emulated

package
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 Go to latest Published: Dec 19, 2023 License: Apache-2.0 Imports: 16 Imported by: 0

Documentation

Overview

Package emulated implements operations over any modulus.

Non-native computation in circuit

Usually, the computations in a SNARK circuit are performed in the 'native' field of the curve. The native field is defined by the scalar field of the underlying curve. This package implements non-native arithmetic on top of the native field to emulate operations in any field and ring.

This package does this by splitting the element into smaller limbs. The parameters for splitting the limb and defining the modulus are stored in types implementing FieldParams type. The elements are parametrized by those types to make compile-time distinction between different emulated fields.

This package defines Element type which stores the element value in split limbs. On top of the Element instance, this package defines typical arithmetic as addition, multiplication and subtraction. If the modulus is a prime (i.e. defines a finite field), then inversion and division operations are also possible.

The results of the operations are not always reduced to be less than the modulus. For consecutive operations it is necessary to manually reduce the value using Field.Reduce method. The number of operations which can be performed without reduction depends when the operations result starts overflowing the limbs.

Element representation

We operate in the scalar field of the SNARK curve (native field). Denote the modulus of the native field as 'q'. Representing the modulus of the native field requires 'n' bits. We wish to emulate operations over modulus 'r'. Modulus r may or may not be a prime. If r is not prime, then we do not have inversion and division operations (the corresponding methods panic). Let the bitlength of r be 'm'. We note that r may be smaller, larger or equal to q.

To represent an element x ∈ N_r, we choose the limb width 'w' such that 

w ≤ (m-1)/2

and write its integer representation as 

x = ∑_{i=0}^k x_i 2^{w i}.

Here, the variable 'x_i' is the w bits of x starting from 2^{w i}, 'k' is the number of limbs and is computed as 

k = (n+w-1)/w,   // NB! integer division

and 'i' is the limb index. In this representation the element is represented in little-endian (least significant limbs first) order. We do not necessarily require that the limb values x_i are less than 2^w. This may happen if the limb values are obtained as a result of arithmetic operations between elements. If we know that the limb values do not overflow 2^w, then we say that the element is in normal form.

In the implementation, we have two functions for splitting an element into limbs and composing an element from limbs -- [decompose] and [recompose]. The [recompose] function also accepts element in non-normal form.

Elements in non-normal form

When an element is initialized, the limbs are in normal form, i.e. the values of the limbs have bitwidth strictly less than w. As addition and multiplication are performed on limbs natively, then the bitwidths of the limbs of the result may be larger than w. We track the number of bits which may exceed the initial width of the limbs. We denote the number of such excess bits as 'f' and call it overflow. The total maximal bitwidth of the limbs is then 

w+f.

Keep in mind that parameter w is global for all emulated elements and f is individual for every individual element.

To compute the overflow for the operations, we consider the arithmetic operations which affect the overflow. In this implementation only addition is done natively (limb-wise addition). When adding two elements, the bitwidth of the result is up to one bit wider than the width of the widest element.

In the context of overflows, if the overflows of the addends are f_0 and f_1 then the overflow value f' for the sum is computed as 

f' = max(f_0, f_1)+1.

Multiplication

The complexity of native limb-wise multiplication is k^2. This translates directly to the complexity in the number of constraints in the constraint system.

For multiplication, we would instead use polynomial representation of the elements: 

x = ∑_{i=0}^k x_i 2^{w i}
y = ∑_{i=0}^k y_i 2^{w i}.

as 

x(X) = ∑_{i=0}^k x_i X^i
y(X) = ∑_{i=0}^k y_i X^i.

If the multiplication result modulo r is c, then the following holds: 

x * y = c + z*r.

We can check the correctness of the multiplication by checking the following identity at a random point: 

x(X) * y(X) = c(X) + z(X) * r(X) + (2^w' - X) e(X),

where e(X) is a polynomial used for carrying the overflows of the left- and right-hand side of the above equation.

Subtraction

We perform subtraction limb-wise between the elements x and y. However, we have to ensure than any limb in the result does not result in overflow, i.e. 

x_i ≥ y_i, ∀ 0≤i<k.

As this does not hold in general, then we need to pad x such that every limb x_i is strictly larger than y_i.

The additional padding 'u' has to be divisible by the emulated modulus r and every limb u_i must be larger than x_i-y_i. Let f' be the overflow of y. We first compute the limbs u'_i as 

u'_i = 1 << (w+f'), ∀ 0≤i<k.

Now, as u' is not divisible by r, we need to compensate for it: 

u'' = u' + regroup(r - (u' % r)),

where regroup() regroups the argument so that it is in normal form (i.e. first applies recompose() and then decompose() method).

We see that u” is now such that it is divisible by r and its every limb is larger than every limb of b. The subtraction is performed as 

z_i = x_i + u''_i - y_i, ∀ 0≤i<k.

Equality checking

The package provides two ways to check equality -- limb-wise equality check and checking equality by value.

In the limb-wise equality check we check that the integer values of the elements x and y are equal. We have to carry the excess using bit decomposition (which makes the computation fairly inefficient). To reduce the number of bit decompositions, we instead carry over the excess of the difference of the limbs instead. As we take the difference, then similarly as computing the padding in subtraction algorithm, we need to add padding to the limbs before subtracting limb-wise to avoid underflows. However, the padding in this case is slightly different -- we do not need the padding to be divisible by the modulus, but instead need that the limb padding is larger than the limb which is being subtracted.

Lets look at the algorithm itself. We assume that the overflow f of x is larger than y. If overflow of y is larger, then we can just swap the arguments and apply the same argumentation. Let 

maxValue = 1 << (k+f), // padding for limbs
maxValueShift = 1 << f.  // carry part of the padding

For every limb we compute the difference as 

diff_0 = maxValue+x_0-y_0,
diff_i = maxValue+carry_i+x_i-y_i-maxValueShift.

We check that the normal part of the difference is zero and carry the rest over to next limb: 

diff_i[0:k] == 0,
carry_{i+1} = diff_i[k:k+f+1] // we also carry over the padding bit.

Finally, after we have compared all the limbs, we still need to check that the final carry corresponds to the padding. We add final check: 

carry_k == maxValueShift.

We can further optimise the limb-wise equality check by first regrouping the limbs. The idea is to group several limbs so that the result would still fit into the scalar field. If 

x = ∑_{i=0}^k x_i 2^{w i},

then we can instead take w' divisible by w such that 

x = ∑_{i=0}^(k/(w'/w)) x'_i 2^{w' i},

where 

x'_j = ∑_{i=0}^(w'/w) x_{j*w'/w+i} 2^{w i}.

For element value equality check, we check that two elements x and y are equal modulo r and for that we need to show that r divides x-y. As mentioned in the subtraction section, we add sufficient padding such that x-y does not underflow and its integer value is always larger than 0. We use hint function to compute z such that 

x-y = z*r,

compute z*r and use limbwise equality checking to show that 

x-y == z*r.

Bitwidth enforcement

When element is computed using hints, we need to ensure that every limb is not wider than k bits. For that, we perform bitwise decomposition of every limb and check that k lower bits are equal to the whole limb. We omit the bitwidth enforcement for multiplication as the correctness of the limbs is ensured using the corresponding system of linear equations.

Additionally, we apply bitwidth enforcement for elements initialized from integers.

Modular reduction

To reduce the integer value of element x to be less than the modulus, we compute the remainder x' of x modulo r using hint, enforce the bitwidth of x' and assert that 

x' == x

using element equality checking.

Values computed using hints

We additionally define functions for computing inverse of an element and ratio of two elements. Both function compute the actual value using hint and then assert the correctness of the operation using multiplication.

Constant values

The package currently does not explicitly differentiate between constant and variable elements. The builder may track some elements as being constants. Some operations have a fast track path for cases when all inputs are constants. There is Field.MulConst, which provides variable by constant multiplication.

Index

Examples

Constants

This section is empty.

Variables

This section is empty.

Functions

func DivHint 

func DivHint(mod *big.Int, inputs []*big.Int, outputs []*big.Int) error

DivHint computes the value z = x/y for inputs x and y and stores z in outputs.

func GetHints 

func GetHints() []solver.Hint

GetHints returns all hint functions used in the package.

func InverseHint 

func InverseHint(mod *big.Int, inputs []*big.Int, outputs []*big.Int) error

InverseHint computes the inverse x^-1 for the input x and stores it in outputs.

func MultiplicationHint 

func MultiplicationHint(mod *big.Int, inputs []*big.Int, outputs []*big.Int) error

MultiplicationHint unpacks the factors and parameters from inputs, computes the product and stores it in output. See internal method computeMultiplicationHint for the input packing.

func QuoHint 

func QuoHint(_ *big.Int, inputs []*big.Int, outputs []*big.Int) error

QuoHint sets z to the quotient x/y for y != 0 and returns z. If y == 0, returns an error. Quo implements truncated division (like Go); see QuoRem for more details.

func RightShift 

func RightShift(_ *big.Int, inputs []*big.Int, outputs []*big.Int) error

RightShift shifts input by the given number of bits. Expects two inputs:

  • first input is the shift, will be represented as uint64;
  • second input is the value to be shifted.

Returns a single output which is the value shifted. Errors if number of inputs is not 2 and number of outputs is not 1.

func SqrtHint 

func SqrtHint(mod *big.Int, inputs []*big.Int, outputs []*big.Int) error

SqrtHint compute square root of the input.

func UnwrapHint 

func UnwrapHint(nativeInputs, nativeOutputs []*big.Int, nonnativeHint solver.Hint) error

UnwrapHint unwraps the native inputs into nonnative inputs. Then it calls nonnativeHint function with nonnative inputs. After nonnativeHint returns, it decomposes the outputs into limbs.

Types

type BLS12377Fp 

type BLS12377Fp = emparams.BLS12377Fp

type BLS12381Fp 

type BLS12381Fp = emparams.BLS12381Fp

type BLS12381Fr 

type BLS12381Fr = emparams.BLS12381Fr

type BN254Fp 

type BN254Fp = emparams.BN254Fp

type BN254Fr 

type BN254Fr = emparams.BN254Fr

type BW6761Fp 

type BW6761Fp = emparams.BW6761Fp

type BW6761Fr 

type BW6761Fr = emparams.BW6761Fr

type Element 

type Element[T FieldParams] struct {
	// Limbs is the decomposition of the integer value into limbs in the native
	// field. To enforce that the limbs are of expected width, use Pack...
	// methods on the Field. Uses little-endian (least significant limb first)
	// encoding.
	Limbs []frontend.Variable
	// contains filtered or unexported fields
}

Element defines an element in the ring of integers modulo n. The integer value of the element is split into limbs of nbBits lengths and represented as a slice of limbs. The type parameter defines the field this element belongs to.

func ValueOf 

func ValueOf[T FieldParams](constant interface{}) Element[T]

ValueOf returns an Element[T] from a constant value. The input is converted to *big.Int and decomposed into limbs and packed into new Element[T].

func (*Element[T]) GnarkInitHook 

func (e *Element[T]) GnarkInitHook()

GnarkInitHook describes how to initialise the element.

type Field 

type Field[T FieldParams] struct {
	// contains filtered or unexported fields
}

Field holds the configuration for non-native field operations. The field parameters (modulus, number of limbs) is given by FieldParams type parameter. If [FieldParams.IsPrime] is true, then allows inverse and division operations.

Example

Example of using Field instance. The witness elements must be Element type. 

package main

import (
	"fmt"

	"github.com/airchains-network/gnark/backend"
	"github.com/airchains-network/gnark/backend/groth16"
	"github.com/airchains-network/gnark/constraint/solver"
	"github.com/airchains-network/gnark/frontend"
	"github.com/airchains-network/gnark/frontend/cs/r1cs"
	"github.com/airchains-network/gnark/std/math/emulated"
	"github.com/consensys/gnark-crypto/ecc"
)

type ExampleFieldCircuit[T emulated.FieldParams] struct {
	In1 emulated.Element[T]
	In2 emulated.Element[T]
	Res emulated.Element[T]
}

func (c *ExampleFieldCircuit[T]) Define(api frontend.API) error {
	f, err := emulated.NewField[T](api)
	if err != nil {
		return fmt.Errorf("new field: %w", err)
	}
	res := f.Mul(&c.In1, &c.In2) // non-reducing
	res = f.Reduce(res)
	f.AssertIsEqual(res, &c.Res)
	return nil
}

// Example of using [Field] instance. The witness elements must be [Element]
// type.
func main() {
	circuit := ExampleFieldCircuit[emulated.BN254Fp]{}
	witness := ExampleFieldCircuit[emulated.BN254Fp]{
		In1: emulated.ValueOf[emulated.BN254Fp](3),
		In2: emulated.ValueOf[emulated.BN254Fp](5),
		Res: emulated.ValueOf[emulated.BN254Fp](15),
	}
	ccs, err := frontend.Compile(ecc.BN254.ScalarField(), r1cs.NewBuilder, &circuit)
	if err != nil {
		panic(err)
	}
	witnessData, err := frontend.NewWitness(&witness, ecc.BN254.ScalarField())
	if err != nil {
		panic(err)
	}
	publicWitnessData, err := witnessData.Public()
	if err != nil {
		panic(err)
	}
	pk, vk, err := groth16.Setup(ccs)
	if err != nil {
		panic(err)
	}
	proof, err := groth16.Prove(ccs, pk, witnessData, backend.WithSolverOptions(solver.WithHints(emulated.GetHints()...)))
	if err != nil {
		panic(err)
	}
	err = groth16.Verify(proof, vk, publicWitnessData)
	if err != nil {
		panic(err)
	}
	fmt.Println("done")
}

Output:

done

func NewField 

func NewField[T FieldParams](native frontend.API) (*Field[T], error)

NewField returns an object to be used in-circuit to perform emulated arithmetic over the field defined by type parameter FieldParams. The operations on this type are defined on Element. There is also another type [FieldAPI] implementing frontend.API which can be used in place of native API for existing circuits.

This is an experimental feature and performing emulated arithmetic in-circuit is extremly costly. See package doc for more info.

func (*Field[T]) Add 

func (f *Field[T]) Add(a, b *Element[T]) *Element[T]

Add computes a+b and returns it. If the result wouldn't fit into Element, then first reduces the inputs (larger first) and tries again. Doesn't mutate inputs.

func (*Field[T]) AssertIsEqual 

func (f *Field[T]) AssertIsEqual(a, b *Element[T])

AssertIsEqual ensures that a is equal to b modulo the modulus.

func (*Field[T]) AssertIsInRange 

func (f *Field[T]) AssertIsInRange(a *Element[T])

AssertIsInRange ensures that a is less than the emulated modulus. When we call [Reduce] then we only ensure that the result is width-constrained, but not actually less than the modulus. This means that the actual value may be either x or x + p. For arithmetic it is sufficient, but for binary comparison it is not. For binary comparison the values have both to be below the modulus.

func (*Field[T]) AssertIsLessOrEqual 

func (f *Field[T]) AssertIsLessOrEqual(e, a *Element[T])

AssertIsLessOrEqual ensures that e is less or equal than a. For proper bitwise comparison first reduce the element using [Reduce] and then assert that its value is less than the modulus using [AssertIsInRange].

func (*Field[T]) AssertLimbsEquality 

func (f *Field[T]) AssertLimbsEquality(a, b *Element[T])

AssertLimbsEquality asserts that the limbs represent a same integer value. This method does not ensure that the values are equal modulo the field order. For strict equality, use AssertIsEqual.

func (*Field[T]) Div 

func (f *Field[T]) Div(a, b *Element[T]) *Element[T]

Div computes a/b and returns it. It uses DivHint as a hint function.

func (*Field[T]) FromBits 

func (f *Field[T]) FromBits(bs ...frontend.Variable) *Element[T]

FromBits returns a new Element given the bits is little-endian order.

func (*Field[T]) Inverse 

func (f *Field[T]) Inverse(a *Element[T]) *Element[T]

Inverse compute 1/a and returns it. It uses InverseHint.

func (*Field[T]) IsZero 

func (f *Field[T]) IsZero(a *Element[T]) frontend.Variable

IsZero returns a boolean indicating if the element is strictly zero. The method internally reduces the element and asserts that the value is less than the modulus.

func (*Field[T]) Lookup2 

func (f *Field[T]) Lookup2(b0, b1 frontend.Variable, a, b, c, d *Element[T]) *Element[T]

Lookup2 performs two-bit lookup between a, b, c, d based on lookup bits b1 and b2 such that:

  • if b0=0 and b1=0, sets to a,
  • if b0=1 and b1=0, sets to b,
  • if b0=0 and b1=1, sets to c,
  • if b0=1 and b1=1, sets to d.

The number of the limbs and overflow in the result is the maximum of the inputs'. If the inputs are very unbalanced, then reduce the result.

func (*Field[T]) Modulus 

func (f *Field[T]) Modulus() *Element[T]

Modulus returns the modulus of the emulated ring as a constant.

func (*Field[T]) Mul 

func (f *Field[T]) Mul(a, b *Element[T]) *Element[T]

Mul computes a*b and reduces it modulo the field order. The returned Element has default number of limbs and zero overflow. If the result wouldn't fit into Element, then locally reduces the inputs first. Doesn't mutate inputs.

For multiplying by a constant, use [Field[T].MulConst] method which is more efficient.

func (*Field[T]) MulConst 

func (f *Field[T]) MulConst(a *Element[T], c *big.Int) *Element[T]

MulConst multiplies a by a constant c and returns it. We assume that the input constant is "small", so that we can compute the product by multiplying all individual limbs with the constant. If it is not small, then use the general [Field[T].Mul] or [Field[T].MulMod] with creating new Element from the constant on-the-fly.

func (*Field[T]) MulMod 

func (f *Field[T]) MulMod(a, b *Element[T]) *Element[T]

MulMod computes a*b and reduces it modulo the field order. The returned Element has default number of limbs and zero overflow.

Equivalent to [Field[T].Mul], kept for backwards compatibility.

func (*Field[T]) Neg 

func (f *Field[T]) Neg(a *Element[T]) *Element[T]

func (*Field[T]) NewElement 

func (f *Field[T]) NewElement(v interface{}) *Element[T]

NewElement builds a new Element[T] from input v.

  • if v is a Element[T] or *Element[T] it clones it
  • if v is a constant this is equivalent to calling emulated.ValueOf[T]
  • if this methods interprets v as being the limbs (frontend.Variable or []frontend.Variable), it constructs a new Element[T] with v as limbs and constraints the limbs to the parameters of the Field[T].

func (*Field[T]) NewHint 

func (f *Field[T]) NewHint(hf solver.Hint, nbOutputs int, inputs ...*Element[T]) ([]*Element[T], error)

NewHint allows to call the emulation hint function hf on inputs, expecting nbOutputs results. This function splits internally the emulated element into limbs and passes these to the hint function. There is UnwrapHint function which performs corresponding recomposition of limbs into integer values (and vice verse for output).

The hint function for this method is defined as: 

func HintFn(mod *big.Int, inputs, outputs []*big.Int) error {
    return emulated.UnwrapHint(inputs, outputs, func(mod *big.Int, inputs, outputs []*big.Int) error { // NB we shadow initial input, output, mod to avoid accidental overwrite!
	    // here all inputs and outputs are modulo nonnative mod. we decompose them automatically
    })}

See the example for full written example.

Example

Example of using hints with emulated elements. 

package main

import (
	"fmt"
	"math/big"

	"github.com/airchains-network/gnark/backend"
	"github.com/airchains-network/gnark/backend/groth16"
	"github.com/airchains-network/gnark/constraint/solver"
	"github.com/airchains-network/gnark/frontend"
	"github.com/airchains-network/gnark/frontend/cs/r1cs"
	"github.com/airchains-network/gnark/std/math/emulated"
	"github.com/consensys/gnark-crypto/ecc"
	"github.com/consensys/gnark-crypto/ecc/bn254/fr"
)

// HintExample is a hint for field emulation which returns the division of the
// first and second input.
func HintExample(nativeMod *big.Int, nativeInputs, nativeOutputs []*big.Int) error {
	// nativeInputs are the limbs of the input non-native elements. We wrap the
	// actual hint function with [emulated.UnwrapHint] to get actual [*big.Int]
	// values of the non-native elements.
	return emulated.UnwrapHint(nativeInputs, nativeOutputs, func(mod *big.Int, inputs, outputs []*big.Int) error {
		// this hint computes the division of first and second input and returns it.
		nominator := inputs[0]
		denominator := inputs[1]
		res := new(big.Int).ModInverse(denominator, mod)
		if res == nil {
			return fmt.Errorf("no modular inverse")
		}
		res.Mul(res, nominator)
		res.Mod(res, mod)
		outputs[0].Set(res)
		return nil
	})
	// when the internal hint function returns, the UnwrapHint function
	// decomposes the non-native value into limbs.
}

type emulationHintCircuit[T emulated.FieldParams] struct {
	Nominator   emulated.Element[T]
	Denominator emulated.Element[T]
	Expected    emulated.Element[T]
}

func (c *emulationHintCircuit[T]) Define(api frontend.API) error {
	field, err := emulated.NewField[T](api)
	if err != nil {
		return err
	}
	res, err := field.NewHint(HintExample, 1, &c.Nominator, &c.Denominator)
	if err != nil {
		return err
	}
	m := field.Mul(res[0], &c.Denominator)
	field.AssertIsEqual(m, &c.Nominator)
	field.AssertIsEqual(res[0], &c.Expected)
	return nil
}

// Example of using hints with emulated elements.
func main() {
	var a, b, c fr.Element
	a.SetRandom()
	b.SetRandom()
	c.Div(&a, &b)

	circuit := emulationHintCircuit[emulated.BN254Fr]{}
	witness := emulationHintCircuit[emulated.BN254Fr]{
		Nominator:   emulated.ValueOf[emulated.BN254Fr](a),
		Denominator: emulated.ValueOf[emulated.BN254Fr](b),
		Expected:    emulated.ValueOf[emulated.BN254Fr](c),
	}
	ccs, err := frontend.Compile(ecc.BN254.ScalarField(), r1cs.NewBuilder, &circuit)
	if err != nil {
		panic(err)
	}
	witnessData, err := frontend.NewWitness(&witness, ecc.BN254.ScalarField())
	if err != nil {
		panic(err)
	}
	publicWitnessData, err := witnessData.Public()
	if err != nil {
		panic(err)
	}
	pk, vk, err := groth16.Setup(ccs)
	if err != nil {
		panic(err)
	}
	proof, err := groth16.Prove(ccs, pk, witnessData, backend.WithSolverOptions(solver.WithHints(HintExample)))
	if err != nil {
		panic(err)
	}
	err = groth16.Verify(proof, vk, publicWitnessData)
	if err != nil {
		panic(err)
	}
	fmt.Println("done")
}

Output:

done

func (*Field[T]) One 

func (f *Field[T]) One() *Element[T]

One returns one as a constant.

func (*Field[T]) Reduce 

func (f *Field[T]) Reduce(a *Element[T]) *Element[T]

Reduce reduces a modulo the field order and returns it.

func (*Field[T]) Select 

func (f *Field[T]) Select(selector frontend.Variable, a, b *Element[T]) *Element[T]

Select sets e to a if selector == 1 and to b otherwise. Sets the number of limbs and overflow of the result to be the maximum of the limb lengths and overflows. If the inputs are strongly unbalanced, then it would better to reduce the result after the operation.

func (*Field[T]) Sqrt 

func (f *Field[T]) Sqrt(a *Element[T]) *Element[T]

Sqrt computes square root of a and returns it. It uses SqrtHint.

func (*Field[T]) Sub 

func (f *Field[T]) Sub(a, b *Element[T]) *Element[T]

Sub subtracts b from a and returns it. Reduces locally if wouldn't fit into Element. Doesn't mutate inputs.

func (*Field[T]) ToBits 

func (f *Field[T]) ToBits(a *Element[T]) []frontend.Variable

ToBits returns the bit representation of the Element in little-endian (LSB first) order. The returned bits are constrained to be 0-1. The number of returned bits is nbLimbs*nbBits+overflow. To obtain the bits of the canonical representation of Element, reduce Element first and take less significant bits corresponding to the bitwidth of the emulated modulus.

func (*Field[T]) Zero 

func (f *Field[T]) Zero() *Element[T]

Zero returns zero as a constant.

type FieldParams 

type FieldParams interface {
	NbLimbs() uint     // number of limbs to represent field element
	BitsPerLimb() uint // number of bits per limb. Top limb may contain less than limbSize bits.
	IsPrime() bool     // indicates if the modulus is prime
	Modulus() *big.Int // returns modulus. Do not modify.
}

FieldParams describes the emulated field characteristics. For a list of included built-in emulation params refer to the emparams package. For backwards compatibility, the current package contains the following parameters:

type Goldilocks 

type Goldilocks = emparams.Goldilocks

type P256Fp 

type P256Fp = emparams.P256Fp

type P256Fr 

type P256Fr = emparams.P256Fr

type P384Fp 

type P384Fp = emparams.P384Fp

type P384Fr 

type P384Fr = emparams.P384Fr

type Secp256k1Fp 

type Secp256k1Fp = emparams.Secp256k1Fp

type Secp256k1Fr 

type Secp256k1Fr = emparams.Secp256k1Fr

Source Files

View all Source files

Directories

Path Synopsis
Package emparams contains emulation parameters for well known fields.
 Package emparams contains emulation parameters for well known fields.

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