returns: Witness indices of the binary basis limbs

Definition at line 948 of file bigfield.hpp.

◆ get_context()

template<typename Builder , typename T >

Builder * bb::stdlib::bigfield< Builder, T >::get_context ( ) const

inline

Definition at line 701 of file bigfield.hpp.

◆ get_maximum_crt_product()

template<typename Builder , typename T >

static constexpr uint1024_t bb::stdlib::bigfield< Builder, T >::get_maximum_crt_product ( )

inlinestaticconstexpr

Compute the maximum product of two bigfield elements in CRT: M = 2^t * n.

When we multiply two bigfield elements a and b, we need to check that: a * b = q * p + r, where q is the quotient, r is the remainder, and p is the size of the non-native field. With the CRT, we should have both sizes less than the maximum product M = 2^t * n.

Returns: uint1024_t Maximum product of two bigfield elements in CRT form

Definition at line 811 of file bigfield.hpp.

◆ get_maximum_unreduced_limb_value()

template<typename Builder , typename T >

static constexpr uint256_t bb::stdlib::bigfield< Builder, T >::get_maximum_unreduced_limb_value ( )

inlinestaticconstexpr

Definition at line 935 of file bigfield.hpp.

◆ get_maximum_unreduced_value()

template<typename Builder , typename T >

static constexpr uint512_t bb::stdlib::bigfield< Builder, T >::get_maximum_unreduced_value ( )

inlinestaticconstexpr

Definition at line 764 of file bigfield.hpp.

◆ get_maximum_value()

template<typename Builder , typename T >

uint512_t bb::stdlib::bigfield< Builder, T >::get_maximum_value ( ) const

Definition at line 305 of file bigfield_impl.hpp.

◆ get_origin_tag()

template<typename Builder , typename T >

bb::OriginTag bb::stdlib::bigfield< Builder, T >::get_origin_tag ( ) const

inline

Definition at line 711 of file bigfield.hpp.

◆ get_prohibited_limb_value()

template<typename Builder , typename T >

static constexpr uint256_t bb::stdlib::bigfield< Builder, T >::get_prohibited_limb_value ( )

inlinestaticconstexpr

Definition at line 938 of file bigfield.hpp.

◆ get_prohibited_value()

template<typename Builder , typename T >

static constexpr uint512_t bb::stdlib::bigfield< Builder, T >::get_prohibited_value ( )

inlinestaticconstexpr

Definition at line 794 of file bigfield.hpp.

◆ get_quotient_max_bits()

template<typename Builder , typename T >

static size_t bb::stdlib::bigfield< Builder, T >::get_quotient_max_bits ( const std::vector< uint1024_t > & remainders_max )

inlinestatic

Compute the maximum number of bits for quotient range proof to protect against CRT underflow.

When multiplying a and b, we need to check that: a * b = q * p + r, and each side of the equation is less than the maximum product M = 2^t * n. The quotient q is a size of the non-native field, and we need to ensure it fits within the available bits. q * p + r < M ==> q < (M - r_max) / p

Parameters

remainders_max Maximum sizes of resulting remainders

Returns: Desired length of range proof

Definition at line 827 of file bigfield.hpp.

◆ get_quotient_reduction_info()

template<typename Builder , typename T >

std::pair< bool, size_t > bb::stdlib::bigfield< Builder, T >::get_quotient_reduction_info	(	const std::vector< uint512_t > &	as_max,
		const std::vector< uint512_t > &	bs_max,
		const std::vector< bigfield< Builder, T > > &	to_add,
		const std::vector< uint1024_t > &	remainders_max = `{ DEFAULT_MAXIMUM_REMAINDER }`
	)

staticprivate

Check for 2 conditions (CRT modulus is overflown or the maximum quotient doesn't fit into range proof). Also returns the length of quotient's range proof if there is no need to reduce.

Parameters

as_max	Vector of left multiplicands' maximum values
bs_max	Vector of right multiplicands' maximum values
to_add	Vector of added bigfield values

Returns: <true, 0> if we need to reduce the product; <false, The length of quotient range proof> if there is no need to reduce the product.

Definition at line 2613 of file bigfield_impl.hpp.

◆ get_value()

template<typename Builder , typename T >

uint512_t bb::stdlib::bigfield< Builder, T >::get_value ( ) const

Definition at line 296 of file bigfield_impl.hpp.

◆ internal_div()

template<typename Builder , typename T >

bigfield< Builder, T > bb::stdlib::bigfield< Builder, T >::internal_div	(	const std::vector< bigfield< Builder, T > > &	numerators,
		const bigfield< Builder, T > &	denominator,
		bool	check_for_zero
	)

static

Division of a sum with an optional check if divisor is zero. Should not be used outside of class.

Parameters

numerators	Vector of numerators
denominator	Denominator
check_for_zero	If the zero check should be enabled

Returns: The result of division

Definition at line 794 of file bigfield_impl.hpp.

◆ invert()

template<typename Builder , typename T >

bigfield bb::stdlib::bigfield< Builder, T >::invert ( ) const

inline

Inverting function with the assumption that the bigfield element we are calling invert on is not zero.

Returns: bigfield

Definition at line 634 of file bigfield.hpp.

◆ is_constant()

template<typename Builder , typename T >

bool bb::stdlib::bigfield< Builder, T >::is_constant ( ) const

inline

Check if the bigfield is constant, i.e. its prime limb is constant.

Returns: true if the bigfield is constant, false otherwise.

We use assertions to ensure that all limbs are consistent in their constant status.

Definition at line 615 of file bigfield.hpp.

◆ madd()

template<typename Builder , typename T >

bigfield< Builder, T > bb::stdlib::bigfield< Builder, T >::madd	(	const bigfield< Builder, T > &	to_mul,
		const std::vector< bigfield< Builder, T > > &	to_add
	)		const

Compute a * b + ...to_add = c mod p.

Parameters

to_mul	Bigfield element to multiply by
to_add	Vector of elements to add

Returns: New bigfield element c

Definition at line 1058 of file bigfield_impl.hpp.

◆ msub_div()

template<typename Builder , typename T >

bigfield< Builder, T > bb::stdlib::bigfield< Builder, T >::msub_div	(	const std::vector< bigfield< Builder, T > > &	mul_left,
		const std::vector< bigfield< Builder, T > > &	mul_right,
		const bigfield< Builder, T > &	divisor,
		const std::vector< bigfield< Builder, T > > &	to_sub,
		bool	enable_divisor_nz_check = `false`
	)

static

multiply, subtract, divide. This method computes:

result = -(\sum{mul_left[i] * mul_right[i]} + ...to_add) / divisor

Algorithm is constructed in this way to ensure that all computed terms are positive

i.e. we evaluate: result * divisor + (\sum{mul_left[i] * mul_right[i]) + ...to_add) = 0

It is critical that ALL the terms on the LHS are positive to eliminate the possiblity of underflows when calling evaluate_multiple_multiply_add

only requires one quotient and remainder + overflow limbs

We proxy this to mult_madd, so it only requires one quotient and remainder + overflow limbs

Definition at line 1490 of file bigfield_impl.hpp.

◆ mul_product_overflows_crt_modulus() [1/2]

template<typename Builder , typename T >

static bool bb::stdlib::bigfield< Builder, T >::mul_product_overflows_crt_modulus	(	const std::vector< uint512_t > &	as_max,
		const std::vector< uint512_t > &	bs_max,
		const std::vector< bigfield< Builder, T > > &	to_add
	)

inlinestatic

Check that the maximum value of a sum of bigfield productc with added values overflows ctf modulus.

Parameters

as_max	Vector of multiplicands' maximum values
b_max	Vector of multipliers' maximum values
to_add	Vector of field elements to be added

Returns: true if there is an overflow, false otherwise

Definition at line 873 of file bigfield.hpp.

◆ mul_product_overflows_crt_modulus() [2/2]

template<typename Builder , typename T >

static bool bb::stdlib::bigfield< Builder, T >::mul_product_overflows_crt_modulus	(	const uint1024_t &	a_max,
		const uint1024_t &	b_max,
		const std::vector< bigfield< Builder, T > > &	to_add
	)

inlinestatic

Check that the maximum value of a bigfield product with added values overflows crt modulus.

Parameters

a_max	multiplicand maximum value
b_max	multiplier maximum value
to_add	vector of field elements to be added

Returns: true if there is an overflow, false otherwise

Definition at line 847 of file bigfield.hpp.

◆ mult_madd()

template<typename Builder , typename T >

bigfield< Builder, T > bb::stdlib::bigfield< Builder, T >::mult_madd	(	const std::vector< bigfield< Builder, T > > &	mul_left,
		const std::vector< bigfield< Builder, T > > &	mul_right,
		const std::vector< bigfield< Builder, T > > &	to_add,
		bool	fix_remainder_to_zero = `false`
	)

static

Evaluate the sum of products and additional values safely.

Parameters

mul_left	Vector of bigfield multiplicands
mul_right	Vector of bigfield multipliers
to_add	Vector of bigfield elements to add to the sum of products

Returns: A reduced value that is the sum of all products and to_add values

Definition at line 1264 of file bigfield_impl.hpp.

◆ one()

template<typename Builder , typename T >

static bigfield bb::stdlib::bigfield< Builder, T >::one ( )

inlinestatic

Create a public one constant

Definition at line 639 of file bigfield.hpp.

◆ operator*()

template<typename Builder , typename T >

bigfield< Builder, T > bb::stdlib::bigfield< Builder, T >::operator* ( const bigfield< Builder, T > & other ) const

Evaluate a non-native field multiplication: (a * b = c mod p) where p == target_basis.modulus.

Parameters

other

Returns: bigfield

We compute quotient term q and remainder c and evaluate that: a * b - q * p - c = 0 mod modulus_u512 (binary basis modulus, currently 2**272) a * b - q * p - c = 0 mod circuit modulus We also check that: a * b < M and q * p - c < M, where M = (2^t * n) is CRT modulus.

Definition at line 708 of file bigfield_impl.hpp.

◆ operator*=()

template<typename Builder , typename T >

bigfield bb::stdlib::bigfield< Builder, T >::operator*= ( const bigfield< Builder, T > & other )

inline

Definition at line 495 of file bigfield.hpp.

◆ operator+()

template<typename Builder , typename T >

bigfield< Builder, T > bb::stdlib::bigfield< Builder, T >::operator+ ( const bigfield< Builder, T > & other ) const

Adds two bigfield elements. Inputs are reduced to the modulus if necessary. Requires 4 gates if both elements are witnesses.

Naive addition of two bigfield elements would require 5 gates: 4 gates to add the binary basis limbs and 1 gate to add the prime basis limbs. However, if both elements are witnesses, we can use an optimized addition trick that uses 4 gates instead of 5. In this case, we add the prime basis limbs and one of the binary basis limbs in a single gate.

Template Parameters

Builder
T

Parameters

other

Returns: bigfield<Builder, T>

Definition at line 355 of file bigfield_impl.hpp.

◆ operator+=()

template<typename Builder , typename T >

bigfield bb::stdlib::bigfield< Builder, T >::operator+= ( const bigfield< Builder, T > & other )

inline

Definition at line 485 of file bigfield.hpp.

◆ operator-() [1/2]

template<typename Builder , typename T >

bigfield bb::stdlib::bigfield< Builder, T >::operator- ( ) const

inline

Negation operator, works by subtracting this from zero.

Returns: bigfield

Definition at line 483 of file bigfield.hpp.

◆ operator-() [2/2]

template<typename Builder , typename T >

bigfield< Builder, T > bb::stdlib::bigfield< Builder, T >::operator- ( const bigfield< Builder, T > & other ) const

Subtraction operator.

Parameters

other

Returns: bigfield

Uses lazy reduction techniques (similar to operator+) to save on field reductions. Instead of computing *this - other, we compute a constant X = s * p and compute: *this + X - other to ensure we do not underflow. Note that NOT enough to ensure that the integer value of *this + X - other does not underflow. We must ALSO ensure that each LIMB of the result does not underflow. Based on this condition, we compute the minimum value of s such that, for each limb i, the following result is positive: *this.limb[i] + X.limb[i] - other.limb[i] ≥ 0.

Plookup bigfield subtractoin

We have a special addition gate we can toggle, that will compute: (w_1 + w_4 - w_4_omega + q_arith = 0) This is in addition to the regular addition gate

We can arrange our wires in memory like this:

| 1 | 2 | 3 | 4 | |--—|--—|--—|--—| | b.p | a.0 | b.0 | c.p | (b.p + c.p - a.p = 0) AND (a.0 - b.0 - c.0 = 0) | a.p | a.1 | b.1 | c.0 | (a.1 - b.1 - c.1 = 0) | a.2 | b.2 | c.2 | c.1 | (a.2 - b.2 - c.2 = 0) | a.3 | b.3 | c.3 | — | (a.3 - b.3 - c.3 = 0)

Step 1: For each limb compute the MAXIMUM value we will have to borrow from the next significant limb

i.e. if we assume that *this = 0 and other = other.maximum_value, how many bits do we need to borrow from the next significant limb to ensure each limb value is positive?

N.B. for this segment maximum_value really refers to maximum NEGATIVE value of the result

Step 2: Compute the constant value X = m * p we must add to the result to ensure EVERY limb is >= 0

We need to find a value X where X.limb[3] > limb_3_maximum_value. As long as the above holds, we can borrow bits from X.limb[3] to ensure less significant limbs are positive

Start by setting constant_to_add = p

Step 3: Compute offset terms t0, t1, t2, t3 that we add to our result to ensure each limb is positive

t3 represents the value we are BORROWING from constant_to_add.limb[3] t2, t1, t0 are the terms we will ADD to constant_to_add.limb[2], constant_to_add.limb[1], constant_to_add.limb[0]

Borrow propagation table: ┌───────┬─────────────────────────────────┬──────────────────────────────────┐ │ Limb │ Value received FROM next limb │ Value given TO previous limb │ ├───────┼─────────────────────────────────┼──────────────────────────────────┤ │ 0 │ 2^limb_0_borrow_shift │ 0 │ │ 1 │ 2^limb_1_borrow_shift │ 2^(limb_0_borrow_shift - L) │ │ 2 │ 2^limb_2_borrow_shift │ 2^(limb_1_borrow_shift - L) │ │ 3 │ 0 │ 2^(limb_2_borrow_shift - L) │ └───────┴─────────────────────────────────┴──────────────────────────────────┘

i.e. The net value we add to constant_to_add is 0. We must ensure that: t3 = t0 + (t1 << NUM_LIMB_BITS) + (t2 << NUM_LIMB_BITS * 2)

e.g. the value we borrow to produce t0 is subtracted from t1, the value we borrow from t1 is subtracted from t2 the value we borrow from t2 is equal to t3

Compute the limbs of constant_to_add, including our offset terms t0, t1, t2, t3 that ensure each result limb is positive

Update the maximum possible value of the result. We assume here that (*this.value) = 0

Compute the binary basis limbs of our result

Compute the prime basis limb of the result

Definition at line 501 of file bigfield_impl.hpp.

◆ operator-=()

template<typename Builder , typename T >

bigfield bb::stdlib::bigfield< Builder, T >::operator-= ( const bigfield< Builder, T > & other )

inline

Definition at line 490 of file bigfield.hpp.

◆ operator/()

template<typename Builder , typename T >

bigfield< Builder, T > bb::stdlib::bigfield< Builder, T >::operator/ ( const bigfield< Builder, T > & other ) const

Division operator: a / b. Creates constraints for division in the circuit and checks b != 0. If you need a variant without the zero check, use div_without_denominator_check.

Parameters

other The denominator.

Returns: The result of the division c = (a / b) mod p.

To evaluate (a / b = c mod p), we instead evaluate (c * b = a mod p), where p is the target modulus. We also check that b != 0.

Definition at line 752 of file bigfield_impl.hpp.

◆ operator/=()

template<typename Builder , typename T >

bigfield bb::stdlib::bigfield< Builder, T >::operator/= ( const bigfield< Builder, T > & other )

inline

Definition at line 500 of file bigfield.hpp.

◆ operator=() [1/2]

template<typename Builder , typename T >

bigfield< Builder, T > & bb::stdlib::bigfield< Builder, T >::operator= ( bigfield< Builder, T > && other )

noexcept

Definition at line 285 of file bigfield_impl.hpp.

◆ operator=() [2/2]

template<typename Builder , typename T >

bigfield< Builder, T > & bb::stdlib::bigfield< Builder, T >::operator= ( const bigfield< Builder, T > & other )

Definition at line 271 of file bigfield_impl.hpp.

◆ operator==()

template<typename Builder , typename T >

bool_t< Builder > bb::stdlib::bigfield< Builder, T >::operator== ( const bigfield< Builder, T > & other ) const

Validate whether two bigfield elements are equal to each other.

To evaluate whether (a == b), we use result boolean r to evaluate the following logic: (n.b all algebra involving bigfield elements is done in the bigfield)

If r == 1 , a - b == 0
If r == 0, a - b posesses an inverse I i.e. (a - b) * I - 1 == 0 We efficiently evaluate this logic by evaluating a single expression (a - b)*X = Y We use conditional assignment logic to define X, Y to be the following: If r == 1 then X = 1, Y = 0 If r == 0 then X = I, Y = 1 This allows us to evaluate operator== using only 1 bigfield multiplication operation. We can check the product equals 0 or 1 by directly evaluating the binary basis/prime basis limbs of Y. i.e. if r == 1 then (a - b)*X should have 0 for all limb values if r == 0 then (a - b)*X should have 1 in the least significant binary basis limb and 0 elsewhere
Template Parameters

Builder

T

Parameters

other

Returns
bool_t<Builder>

Definition at line 1718 of file bigfield_impl.hpp.

◆ perform_reductions_for_mult_madd()

template<typename Builder , typename T >

void bb::stdlib::bigfield< Builder, T >::perform_reductions_for_mult_madd	(	std::vector< bigfield< Builder, T > > &	mul_left,
		std::vector< bigfield< Builder, T > > &	mul_right,
		const std::vector< bigfield< Builder, T > > &	to_add
	)

static

Performs individual reductions on the supplied elements as well as more complex reductions to prevent CRT modulus overflow and to fit the quotient inside the range proof.

Template Parameters

Builder	builder
T	basefield

Parameters

mul_left
mul_right
to_add

Definition at line 1133 of file bigfield_impl.hpp.

◆ pow()

template<typename Builder , typename T >

bigfield< Builder, T > bb::stdlib::bigfield< Builder, T >::pow ( const uint32_t exponent ) const

Raise the bigfield element to the power of (out-of-circuit) exponent.

Parameters

exponent The exponent to raise the bigfield element to.

Returns: this ** (exponent)

Uses the square-and-multiply algorithm to compute a^exponent mod p.

NOTE(https://github.com/AztecProtocol/barretenberg/issues/1014) Improve the efficiency of this function using sliding window method.

Definition at line 1005 of file bigfield_impl.hpp.

◆ reconstruct_from_public()

template<typename Builder , typename T >

static bigfield bb::stdlib::bigfield< Builder, T >::reconstruct_from_public ( const std::span< const field_ct, PUBLIC_INPUTS_SIZE > & limbs )

inlinestatic

Reconstruct a bigfield from limbs (generally stored in the public inputs)

Definition at line 759 of file bigfield.hpp.

◆ reduction_check()

template<typename Builder , typename T >

void bb::stdlib::bigfield< Builder, T >::reduction_check ( ) const

private

Check if the bigfield element needs to be reduced.

This function checks if the bigfield element is within the valid range and reduces it if necessary. When multiplying bigfield elements a and b, we need to ensure that each side of the equation: a * b = q * p + r (a) holds modulo the size of the native field n, (b) holds modulo the size of the bigger ring 2^t. (c) is less than the maximum value: M = 2^t * n. This implies a, b must always be less than √M, and their limbs must be less than the maximum limb value.

Given a bigfield element c, this function applies these checks: (i) c < √M (see note below). (ii) each limb (binary basis) of c is less than the maximum limb value.

These checks prevent our field arithmetic from overflowing the native modulus boundary, whilst ensuring we can still use the chinese remainder theorem to validate field multiplications with a reduced number of range checks.

Note: We actually apply a stricter bound, see get_maximum_unreduced_value for an explanation.

Definition at line 1759 of file bigfield_impl.hpp.

◆ sanity_check()

template<typename Builder , typename T >

void bb::stdlib::bigfield< Builder, T >::sanity_check ( ) const

private

Perform a sanity check on a value that is about to interact with another value.

ASSERTs that the value of all limbs is less than or equal to the prohibited maximum value. Checks that the maximum value of the whole element is also less than a prohibited maximum value.

Definition at line 1798 of file bigfield_impl.hpp.

◆ self_reduce()

template<typename Builder , typename T >

void bb::stdlib::bigfield< Builder, T >::self_reduce ( ) const

Definition at line 1990 of file bigfield_impl.hpp.

◆ set_free_witness_tag()

template<typename Builder , typename T >

void bb::stdlib::bigfield< Builder, T >::set_free_witness_tag ( )

inline

Set the free witness flag for the bigfield.

Definition at line 723 of file bigfield.hpp.

◆ set_origin_tag()

template<typename Builder , typename T >

void bb::stdlib::bigfield< Builder, T >::set_origin_tag ( const bb::OriginTag & tag ) const

inline

Definition at line 703 of file bigfield.hpp.

◆ set_public()

template<typename Builder , typename T >

uint32_t bb::stdlib::bigfield< Builder, T >::set_public ( ) const

inline

Set the witness indices of the binary basis limbs to public.

Returns: uint32_t The public input index at which the representation of the bigfield starts

Definition at line 746 of file bigfield.hpp.

◆ sqr()

template<typename Builder , typename T >

bigfield< Builder, T > bb::stdlib::bigfield< Builder, T >::sqr ( ) const

Square operator, computes a * a = c mod p.

Returns: bigfield

Costs the same as operator* as it just sets a = b. NOTE(https://github.com/AztecProtocol/aztec-packages/issues/15089): Can optimize this further to save a gate.

Definition at line 903 of file bigfield_impl.hpp.

◆ sqradd()

template<typename Builder , typename T >

bigfield< Builder, T > bb::stdlib::bigfield< Builder, T >::sqradd ( const std::vector< bigfield< Builder, T > > & to_add ) const

Square and add operator, computes a * a + ...to_add = c mod p.

Parameters

to_add The bigfield element to add to the square.

Returns: bigfield

We can chain multiple additions to a square/multiply with a single quotient/remainder. Chaining the additions here is cheaper than calling operator+ because we can combine some gates in evaluate_multiply_add

Definition at line 934 of file bigfield_impl.hpp.

◆ sum()

template<typename Builder , typename T >

bigfield< Builder, T > bb::stdlib::bigfield< Builder, T >::sum ( const std::vector< bigfield< Builder, T > > & terms )

static

Create constraints for summing these terms.

Template Parameters

Builder
T

Parameters

terms

Returns: The sum of terms

Definition at line 766 of file bigfield_impl.hpp.

◆ to_byte_array()

template<typename Builder , typename T >

byte_array< Builder > bb::stdlib::bigfield< Builder, T >::to_byte_array ( ) const

inline

Convert the bigfield element to a byte array. Concatenates byte arrays of the high (2L bits) and low (2L bits) parts of the bigfield element.

Assumes that 2L is divisible by 8, i.e. (NUM_LIMB_BITS * 2) % 8 == 0. Also we check that the bigfield element is in the target field.

Returns: byte_array<Builder>

Definition at line 363 of file bigfield.hpp.

◆ unreduced_zero()

template<typename Builder , typename T >

static constexpr bigfield bb::stdlib::bigfield< Builder, T >::unreduced_zero ( )

inlinestaticconstexpr

Create an unreduced 0 ~ p*k, where p*k is the minimal multiple of modulus that should be reduced.

We need it for division. If we always add this element during division, then we never run into the formula-breaking situation

Definition at line 660 of file bigfield.hpp.

◆ unsafe_construct_from_limbs() [1/2]

template<typename Builder , typename T >

static bigfield bb::stdlib::bigfield< Builder, T >::unsafe_construct_from_limbs	(	const field_t< Builder > &	a,
		const field_t< Builder > &	b,
		const field_t< Builder > &	c,
		const field_t< Builder > &	d,
		const bool	can_overflow = `false`
	)

inlinestatic

Construct a bigfield element from binary limbs that are already reduced.

This API should only be used by bigfield and other stdlib members for efficiency and with extreme care. We need it in cases where we precompute and reduce the elements, for example, and then put them in a table

Definition at line 157 of file bigfield.hpp.

◆ unsafe_construct_from_limbs() [2/2]

template<typename Builder , typename T >

static bigfield bb::stdlib::bigfield< Builder, T >::unsafe_construct_from_limbs	(	const field_t< Builder > &	a,
		const field_t< Builder > &	b,
		const field_t< Builder > &	c,
		const field_t< Builder > &	d,
		const field_t< Builder > &	prime_limb,
		const bool	can_overflow = `false`
	)

inlinestatic

Construct a bigfield element from binary limbs and a prime basis limb that are already reduced.

This API should only be used by bigfield and other stdlib members for efficiency and with extreme care. We need it in cases where we precompute and reduce the elements, for example, and then put them in a table

Definition at line 230 of file bigfield.hpp.

◆ unsafe_evaluate_multiple_multiply_add()

template<typename Builder , typename T >

void bb::stdlib::bigfield< Builder, T >::unsafe_evaluate_multiple_multiply_add	(	const std::vector< bigfield< Builder, T > > &	input_left,
		const std::vector< bigfield< Builder, T > > &	input_right,
		const std::vector< bigfield< Builder, T > > &	to_add,
		const bigfield< Builder, T > &	input_quotient,
		const std::vector< bigfield< Builder, T > > &	input_remainders
	)

staticprivate

Evaluate a relation involving multiple multiplications and additions.

Parameters

input_left	Vector of left multiplication operands.
input_right	Vector of right multiplication operands.
to_add	Vector of elements to add to the product.
input_quotient	Quotient term.
input_remainders	Vector of remainders.

This function evaluates the relationship: (a[0] * b[0] + ... + (a[-1] * b[-1])) + (to_add[0] + .. + to_add[-1]) - q * p - (r[0] + .. + r[-1]) = 0 mod 2^t (a[0] * b[0] + ... + (a[-1] * b[-1])) + (to_add[0] + .. + to_add[-1]) - q * p - (r[0] + .. + r[-1]) = 0 mod n

Note: This method supports multiple "remainders" because, when evaluating division of the form: (n[0] * n[1] + ... + n[-1]) / d, the numerator (which becomes the remainder term) can be a sum of multiple elements. See msub_div for more details on how this is used.

Warning: THIS FUNCTION IS UNSAFE TO USE IN CIRCUITS AS IT DOES NOT PROTECT AGAINST CRT OVERFLOWS.

Step 1: Compute the maximum potential value of our product limbs

max_lo = maximum value of limb products that span the range 0 - 2^{3t} max_hi = maximum value of limb products that span the range 2^{2t} - 2^{5t} (t = NUM_LIMB_BITS)

Definition at line 2245 of file bigfield_impl.hpp.

◆ unsafe_evaluate_multiply_add()

template<typename Builder , typename T >

void bb::stdlib::bigfield< Builder, T >::unsafe_evaluate_multiply_add	(	const bigfield< Builder, T > &	input_left,
		const bigfield< Builder, T > &	input_to_mul,
		const std::vector< bigfield< Builder, T > > &	to_add,
		const bigfield< Builder, T > &	input_quotient,
		const std::vector< bigfield< Builder, T > > &	input_remainders
	)

staticprivate

Evaluate a multiply add identity with several added elements and several remainders.

Parameters

left	Left multiplicand
right	Right multiplicand
to_add	Vector of elements to add
quotient	Quotient term
remainders	Vector of remainders

This function evaluates the relationship: a * b + (to_add[0] + .. + to_add[-1]) - q * p - (r[0] + .. + r[-1]) = 0 mod 2^t (binary basis modulus) a * b + (to_add[0] + .. + to_add[-1]) - q * p - (r[0] + .. + r[-1]) = 0 mod n (circuit modulus)

Note: This method supports multiple "remainders" because, when evaluating division of the form: (n[0] * n[1] + ... + n[-1]) / d, the numerator (which becomes the remainder term) can be a sum of multiple elements.

Warning: THIS FUNCTION IS UNSAFE TO USE IN CIRCUITS AS IT DOES NOT PROTECT AGAINST CRT OVERFLOWS.

Definition at line 2037 of file bigfield_impl.hpp.

◆ unsafe_evaluate_square_add()

template<typename Builder , typename T >

void bb::stdlib::bigfield< Builder, T >::unsafe_evaluate_square_add	(	const bigfield< Builder, T > &	left,
		const std::vector< bigfield< Builder, T > > &	to_add,
		const bigfield< Builder, T > &	quotient,
		const bigfield< Builder, T > &	remainder
	)

staticprivate

Evaluate a square with several additions.

Parameters

left	Left multiplicand
to_add	Vector of elements to add
quotient	Quotient term
remainder	Remainder term

This function evaluates the relationship: (a * a) + (to_add[0] + .. + to_add[-1]) - q * p - r = 0 mod 2^t (binary basis modulus) (a * a) + (to_add[0] + .. + to_add[-1]) - q * p - r = 0 mod n (circuit modulus)

Warning: THIS FUNCTION IS UNSAFE TO USE IN CIRCUITS AS IT DOES NOT PROTECT AGAINST CRT OVERFLOWS.

Definition at line 2544 of file bigfield_impl.hpp.

◆ unset_free_witness_tag()

template<typename Builder , typename T >

void bb::stdlib::bigfield< Builder, T >::unset_free_witness_tag ( )

inline

Unset the free witness flag for the bigfield.

Definition at line 734 of file bigfield.hpp.

◆ zero()

template<typename Builder , typename T >

static bigfield bb::stdlib::bigfield< Builder, T >::zero ( )

inlinestatic

Create a public zero constant

Definition at line 648 of file bigfield.hpp.

Member Data Documentation

◆ binary_basis

template<typename Builder , typename T >

constexpr Basis bb::stdlib::bigfield< Builder, T >::binary_basis { uint512_t(1) << LOG2_BINARY_MODULUS, LOG2_BINARY_MODULUS }

staticconstexpr

Definition at line 332 of file bigfield.hpp.

◆ binary_basis_limbs

template<typename Builder , typename T >

std::array<Limb, NUM_LIMBS> bb::stdlib::bigfield< Builder, T >::binary_basis_limbs

mutable

Represents a bigfield element in the binary basis. A bigfield element is represented as a combination of 4 binary basis limbs: a = a[0] + a[1] * 2^L + a[2] * 2^2L + a[3] * 2^3L.

Definition at line 80 of file bigfield.hpp.

◆ context

template<typename Builder , typename T >

Builder* bb::stdlib::bigfield< Builder, T >::context

Definition at line 74 of file bigfield.hpp.

◆ DEFAULT_MAXIMUM_LIMB

template<typename Builder , typename T >

constexpr uint256_t bb::stdlib::bigfield< Builder, T >::DEFAULT_MAXIMUM_LIMB = (uint256_t(1) << NUM_LIMB_BITS) - uint256_t(1)

staticconstexpr

Definition at line 319 of file bigfield.hpp.

◆ DEFAULT_MAXIMUM_MOST_SIGNIFICANT_LIMB

template<typename Builder , typename T >

constexpr uint256_t bb::stdlib::bigfield< Builder, T >::DEFAULT_MAXIMUM_MOST_SIGNIFICANT_LIMB

staticconstexpr

Initial value:

=

(uint256_t(1) << NUM_LAST_LIMB_BITS) - uint256_t(1)

bb::stdlib::bigfield::NUM_LAST_LIMB_BITS

static constexpr uint64_t NUM_LAST_LIMB_BITS

Definition bigfield.hpp:311

Definition at line 320 of file bigfield.hpp.

◆ DEFAULT_MAXIMUM_REMAINDER

template<typename Builder , typename T >

const uint1024_t bb::stdlib::bigfield< Builder, T >::DEFAULT_MAXIMUM_REMAINDER

inlinestatic

Initial value:

=

(uint1024_t(1) << (NUM_LIMB_BITS * 3 + NUM_LAST_LIMB_BITS)) - uint1024_t(1)

bb::stdlib::bigfield::NUM_LIMB_BITS

static constexpr uint64_t NUM_LIMB_BITS

Definition bigfield.hpp:310

bb::numeric::uint1024_t

uintx< uint512_t > uint1024_t

Definition uintx.hpp:309

Definition at line 317 of file bigfield.hpp.

◆ is_composite

template<typename Builder , typename T >

constexpr bool bb::stdlib::bigfield< Builder, T >::is_composite = true

staticconstexpr

Definition at line 323 of file bigfield.hpp.

◆ LOG2_BINARY_MODULUS

template<typename Builder , typename T >

constexpr uint64_t bb::stdlib::bigfield< Builder, T >::LOG2_BINARY_MODULUS = NUM_LIMB_BITS * NUM_LIMBS

staticconstexpr

Definition at line 322 of file bigfield.hpp.

◆ MAX_ADDITION_LOG

template<typename Builder , typename T >

constexpr uint64_t bb::stdlib::bigfield< Builder, T >::MAX_ADDITION_LOG = 10

staticconstexpr

Definition at line 897 of file bigfield.hpp.

◆ MAX_UNREDUCED_LIMB_BITS

template<typename Builder , typename T >

constexpr uint64_t bb::stdlib::bigfield< Builder, T >::MAX_UNREDUCED_LIMB_BITS = NUM_LIMB_BITS + 10

staticconstexpr

Definition at line 928 of file bigfield.hpp.

◆ MAXIMUM_LIMB_SIZE_THAT_WOULDNT_OVERFLOW

template<typename Builder , typename T >

constexpr uint64_t bb::stdlib::bigfield< Builder, T >::MAXIMUM_LIMB_SIZE_THAT_WOULDNT_OVERFLOW

staticconstexpr

Initial value:

=

(bb::fr::modulus.get_msb() - MAX_ADDITION_LOG - NUM_LIMB_BITS - 2) / 2

bb::numeric::uint256_t::get_msb

constexpr uint64_t get_msb() const

Definition uint256_impl.hpp:329

bb::stdlib::bigfield::MAX_ADDITION_LOG

static constexpr uint64_t MAX_ADDITION_LOG

Definition bigfield.hpp:897

bb::field< Bn254FrParams >::modulus

static constexpr uint256_t modulus

Definition field_declarations.hpp:197

Definition at line 922 of file bigfield.hpp.

◆ MAXIMUM_SUMMAND_COUNT

template<typename Builder , typename T >

constexpr size_t bb::stdlib::bigfield< Builder, T >::MAXIMUM_SUMMAND_COUNT = 1 << MAXIMUM_SUMMAND_COUNT_LOG

staticconstexpr

Definition at line 327 of file bigfield.hpp.

◆ MAXIMUM_SUMMAND_COUNT_LOG

template<typename Builder , typename T >

constexpr size_t bb::stdlib::bigfield< Builder, T >::MAXIMUM_SUMMAND_COUNT_LOG = 4

staticconstexpr

Definition at line 326 of file bigfield.hpp.

◆ modulus

template<typename Builder , typename T >

constexpr uint256_t bb::stdlib::bigfield< Builder, T >::modulus = (uint256_t(T::modulus_0, T::modulus_1, T::modulus_2, T::modulus_3))

staticconstexpr

Definition at line 308 of file bigfield.hpp.

◆ modulus_u512

template<typename Builder , typename T >

constexpr uint512_t bb::stdlib::bigfield< Builder, T >::modulus_u512 = static_cast<uint512_t>(modulus)

staticconstexpr

Definition at line 309 of file bigfield.hpp.

◆ neg_modulus_limbs

template<typename Builder , typename T >

constexpr std::array<bb::fr, NUM_LIMBS> bb::stdlib::bigfield< Builder, T >::neg_modulus_limbs

staticconstexpr

Initial value:

{
        bb::fr(negative_prime_modulus.slice(0, NUM_LIMB_BITS).lo),
        bb::fr(negative_prime_modulus.slice(NUM_LIMB_BITS, NUM_LIMB_BITS * 2).lo),
        bb::fr(negative_prime_modulus.slice(NUM_LIMB_BITS * 2, NUM_LIMB_BITS * 3).lo),
        bb::fr(negative_prime_modulus.slice(NUM_LIMB_BITS * 3, NUM_LIMB_BITS * 4).lo),
    }

Definition at line 347 of file bigfield.hpp.

◆ neg_modulus_limbs_u256

template<typename Builder , typename T >

constexpr std::array<uint256_t, NUM_LIMBS> bb::stdlib::bigfield< Builder, T >::neg_modulus_limbs_u256

staticconstexpr

Initial value:

{
        negative_prime_modulus.slice(0, NUM_LIMB_BITS).lo,
        negative_prime_modulus.slice(NUM_LIMB_BITS, NUM_LIMB_BITS * 2).lo,
        negative_prime_modulus.slice(NUM_LIMB_BITS * 2, NUM_LIMB_BITS * 3).lo,
        negative_prime_modulus.slice(NUM_LIMB_BITS * 3, NUM_LIMB_BITS * 4).lo,
    }

Definition at line 341 of file bigfield.hpp.

◆ negative_prime_modulus

template<typename Builder , typename T >

constexpr uint512_t bb::stdlib::bigfield< Builder, T >::negative_prime_modulus = binary_basis.modulus - target_basis.modulus

staticconstexpr

Definition at line 340 of file bigfield.hpp.

◆ negative_prime_modulus_mod_binary_basis

template<typename Builder , typename T >

constexpr bb::fr bb::stdlib::bigfield< Builder, T >::negative_prime_modulus_mod_binary_basis = -bb::fr(uint256_t(modulus_u512))

staticconstexpr

Definition at line 339 of file bigfield.hpp.

◆ NUM_LAST_LIMB_BITS

template<typename Builder , typename T >

constexpr uint64_t bb::stdlib::bigfield< Builder, T >::NUM_LAST_LIMB_BITS = modulus_u512.get_msb() + 1 - (NUM_LIMB_BITS * 3)

staticconstexpr

Definition at line 311 of file bigfield.hpp.

◆ NUM_LIMB_BITS

template<typename Builder , typename T >

constexpr uint64_t bb::stdlib::bigfield< Builder, T >::NUM_LIMB_BITS = NUM_LIMB_BITS_IN_FIELD_SIMULATION

staticconstexpr

Definition at line 310 of file bigfield.hpp.

◆ NUM_LIMBS

template<typename Builder , typename T >

constexpr size_t bb::stdlib::bigfield< Builder, T >::NUM_LIMBS = 4

staticconstexpr

Definition at line 72 of file bigfield.hpp.

◆ prime_basis

template<typename Builder , typename T >

constexpr Basis bb::stdlib::bigfield< Builder, T >::prime_basis { uint512_t(bb::fr::modulus), bb::fr::modulus.get_msb() + 1 }

staticconstexpr

Definition at line 331 of file bigfield.hpp.

◆ prime_basis_limb

template<typename Builder , typename T >

field_t<Builder> bb::stdlib::bigfield< Builder, T >::prime_basis_limb

mutable

Represents a bigfield element in the prime basis: (a mod n) where n is the native modulus.

Definition at line 85 of file bigfield.hpp.

◆ prime_basis_maximum_limb

template<typename Builder , typename T >

constexpr uint256_t bb::stdlib::bigfield< Builder, T >::prime_basis_maximum_limb

staticconstexpr

Initial value:

=

modulus_u512.slice(NUM_LIMB_BITS * (NUM_LIMBS - 1), NUM_LIMB_BITS* NUM_LIMBS).lo

bb::stdlib::bigfield::NUM_LIMBS

static constexpr size_t NUM_LIMBS

Definition bigfield.hpp:72

bb::stdlib::bigfield::modulus_u512

static constexpr uint512_t modulus_u512

Definition bigfield.hpp:309

Definition at line 329 of file bigfield.hpp.

◆ PROHIBITED_LIMB_BITS

template<typename Builder , typename T >

constexpr uint64_t bb::stdlib::bigfield< Builder, T >::PROHIBITED_LIMB_BITS = MAX_UNREDUCED_LIMB_BITS + 5

staticconstexpr

Definition at line 932 of file bigfield.hpp.

◆ PUBLIC_INPUTS_SIZE

template<typename Builder , typename T >

constexpr size_t bb::stdlib::bigfield< Builder, T >::PUBLIC_INPUTS_SIZE = BIGFIELD_PUBLIC_INPUTS_SIZE

staticconstexpr

Definition at line 31 of file bigfield.hpp.

◆ shift_1

template<typename Builder , typename T >

constexpr bb::fr bb::stdlib::bigfield< Builder, T >::shift_1 = bb::fr(uint256_t(1) << NUM_LIMB_BITS)

staticconstexpr

Definition at line 334 of file bigfield.hpp.

◆ shift_2

template<typename Builder , typename T >

constexpr bb::fr bb::stdlib::bigfield< Builder, T >::shift_2 = bb::fr(uint256_t(1) << (NUM_LIMB_BITS * 2))

staticconstexpr

Definition at line 335 of file bigfield.hpp.

◆ shift_3

template<typename Builder , typename T >

constexpr bb::fr bb::stdlib::bigfield< Builder, T >::shift_3 = bb::fr(uint256_t(1) << (NUM_LIMB_BITS * 3))

staticconstexpr

Definition at line 336 of file bigfield.hpp.

◆ shift_right_1

template<typename Builder , typename T >

constexpr bb::fr bb::stdlib::bigfield< Builder, T >::shift_right_1 = bb::fr(1) / shift_1

staticconstexpr

Definition at line 337 of file bigfield.hpp.

◆ shift_right_2

template<typename Builder , typename T >

constexpr bb::fr bb::stdlib::bigfield< Builder, T >::shift_right_2 = bb::fr(1) / shift_2

staticconstexpr

Definition at line 338 of file bigfield.hpp.

◆ target_basis

template<typename Builder , typename T >

constexpr Basis bb::stdlib::bigfield< Builder, T >::target_basis { modulus_u512, static_cast<size_t>(modulus_u512.get_msb() + 1) }

staticconstexpr

Definition at line 333 of file bigfield.hpp.

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

src/barretenberg/stdlib/primitives/bigfield/bigfield.hpp
src/barretenberg/stdlib/primitives/bigfield/bigfield_impl.hpp

Classes

Public Types

Public Member Functions

Static Public Member Functions

Public Attributes

Static Public Attributes

Private Member Functions

Static Private Member Functions

Detailed Description

Member Typedef Documentation

◆ CoefficientAccumulator

◆ field_ct

◆ native

◆ TParams

◆ View

Constructor & Destructor Documentation

◆ bigfield() [1/11]

◆ bigfield() [2/11]

◆ bigfield() [3/11]

◆ bigfield() [4/11]

◆ bigfield() [5/11]

◆ bigfield() [6/11]

◆ bigfield() [7/11]

◆ bigfield() [8/11]

◆ bigfield() [9/11]

◆ bigfield() [10/11]

◆ bigfield() [11/11]

◆ ~bigfield()

Member Function Documentation

◆ add_to_lower_limb()

◆ add_two()

◆ assert_equal()

◆ assert_is_in_field()

◆ assert_is_not_equal()

◆ assert_less_than()

◆ compute_maximum_quotient_value()

◆ compute_partial_schoolbook_multiplication()

◆ compute_quotient_remainder_values()

◆ conditional_assign()

◆ conditional_negate()

◆ conditional_select()

◆ construct_from_limbs()

◆ convert_constant_to_fixed_witness()

◆ create_from_u512_as_witness()

◆ div_check_denominator_nonzero()

◆ div_without_denominator_check() [1/2]

◆ div_without_denominator_check() [2/2]

◆ dual_madd()

◆ fix_witness()

◆ from_witness()

◆ get_binary_basis_limb_maximums()

◆ get_binary_basis_limb_witness_indices()

◆ get_context()

◆ get_maximum_crt_product()

◆ get_maximum_unreduced_limb_value()

◆ get_maximum_unreduced_value()

◆ get_maximum_value()

◆ get_origin_tag()

◆ get_prohibited_limb_value()

◆ get_prohibited_value()

◆ get_quotient_max_bits()

◆ get_quotient_reduction_info()

◆ get_value()

◆ internal_div()

◆ invert()

◆ is_constant()

◆ madd()

◆ msub_div()

◆ mul_product_overflows_crt_modulus() [1/2]

◆ mul_product_overflows_crt_modulus() [2/2]

◆ mult_madd()

◆ one()

◆ operator*()

◆ operator*=()

◆ operator+()

◆ operator+=()

◆ operator-() [1/2]

◆ operator-() [2/2]

◆ operator-=()

◆ operator/()