num

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 Go to latest Published: Feb 16, 2026 License: Apache-2.0 Imports: 14 Imported by: 0

README

num

Package num provides arbitrary-precision arithmetic for cryptographic applications. It is built over the constant-time numct package and is suitable for high-level applications.

Overview

This package provides immutable, strongly-typed number representations that implement standard algebraic interfaces. Unlike numct, operations in num are not constant-time; Use numct directly when timing side-channels must be avoided.

Key Types

  • NatPlus (NPlus()): Positive natural numbers (excludes zero).
  • Nat (N()): Natural numbers (non-negative integers).
  • Int (Z()): Signed integers.
  • Rat (Q()): Rational numbers (fractions).
  • Uint / ZMod: Integers modulo n. Implements algebra.ZModLike.

Architecture

Each numeric type has an associated structure type (e.g., NaturalNumbers, Integers, Rationals, ZMod) accessible via singleton functions (N(), Z(), Q(), NewZMod()). These structures provide constructors, constants, and implement algebraic interfaces from the algebra package.

All types wrap numct primitives internally and provide:

  • Conversion to/from *big.Int and *big.Rat
  • CBOR serialization
  • Hash codes for use in maps and sets
  • Random sampling within ranges

Usage Notes

  • Types are immutable; all operations return new values.
  • Use Canonical() on Rat to reduce fractions to lowest terms.
  • ZMod caches Montgomery context for efficient modular arithmetic.
  • Random sampling uses half-open intervals [low, high).

Documentation

Overview

Package num provides arbitrary-precision arithmetic for cryptographic applications. It is built over the constant-time numct package and is suitable for high-level applications.

See README.md for details.

Index

Constants

This section is empty.

Variables

View Source 
var (
	ErrInexactDivision = errs.New("inexact division")
	ErrUndefined       = errs.New("operation is undefined")
	ErrOutOfRange      = errs.New("value is out of range")
	ErrUnequalModuli   = errs.New("moduli are unequal")
	ErrIsNil           = errs.New("value must not be nil")
	ErrDivisionByZero  = errs.New("division by zero")
)

Functions

This section is empty.

Types

type Int 

type Int struct {
	// contains filtered or unexported fields
}

Int represents an integer in the ring of integers Z.

func (*Int) Abs 

func (i *Int) Abs() *Nat

Abs returns the absolute value of the integer.

func (*Int) Add 

func (i *Int) Add(other *Int) *Int

Add performs addition of two integers.

func (*Int) AnnouncedLen 

func (i *Int) AnnouncedLen() int

AnnouncedLen returns the announced length of the integer in bytes.

func (*Int) Big 

func (i *Int) Big() *big.Int

Big converts the integer to a big.Int.

func (*Int) Bytes 

func (i *Int) Bytes() []byte

Bytes returns the byte representation of the integer.

func (*Int) Cardinal 

func (i *Int) Cardinal() cardinal.Cardinal

Cardinal returns the cardinality of the absolute value of the integer.

func (*Int) Clone 

func (i *Int) Clone() *Int

Clone creates a copy of the integer.

func (*Int) Compare 

func (i *Int) Compare(other *Int) base.Ordering

Compare compares the integer with another integer.

func (*Int) Coprime 

func (i *Int) Coprime(other *Int) bool

Coprime checks if two integers are coprime.

func (*Int) Decrement 

func (i *Int) Decrement() *Int

Decrement returns the integer decremented by one.

func (*Int) DivRound 

func (i *Int) DivRound(other *Int) (*Int, error)

DivRound performs the division of the integer by another integer returning the quotient rounded towards zero.

func (*Int) DivRoundVarTime 

func (i *Int) DivRoundVarTime(other *Int) (*Int, error)

DivRoundVarTime performs the division of the integer by another integer returning the quotient rounded towards zero. It is not constant-time due to having to generate montgomery parameters for the divisor (i.e., leaks divisor).

func (*Int) Double 

func (i *Int) Double() *Int

Double returns the integer multiplied by 2.

func (*Int) Equal 

func (i *Int) Equal(other *Int) bool

Equal checks if the integer is equal to another integer.

func (*Int) EuclideanDiv 

func (i *Int) EuclideanDiv(other *Int) (quot, rem *Int, err error)

EuclideanDiv performs Euclidean division of the integer by another integer.

func (*Int) EuclideanDivVarTime 

func (i *Int) EuclideanDivVarTime(other *Int) (quot, rem *Int, err error)

EuclideanDivVarTime performs Euclidean division of the integer by another integer. It is not constant-time due to having to generate montgomery parameters for the divisor (i.e., leaks divisor).

func (*Int) EuclideanValuation 

func (i *Int) EuclideanValuation() algebra.Cardinal

EuclideanValuation returns the Euclidean valuation of the integer.

func (*Int) HashCode 

func (i *Int) HashCode() base.HashCode

HashCode returns the hash code of the integer.

func (*Int) Increment 

func (i *Int) Increment() *Int

Increment returns the integer incremented by one.

func (*Int) IsEven 

func (i *Int) IsEven() bool

IsEven checks if the integer is even.

func (*Int) IsInRange 

func (i *Int) IsInRange(modulus *NatPlus) bool

IsInRange checks if the integer is within the range defined by the modulus i.e., 0 <= x < m.

func (*Int) IsInRangeSymmetric 

func (i *Int) IsInRangeSymmetric(modulus *NatPlus) bool

IsInRangeSymmetric checks if the integer is within the symmetric range defined by the modulus i.e., -m/2 <= x < m/2.

func (*Int) IsLessThanOrEqual 

func (i *Int) IsLessThanOrEqual(other *Int) bool

IsLessThanOrEqual checks if the integer is less than or equal to another integer.

func (*Int) IsNegative 

func (i *Int) IsNegative() bool

IsNegative checks if the integer is negative.

func (*Int) IsOdd 

func (i *Int) IsOdd() bool

IsOdd checks if the integer is odd.

func (*Int) IsOne 

func (i *Int) IsOne() bool

IsOne checks if the integer is one.

func (*Int) IsOpIdentity 

func (i *Int) IsOpIdentity() bool

IsOpIdentity checks if the integer is the additive identity (zero).

func (*Int) IsPositive 

func (i *Int) IsPositive() bool

IsPositive checks if the integer is positive.

func (*Int) IsProbablyPrime 

func (i *Int) IsProbablyPrime() bool

IsProbablyPrime checks if the integer is probably prime.

func (*Int) IsTorsionFree 

func (*Int) IsTorsionFree() bool

IsTorsionFree returns true, indicating that the integers are torsion-free.

func (*Int) IsUnit 

func (i *Int) IsUnit(modulus *NatPlus) bool

IsUnit checks if the integer is a unit modulo the given NatPlus modulus.

func (*Int) IsZero 

func (i *Int) IsZero() bool

IsZero checks if the integer is zero.

func (*Int) Lift 

func (i *Int) Lift() *Int

Lift returns a copy of the integer (identity function for integers).

func (*Int) Lsh 

func (i *Int) Lsh(shift uint) *Int

Lsh performs a left shift operation on the integer.

func (*Int) MarshalCBOR 

func (i *Int) MarshalCBOR() ([]byte, error)

func (*Int) Mod 

func (i *Int) Mod(modulus *NatPlus) *Uint

Mod returns the integer modulo the given NatPlus modulus.

func (*Int) Mul 

func (i *Int) Mul(other *Int) *Int

Mul performs multiplication of two integers.

func (*Int) Neg 

func (i *Int) Neg() *Int

Neg computes the negation of the integer.

func (*Int) Op 

func (i *Int) Op(other *Int) *Int

Op performs addition of two integers.

func (*Int) OpInv 

func (i *Int) OpInv() *Int

OpInv returns the additive inverse of the integer.

func (*Int) OtherOp 

func (i *Int) OtherOp(other *Int) *Int

OtherOp performs multiplication of two integers.

func (*Int) Rat 

func (i *Int) Rat() *Rat

Rat converts the integer to a rational number.

func (*Int) Rsh 

func (i *Int) Rsh(shift uint) *Int

Rsh performs a right shift operation on the integer.

func (*Int) ScalarMul 

func (i *Int) ScalarMul(other *Int) *Int

ScalarMul performs scalar multiplication of the integer by another integer.

func (*Int) ScalarOp 

func (i *Int) ScalarOp(other *Int) *Int

ScalarOp performs scalar multiplication of the integer by another integer.

func (*Int) Square 

func (i *Int) Square() *Int

Square returns the square of the integer.

func (*Int) String 

func (i *Int) String() string

String returns the string representation of the integer.

func (*Int) Structure 

func (*Int) Structure() algebra.Structure[*Int]

Structure returns the algebraic structure of the integers.

func (*Int) Sub 

func (i *Int) Sub(other *Int) *Int

Sub performs subtraction of two integers.

func (*Int) TrueLen 

func (i *Int) TrueLen() int

TrueLen returns the true length of the integer in bytes.

func (*Int) TryDiv 

func (i *Int) TryDiv(other *Int) (*Int, error)

TryDiv performs the exact division of the integer by another integer.

func (*Int) TryDivVarTime 

func (i *Int) TryDivVarTime(other *Int) (*Int, error)

TryDivVarTime performs the exact division of the integer by another integer. It is not constant-time due to having to generate montgomery parameters for the divisor (i.e., leaks divisor).

func (*Int) TryInv 

func (i *Int) TryInv() (*Int, error)

TryInv attempts to compute the multiplicative inverse of the integer.

func (*Int) TryNeg 

func (i *Int) TryNeg() (*Int, error)

TryNeg attempts to compute the negation of the integer. It never fails.

func (*Int) TryOpInv 

func (i *Int) TryOpInv() (*Int, error)

TryOpInv returns the additive inverse of the integer.

func (*Int) TrySub 

func (i *Int) TrySub(other *Int) (*Int, error)

TrySub performs subtraction of two integers.

func (*Int) UnmarshalCBOR 

func (i *Int) UnmarshalCBOR(data []byte) error

func (*Int) Value 

func (i *Int) Value() *numct.Int

Value returns the underlying numct.Int value of the integer.

type Integers 

type Integers struct{}

Integers implements the algebra.Structure interface for the ring of integers Z.

func Z 

func Z() *Integers

Z returns the singleton instance of the Integers structure.

func (*Integers) Characteristic 

func (*Integers) Characteristic() cardinal.Cardinal

Characteristic returns the characteristic of the integers, which is 0.

func (*Integers) ElementSize 

func (*Integers) ElementSize() int

ElementSize returns -1 indicating that the size of integer elements is unbounded.

func (*Integers) FromBig 

func (zs *Integers) FromBig(value *big.Int) (*Int, error)

FromBig creates an integer from a big.Int value.

func (*Integers) FromBytes 

func (*Integers) FromBytes(input []byte) (*Int, error)

FromBytes creates an integer from its byte representation.

func (*Integers) FromCardinal 

func (zs *Integers) FromCardinal(value cardinal.Cardinal) (*Int, error)

FromCardinal creates an integer from a cardinal.Cardinal value.

func (*Integers) FromInt64 

func (*Integers) FromInt64(value int64) *Int

FromInt64 creates an integer from an int64 value.

func (*Integers) FromIntCT 

func (*Integers) FromIntCT(value *numct.Int) (*Int, error)

FromIntCT creates an integer from a numct.Int value.

func (*Integers) FromNat 

func (zs *Integers) FromNat(value *Nat) (*Int, error)

FromNat creates an integer from a Nat value.

func (*Integers) FromNatCT 

func (*Integers) FromNatCT(value *numct.Nat) (*Int, error)

FromNatCT creates an integer from a numct.Nat value.

func (*Integers) FromNatPlus 

func (zs *Integers) FromNatPlus(value *NatPlus) (*Int, error)

FromNatPlus creates an integer from a NatPlus value.

func (*Integers) FromRat 

func (*Integers) FromRat(input *Rat) (*Int, error)

FromRat creates an integer from a Rat value, if the Rat is an integer.

func (*Integers) FromUint 

func (zs *Integers) FromUint(input *Uint) (*Int, error)

FromUint creates an integer from a Uint value.

func (*Integers) FromUint64 

func (*Integers) FromUint64(value uint64) *Int

FromUint64 creates an integer from a uint64 value.

func (*Integers) FromUintSymmetric 

func (*Integers) FromUintSymmetric(input *Uint) (*Int, error)

FromUintSymmetric creates an integer from a Uint value using symmetric representation.

func (*Integers) IsDomain 

func (*Integers) IsDomain() bool

IsDomain returns true, indicating that the integers form a domain (no zero divisors).

func (*Integers) Name 

func (*Integers) Name() string

func (*Integers) One 

func (*Integers) One() *Int

One returns the multiplicative identity element (one) of the integers.

func (*Integers) OpIdentity 

func (zs *Integers) OpIdentity() *Int

OpIdentity returns the additive identity element (zero) of the integers.

func (*Integers) Order 

func (*Integers) Order() cardinal.Cardinal

Order returns the (infinite) order of the integers.

func (*Integers) Random 

func (*Integers) Random(lowInclusive, highExclusive *Int, prng io.Reader) (*Int, error)

Random generates a random integer in the range [lowInclusive, highExclusive).

func (*Integers) ScalarStructure 

func (*Integers) ScalarStructure() algebra.Structure[*Int]

ScalarStructure returns the structure of the scalars, which is also the integers.

func (*Integers) Zero 

func (*Integers) Zero() *Int

Zero returns the zero element of the integers.

type Nat 

type Nat struct {
	// contains filtered or unexported fields
}

Nat represents a natural number (non-negative integer).

func (*Nat) Add 

func (n *Nat) Add(other *Nat) *Nat

Add performs the addition of two Nat values.

func (*Nat) AnnouncedLen 

func (n *Nat) AnnouncedLen() int

AnnouncedLen returns the announced length of the Nat in bytes.

func (*Nat) Big 

func (n *Nat) Big() *big.Int

Big returns the big.Int representation of the Nat.

func (*Nat) Bit 

func (n *Nat) Bit(i uint) byte

Bit returns the value of the i-th bit of the Nat.

func (*Nat) Byte 

func (n *Nat) Byte(i uint) byte

Byte returns the value of the i-th byte of the Nat.

func (*Nat) Bytes 

func (n *Nat) Bytes() []byte

Bytes returns the byte slice representation of the Nat.

func (*Nat) BytesBE 

func (n *Nat) BytesBE() []byte

BytesBE returns the big-endian byte slice representation of the Nat.

func (*Nat) Cardinal 

func (n *Nat) Cardinal() cardinal.Cardinal

Cardinal returns the cardinal representation of the Nat.

func (*Nat) Clone 

func (n *Nat) Clone() *Nat

Clone creates a copy of the Nat.

func (*Nat) Compare 

func (n *Nat) Compare(other *Nat) base.Ordering

Compare compares the Nat with another Nat, returning an ordering result.

func (*Nat) Coprime 

func (n *Nat) Coprime(other *Nat) bool

Coprime checks if the Nat is coprime with another Nat.

func (*Nat) Decrement 

func (n *Nat) Decrement() (*Nat, error)

Decrement returns the Nat decremented by 1, returning an error if the result would be negative.

func (*Nat) DivRound 

func (n *Nat) DivRound(other *Nat) (*Nat, error)

DivRound divides the Nat by another Nat returning quotient rounded towards zero. It returns an error if the division is not exact.

func (*Nat) DivRoundVarTime 

func (n *Nat) DivRoundVarTime(other *Nat) (*Nat, error)

DivRoundVarTime divides the Nat by another Nat returning quotient rounded towards zero. It returns an error if the division is not exact. It is not constant-time due to having to generate montgomery parameters for the divisor (i.e., leaks divisor).

func (*Nat) Double 

func (n *Nat) Double() *Nat

Double returns the Nat doubled.

func (*Nat) Equal 

func (n *Nat) Equal(other *Nat) bool

Equal checks if the Nat is equal to another Nat.

func (*Nat) EuclideanDiv 

func (n *Nat) EuclideanDiv(other *Nat) (quot, rem *Nat, err error)

EuclideanDiv performs Euclidean division of the Nat by another Nat, returning the quotient and remainder.

func (*Nat) EuclideanDivVarTime 

func (n *Nat) EuclideanDivVarTime(other *Nat) (quot, rem *Nat, err error)

EuclideanDivVarTime performs Euclidean division of the Nat by another Nat, returning the quotient and remainder. It is not constant-time due to having to generate montgomery parameters for the divisor (i.e., leaks divisor).

func (*Nat) EuclideanValuation 

func (n *Nat) EuclideanValuation() cardinal.Cardinal

EuclideanValuation computes the Euclidean valuation of the Nat.

func (*Nat) GCD 

func (n *Nat) GCD(other *Nat) *Nat

GCD computes the greatest common divisor (GCD) of the Nat and another Nat.

func (*Nat) HashCode 

func (n *Nat) HashCode() base.HashCode

HashCode computes a hash code for the Nat.

func (*Nat) Increment 

func (n *Nat) Increment() *Nat

Increment returns the Nat incremented by 1.

func (*Nat) IsBottom 

func (n *Nat) IsBottom() bool

IsBottom checks if the Nat is the bottom element (0).

func (*Nat) IsEven 

func (n *Nat) IsEven() bool

IsEven checks if the Nat is even.

func (*Nat) IsLessThanOrEqual 

func (n *Nat) IsLessThanOrEqual(other *Nat) bool

IsLessThanOrEqual checks if the Nat is less than or equal to another Nat.

func (*Nat) IsOdd 

func (n *Nat) IsOdd() bool

IsOdd checks if the Nat is odd.

func (*Nat) IsOne 

func (n *Nat) IsOne() bool

IsOne checks if the Nat is one.

func (*Nat) IsOpIdentity 

func (n *Nat) IsOpIdentity() bool

IsOpIdentity checks if the Nat is the additive identity (0).

func (*Nat) IsPositive 

func (n *Nat) IsPositive() bool

IsPositive checks if the Nat is positive (greater than 0).

func (*Nat) IsProbablyPrime 

func (n *Nat) IsProbablyPrime() bool

IsProbablyPrime checks if the Nat is probably prime.

func (*Nat) IsTorsionFree 

func (*Nat) IsTorsionFree() bool

IsTorsionFree checks if the Nat is torsion-free under addition, which is always true for natural numbers.

func (*Nat) IsUnit 

func (n *Nat) IsUnit(modulus *NatPlus) bool

IsUnit checks if the Nat is a unit modulo the given NatPlus modulus.

func (*Nat) IsZero 

func (n *Nat) IsZero() bool

IsZero checks if the Nat is zero.

func (*Nat) Lift 

func (n *Nat) Lift() *Int

Lift converts the Nat to an Int.

func (*Nat) Lsh 

func (n *Nat) Lsh(shift uint) *Nat

Lsh performs a left shift operation on the Nat by the specified number of bits.

func (*Nat) MarshalCBOR 

func (n *Nat) MarshalCBOR() ([]byte, error)

func (*Nat) Mod 

func (n *Nat) Mod(modulus *NatPlus) *Uint

Mod computes the Nat modulo the given NatPlus modulus.

func (*Nat) Mul 

func (n *Nat) Mul(other *Nat) *Nat

Mul performs the multiplication of two Nat values.

func (*Nat) Op 

func (n *Nat) Op(other *Nat) *Nat

Op performs the addition operation on two Nat values.

func (*Nat) OtherOp 

func (n *Nat) OtherOp(other *Nat) *Nat

OtherOp performs the multiplication operation on two Nat values.

func (*Nat) Rsh 

func (n *Nat) Rsh(shift uint) *Nat

Rsh performs a right shift operation on the Nat by the specified number of bits.

func (*Nat) ScalarMul 

func (n *Nat) ScalarMul(sc *Nat) *Nat

ScalarMul performs scalar multiplication of the Nat by another Nat.

func (*Nat) ScalarOp 

func (n *Nat) ScalarOp(sc *Nat) *Nat

ScalarOp performs scalar multiplication of the Nat by another Nat.

func (*Nat) Sqrt 

func (n *Nat) Sqrt() (*Nat, error)

Sqrt computes the square root of the Nat, returning an error if the square root is not defined.

func (*Nat) Square 

func (n *Nat) Square() *Nat

Square returns the square of the Nat.

func (*Nat) String 

func (n *Nat) String() string

String returns the string representation of the Nat.

func (*Nat) Structure 

func (*Nat) Structure() algebra.Structure[*Nat]

Structure returns the NaturalNumbers structure.

func (*Nat) TrueLen 

func (n *Nat) TrueLen() int

TrueLen returns the true length of the Nat in bytes.

func (*Nat) TryDiv 

func (n *Nat) TryDiv(other *Nat) (*Nat, error)

TryDiv attempts to divide the Nat by another Nat. It returns an error if the division is not exact.

func (*Nat) TryDivVarTime 

func (n *Nat) TryDivVarTime(other *Nat) (*Nat, error)

TryDivVarTime attempts to divide the Nat by another Nat. It returns an error if the division is not exact. It is not constant-time due to having to generate montgomery parameters for the divisor (i.e., leaks divisor).

func (*Nat) TryInv 

func (n *Nat) TryInv() (*Nat, error)

TryInv attempts to compute the multiplicative inverse of the Nat. It returns an error unless the Nat is 1.

func (*Nat) TryNeg 

func (*Nat) TryNeg() (*Nat, error)

TryNeg attempts to compute the negation of the Nat. It will always return an error since natural numbers do not have negation.

func (*Nat) TryOpInv 

func (n *Nat) TryOpInv() (*Nat, error)

TryOpInv attempts to compute the additive inverse of the Nat. It will always return an error since natural numbers do not have additive inverses.

func (*Nat) TrySub 

func (n *Nat) TrySub(other *Nat) (*Nat, error)

TrySub attempts to subtract another Nat from the current Nat. It returns an error if the result would not be a natural number.

func (*Nat) Uint64 

func (n *Nat) Uint64() uint64

Uint64 returns the uint64 representation of the Nat. It wraps around if the Nat is too large.

func (*Nat) UnmarshalCBOR 

func (n *Nat) UnmarshalCBOR(data []byte) error

func (*Nat) Value 

func (n *Nat) Value() *numct.Nat

Value returns the underlying numct.Nat value of the Nat.

type NatPlus 

type NatPlus struct {
	// contains filtered or unexported fields
}

NatPlus represents a positive natural number (N\{0}).

func (*NatPlus) Abs 

func (np *NatPlus) Abs() *NatPlus

Abs returns the absolute value of the NatPlus, which is itself.

func (*NatPlus) Add 

func (np *NatPlus) Add(other *NatPlus) *NatPlus

Add performs addition of two NatPlus elements.

func (*NatPlus) AnnouncedLen 

func (np *NatPlus) AnnouncedLen() int

AnnouncedLen returns the announced length of the NatPlus in bytes.

func (*NatPlus) Big 

func (np *NatPlus) Big() *big.Int

Big returns the big.Int representation of the NatPlus.

func (*NatPlus) Bit 

func (np *NatPlus) Bit(i uint) byte

Bit returns the value of the i-th bit of the NatPlus.

func (*NatPlus) Byte 

func (np *NatPlus) Byte(i uint) byte

Byte returns the value of the i-th byte of the NatPlus.

func (*NatPlus) Bytes 

func (np *NatPlus) Bytes() []byte

Bytes returns the big-endian byte representation of the NatPlus.

func (*NatPlus) BytesBE 

func (np *NatPlus) BytesBE() []byte

BytesBE returns the big-endian byte representation of the NatPlus.

func (*NatPlus) Cardinal 

func (np *NatPlus) Cardinal() cardinal.Cardinal

Cardinal returns the cardinal representation of the NatPlus.

func (*NatPlus) Clone 

func (np *NatPlus) Clone() *NatPlus

Clone creates a copy of the NatPlus.

func (*NatPlus) Compare 

func (np *NatPlus) Compare(other *NatPlus) base.Ordering

Compare compares the NatPlus with another NatPlus, returning the ordering result.

func (*NatPlus) Decrement 

func (np *NatPlus) Decrement() (*NatPlus, error)

Decrement returns the NatPlus decremented by 1, returning an error if the result would be less than 1.

func (*NatPlus) Double 

func (np *NatPlus) Double() *NatPlus

Double returns the result of multiplying the NatPlus by 2.

func (*NatPlus) Equal 

func (np *NatPlus) Equal(other *NatPlus) bool

Equal checks if the NatPlus is equal to another NatPlus.

func (*NatPlus) HashCode 

func (np *NatPlus) HashCode() base.HashCode

HashCode computes the hash code of the NatPlus.

func (*NatPlus) Increment 

func (np *NatPlus) Increment() *NatPlus

Increment returns the NatPlus incremented by 1.

func (*NatPlus) IsBottom 

func (np *NatPlus) IsBottom() bool

IsBottom checks if the NatPlus is the smallest element (1).

func (*NatPlus) IsEven 

func (np *NatPlus) IsEven() bool

IsEven checks if the NatPlus is even.

func (*NatPlus) IsLessThanOrEqual 

func (np *NatPlus) IsLessThanOrEqual(other *NatPlus) bool

IsLessThanOrEqual checks if the NatPlus is less than or equal to another NatPlus.

func (*NatPlus) IsOdd 

func (np *NatPlus) IsOdd() bool

IsOdd checks if the NatPlus is odd.

func (*NatPlus) IsOne 

func (np *NatPlus) IsOne() bool

IsOne checks if the NatPlus is equal to 1.

func (*NatPlus) IsOpIdentity 

func (np *NatPlus) IsOpIdentity() bool

IsOpIdentity checks if the NatPlus is the multiplicative identity (1).

func (*NatPlus) IsProbablyPrime 

func (np *NatPlus) IsProbablyPrime() bool

IsProbablyPrime checks if the NatPlus is probably prime.

func (*NatPlus) IsUnit 

func (np *NatPlus) IsUnit(modulus *NatPlus) bool

IsUnit checks if the NatPlus is a unit with respect to the given modulus.

func (*NatPlus) Lift 

func (np *NatPlus) Lift() *Int

Lift converts the NatPlus to an Int.

func (*NatPlus) Lsh 

func (np *NatPlus) Lsh(shift uint) *NatPlus

Lsh performs a left shift operation on the NatPlus.

func (*NatPlus) MarshalCBOR 

func (np *NatPlus) MarshalCBOR() ([]byte, error)

func (*NatPlus) Mod 

func (np *NatPlus) Mod(modulus *NatPlus) *Uint

Mod computes the modulus of the NatPlus with respect to another NatPlus.

func (*NatPlus) ModulusCT 

func (np *NatPlus) ModulusCT() *numct.Modulus

ModulusCT returns the cached modulus or computes it if not cached.

func (*NatPlus) Mul 

func (np *NatPlus) Mul(other *NatPlus) *NatPlus

Mul performs multiplication of two NatPlus elements.

func (*NatPlus) Nat 

func (np *NatPlus) Nat() *Nat

Nat returns the Nat representation of the NatPlus.

func (*NatPlus) Op 

func (np *NatPlus) Op(other *NatPlus) *NatPlus

Op performs multiplication of two NatPlus elements.

func (*NatPlus) OtherOp 

func (np *NatPlus) OtherOp(other *NatPlus) *NatPlus

OtherOp performs addition of two NatPlus elements.

func (*NatPlus) Rsh 

func (np *NatPlus) Rsh(shift uint) *NatPlus

Rsh performs a right shift operation on the NatPlus. Panics if the result would be zero.

func (*NatPlus) Square 

func (np *NatPlus) Square() *NatPlus

Square returns the result of squaring the NatPlus.

func (*NatPlus) String 

func (np *NatPlus) String() string

String returns the string representation of the NatPlus.

func (*NatPlus) Structure 

func (*NatPlus) Structure() algebra.Structure[*NatPlus]

Structure returns the algebraic structure of NatPlus, which is PositiveNaturalNumbers.

func (*NatPlus) TrueLen 

func (np *NatPlus) TrueLen() int

TrueLen returns the true length of the NatPlus in bytes.

func (*NatPlus) TryDiv 

func (np *NatPlus) TryDiv(other *NatPlus) (*NatPlus, error)

TryDiv attempts to divide the NatPlus by another NatPlus, returning an error if the division is not exact.

func (*NatPlus) TryInv 

func (np *NatPlus) TryInv() (*NatPlus, error)

TryInv attempts to compute the multiplicative inverse of the NatPlus, returning an error since it does not exist.

func (*NatPlus) TryOpInv 

func (np *NatPlus) TryOpInv() (*NatPlus, error)

TryOpInv attempts to compute the multiplicative inverse of the NatPlus, returning an error since it does not exist.

func (*NatPlus) TryRsh 

func (np *NatPlus) TryRsh(shift uint) (*NatPlus, error)

TryRsh attempts to right shift the NatPlus, returning an error if the result would be zero.

func (*NatPlus) TrySub 

func (np *NatPlus) TrySub(other *NatPlus) (*NatPlus, error)

TrySub attempts to subtract another NatPlus from the NatPlus, returning an error if the result is not a positive natural number.

func (*NatPlus) Uint64 

func (np *NatPlus) Uint64() uint64

func (*NatPlus) UnmarshalCBOR 

func (np *NatPlus) UnmarshalCBOR(data []byte) error

func (*NatPlus) Value 

func (np *NatPlus) Value() *numct.Nat

Value returns the underlying numct.Nat value of the NatPlus.

type NaturalNumbers 

type NaturalNumbers struct{}

NaturalNumbers represents the set of natural numbers (non-negative integers).

func N 

func N() *NaturalNumbers

N returns the singleton instance of the NaturalNumbers structure.

func (*NaturalNumbers) Bottom 

func (ns *NaturalNumbers) Bottom() *Nat

Bottom returns the smallest element in the NaturalNumbers structure, which is 0.

func (*NaturalNumbers) Characteristic 

func (*NaturalNumbers) Characteristic() cardinal.Cardinal

Characteristic returns the characteristic of the NaturalNumbers structure, which is 0.

func (*NaturalNumbers) ElementSize 

func (*NaturalNumbers) ElementSize() int

ElementSize returns -1 indicating that elements of NaturalNumbers do not have a fixed size.

func (*NaturalNumbers) FromBig 

func (ns *NaturalNumbers) FromBig(value *big.Int) (*Nat, error)

FromBig creates a Nat from a big.Int value, returning an error if the input is nil or negative.

func (*NaturalNumbers) FromBytes 

func (*NaturalNumbers) FromBytes(input []byte) (*Nat, error)

FromBytes creates a Nat from a byte slice, returning an error if the input is nil.

func (*NaturalNumbers) FromBytesBE 

func (ns *NaturalNumbers) FromBytesBE(input []byte) (*Nat, error)

FromBytesBE creates a Nat from a big-endian byte slice, returning an error if the input is nil.

func (*NaturalNumbers) FromCardinal 

func (ns *NaturalNumbers) FromCardinal(value cardinal.Cardinal) (*Nat, error)

FromCardinal creates a Nat from a cardinal.Cardinal value, returning an error if the input is nil or infinite.

func (*NaturalNumbers) FromInt 

func (ns *NaturalNumbers) FromInt(value *Int) (*Nat, error)

FromInt creates a Nat from an Int value, returning an error if the input is nil or negative.

func (*NaturalNumbers) FromNatCT 

func (*NaturalNumbers) FromNatCT(value *numct.Nat) (*Nat, error)

FromNatCT creates a Nat from a numct.Nat value, returning an error if the input is nil.

func (*NaturalNumbers) FromNatPlus 

func (*NaturalNumbers) FromNatPlus(value *NatPlus) (*Nat, error)

FromNatPlus creates a Nat from a NatPlus value, returning an error if the input is nil.

func (*NaturalNumbers) FromRat 

func (ns *NaturalNumbers) FromRat(value *Rat) (*Nat, error)

FromRat creates a Nat from a Rat value, returning an error if Rat is not a non-negative integer.

func (*NaturalNumbers) FromUint64 

func (*NaturalNumbers) FromUint64(value uint64) *Nat

FromUint64 creates a Nat from a uint64 value.

func (*NaturalNumbers) Name 

func (*NaturalNumbers) Name() string

Name returns the name of the structure: "N".

func (*NaturalNumbers) One 

func (*NaturalNumbers) One() *Nat

One returns the multiplicative identity element of the NaturalNumbers structure.

func (*NaturalNumbers) OpIdentity 

func (ns *NaturalNumbers) OpIdentity() *Nat

OpIdentity returns the identity element for the addition operation in the NaturalNumbers structure.

func (*NaturalNumbers) Order 

func (*NaturalNumbers) Order() cardinal.Cardinal

Order returns the order of the NaturalNumbers structure, which is infinite.

func (*NaturalNumbers) Random 

func (ns *NaturalNumbers) Random(lowInclusive, highExclusive *Nat, prng io.Reader) (*Nat, error)

Random generates a random Nat in the range [lowInclusive, highExclusive), returning an error if highExclusive is nil.

func (*NaturalNumbers) ScalarStructure 

func (*NaturalNumbers) ScalarStructure() algebra.Structure[*Nat]

ScalarStructure returns the regular semi-module structure of NaturalNumbers.

func (*NaturalNumbers) Zero 

func (*NaturalNumbers) Zero() *Nat

Zero returns the additive identity element of the NaturalNumbers structure.

type PositiveNaturalNumbers 

type PositiveNaturalNumbers struct{}

PositiveNaturalNumbers represents the set of positive natural numbers (N\{0}).

func NPlus 

func NPlus() *PositiveNaturalNumbers

NPlus returns the singleton instance of PositiveNaturalNumbers.

func (*PositiveNaturalNumbers) Bottom 

func (nps *PositiveNaturalNumbers) Bottom() *NatPlus

Bottom returns the smallest element of PositiveNaturalNumbers, which is 1.

func (*PositiveNaturalNumbers) Characteristic 

func (*PositiveNaturalNumbers) Characteristic() cardinal.Cardinal

Characteristic returns the characteristic of PositiveNaturalNumbers, which is 0.

func (*PositiveNaturalNumbers) ElementSize 

func (*PositiveNaturalNumbers) ElementSize() int

ElementSize returns -1 indicating that NatPlus does not have a fixed element size.

func (*PositiveNaturalNumbers) FromBig 

func (nps *PositiveNaturalNumbers) FromBig(b *big.Int) (*NatPlus, error)

FromBig creates a NatPlus from the given big.Int, returning an error if the integer is nil or not positive.

func (*PositiveNaturalNumbers) FromBytes 

func (*PositiveNaturalNumbers) FromBytes(input []byte) (*NatPlus, error)

FromBytes creates a NatPlus from the given big-endian byte slice, returning an error if the input is empty or represents zero.

func (*PositiveNaturalNumbers) FromBytesBE 

func (nps *PositiveNaturalNumbers) FromBytesBE(input []byte) (*NatPlus, error)

FromBytesBE creates a NatPlus from the given big-endian byte slice, returning an error if the input is empty or represents zero.

func (*PositiveNaturalNumbers) FromCardinal 

func (*PositiveNaturalNumbers) FromCardinal(c algebra.Cardinal) (*NatPlus, error)

FromCardinal creates a NatPlus from the given cardinal, returning an error if the cardinal is zero.

func (*PositiveNaturalNumbers) FromInt 

func (*PositiveNaturalNumbers) FromInt(value *Int) (*NatPlus, error)

FromInt creates a NatPlus from the given Int, returning an error if the Int is nil, zero, or negative.

func (*PositiveNaturalNumbers) FromModulusCT 

func (*PositiveNaturalNumbers) FromModulusCT(m *numct.Modulus) *NatPlus

FromModulusCT creates a NatPlus from the given numct.Modulus.

func (*PositiveNaturalNumbers) FromNat 

func (*PositiveNaturalNumbers) FromNat(value *Nat) (*NatPlus, error)

FromNat creates a NatPlus from the given Nat, returning an error if the Nat is nil or zero.

func (*PositiveNaturalNumbers) FromNatCT 

func (*PositiveNaturalNumbers) FromNatCT(value *numct.Nat) (*NatPlus, error)

FromNatCT creates a NatPlus from the given numct.Nat, returning an error if the value is nil or zero.

func (*PositiveNaturalNumbers) FromRat 

func (nps *PositiveNaturalNumbers) FromRat(v *Rat) (*NatPlus, error)

FromRat creates a NatPlus from the given Rat, returning an error if the Rat is not a positive integer.

func (*PositiveNaturalNumbers) FromUint64 

func (*PositiveNaturalNumbers) FromUint64(value uint64) (*NatPlus, error)

FromUint64 creates a NatPlus from the given uint64, returning an error if the value is zero.

func (*PositiveNaturalNumbers) Name 

func (*PositiveNaturalNumbers) Name() string

Name returns the name of the structure: "N\{0}".

func (*PositiveNaturalNumbers) One 

func (*PositiveNaturalNumbers) One() *NatPlus

One returns the multiplicative identity element of PositiveNaturalNumbers, which is 1.

func (*PositiveNaturalNumbers) OpIdentity 

func (nps *PositiveNaturalNumbers) OpIdentity() *NatPlus

OpIdentity returns the multiplicative identity element of PositiveNaturalNumbers, which is 1. Note that this OpIdentity isn't standard, as it considers (N\{0}, *, +) to be a hemi ring, NOT the usual (N\{0}, +, *).

func (*PositiveNaturalNumbers) Order 

func (*PositiveNaturalNumbers) Order() cardinal.Cardinal

Order returns the order of PositiveNaturalNumbers, which is infinite.

func (*PositiveNaturalNumbers) Random 

func (nps *PositiveNaturalNumbers) Random(lowInclusive, highExclusive *NatPlus, prng io.Reader) (*NatPlus, error)

Random generates a random NatPlus in the range [lowInclusive, highExclusive), returning an error if highExclusive is nil.

type Rat 

type Rat struct {
	// contains filtered or unexported fields
}

Rat represents an element of the field of rational numbers Q.

func (*Rat) Add 

func (r *Rat) Add(rhs *Rat) *Rat

Add performs addition of two Rat elements.

func (*Rat) Big 

func (r *Rat) Big() *big.Rat

func (*Rat) Bytes 

func (r *Rat) Bytes() []byte

Bytes serialises the Rat element to a byte slice.

func (*Rat) Canonical 

func (r *Rat) Canonical() *Rat

Canonical returns the canonical form of the Rat element.

func (*Rat) Ceil 

func (r *Rat) Ceil() (*Int, error)

Ceil returns the smallest integer greater than or equal to the Rat element.

func (*Rat) Clone 

func (r *Rat) Clone() *Rat

Clone creates a deep copy of the Rat element.

func (*Rat) Denominator 

func (r *Rat) Denominator() *NatPlus

Denominator returns the denominator of the Rat element.

func (*Rat) Double 

func (r *Rat) Double() *Rat

Double returns the result of adding the Rat element to itself.

func (*Rat) Equal 

func (r *Rat) Equal(rhs *Rat) bool

Equal checks if the Rat element is equal to another Rat element.

func (*Rat) EuclideanDiv 

func (r *Rat) EuclideanDiv(rhs *Rat) (quo, rem *Rat, err error)

EuclideanDiv performs Euclidean division of two Rat elements.

func (*Rat) EuclideanValuation 

func (r *Rat) EuclideanValuation() cardinal.Cardinal

EuclideanValuation returns the Euclidean valuation of the Rat element.

func (*Rat) Floor 

func (r *Rat) Floor() (*Int, error)

Floor returns the largest integer less than or equal to the Rat element.

func (*Rat) HashCode 

func (r *Rat) HashCode() base.HashCode

HashCode computes the hash code of the Rat element.

func (*Rat) IsInt 

func (r *Rat) IsInt() bool

IsInt checks if the Rat element is an integer.

func (*Rat) IsLessThanOrEqual 

func (r *Rat) IsLessThanOrEqual(rhs *Rat) bool

IsLessThanOrEqual checks if the Rat element is less than another Rat element.

func (*Rat) IsNegative 

func (r *Rat) IsNegative() bool

IsNegative checks if the Rat element is negative.

func (*Rat) IsOne 

func (r *Rat) IsOne() bool

IsOne checks if the Rat element is one.

func (*Rat) IsOpIdentity 

func (r *Rat) IsOpIdentity() bool

IsOpIdentity checks if the Rat element is the additive identity (zero).

func (*Rat) IsPositive 

func (r *Rat) IsPositive() bool

IsPositive checks if the Rat element is positive.

func (*Rat) IsProbablyPrime 

func (r *Rat) IsProbablyPrime() bool

IsProbablyPrime checks if the Rat element is probably prime.

func (*Rat) IsZero 

func (r *Rat) IsZero() bool

IsZero checks if the Rat element is zero.

func (*Rat) MarshalCBOR 

func (r *Rat) MarshalCBOR() ([]byte, error)

func (*Rat) Mul 

func (r *Rat) Mul(rhs *Rat) *Rat

Mul performs multiplication of two Rat elements.

func (*Rat) Neg 

func (r *Rat) Neg() *Rat

Neg returns the additive inverse of the Rat element.

func (*Rat) Numerator 

func (r *Rat) Numerator() *Int

Numerator returns the numerator of the Rat element.

func (*Rat) Op 

func (r *Rat) Op(rhs *Rat) *Rat

Op performs addition of two Rat elements.

func (*Rat) OpInv 

func (r *Rat) OpInv() *Rat

OpInv returns the additive inverse of the Rat element.

func (*Rat) OtherOp 

func (r *Rat) OtherOp(rhs *Rat) *Rat

OtherOp performs multiplication of two Rat elements.

func (*Rat) Square 

func (r *Rat) Square() *Rat

Square returns the square of the Rat element.

func (*Rat) String 

func (r *Rat) String() string

String returns the string representation of the Rat element.

func (*Rat) Structure 

func (*Rat) Structure() algebra.Structure[*Rat]

Structure returns the algebraic structure to which the Rat element belongs.

func (*Rat) Sub 

func (r *Rat) Sub(rhs *Rat) *Rat

Sub performs subtraction of two Rat elements.

func (*Rat) TryDiv 

func (r *Rat) TryDiv(rhs *Rat) (*Rat, error)

TryDiv performs division of two Rat elements.

func (*Rat) TryInv 

func (r *Rat) TryInv() (*Rat, error)

TryInv returns the multiplicative inverse of the Rat element.

func (*Rat) TryNeg 

func (r *Rat) TryNeg() (*Rat, error)

TryNeg returns the additive inverse of the Rat element.

func (*Rat) TryOpInv 

func (r *Rat) TryOpInv() (*Rat, error)

TryOpInv returns the additive inverse of the Rat element.

func (*Rat) TrySub 

func (r *Rat) TrySub(rhs *Rat) (*Rat, error)

TrySub performs subtraction of two Rat elements.

func (*Rat) UnmarshalCBOR 

func (r *Rat) UnmarshalCBOR(data []byte) error

type Rationals 

type Rationals struct{}

Rationals represents the field of rational numbers Q.

func Q 

func Q() *Rationals

Q returns the singleton instance of the Rationals structure.

func (*Rationals) Characteristic 

func (*Rationals) Characteristic() algebra.Cardinal

Characteristic returns the characteristic of the field Q, which is 0.

func (*Rationals) ElementSize 

func (*Rationals) ElementSize() int

ElementSize returns -1 to indicate that elements of Q do not have a fixed size.

func (*Rationals) ExtensionDegree 

func (*Rationals) ExtensionDegree() uint

ExtensionDegree returns the extension degree of Q over itself, which is 1.

func (*Rationals) FromBig 

func (*Rationals) FromBig(n *big.Int) (*Rat, error)

FromBig creates a *Rat element from a *big.Int value.

func (*Rationals) FromBigRat 

func (*Rationals) FromBigRat(n *big.Rat) (*Rat, error)

FromBigRat creates a *Rat element from a *big.Rat value.

func (*Rationals) FromBytes 

func (*Rationals) FromBytes(data []byte) (*Rat, error)

FromBytes deserializes a Rat element from the given byte slice.

func (*Rationals) FromInt 

func (*Rationals) FromInt(n *Int) (*Rat, error)

FromInt creates a Rat element from an Int value.

func (*Rationals) FromInt64 

func (*Rationals) FromInt64(n int64) *Rat

FromInt64 creates a Rat element from an int64 value.

func (*Rationals) FromNat 

func (*Rationals) FromNat(n *Nat) (*Rat, error)

FromNat creates a Rat element from a Nat value.

func (*Rationals) FromNatPlus 

func (*Rationals) FromNatPlus(n *NatPlus) (*Rat, error)

FromNatPlus creates a Rat element from a NatPlus value.

func (*Rationals) FromUint 

func (*Rationals) FromUint(n *Uint) (*Rat, error)

FromUint creates a Rat element from a Uint value.

func (*Rationals) FromUint64 

func (*Rationals) FromUint64(n uint64) *Rat

FromUint64 creates a Rat element from a uint64 value.

func (*Rationals) IsDomain 

func (*Rationals) IsDomain() bool

IsDomain indicates that Q is a domain.

func (*Rationals) Name 

func (*Rationals) Name() string

Name returns the name of the structure.

func (*Rationals) New 

func (*Rationals) New(a *Int, b *NatPlus) (*Rat, error)

New creates a new Rat element with the given numerator and denominator.

func (*Rationals) One 

func (*Rationals) One() *Rat

One returns the multiplicative identity element of Q.

func (*Rationals) OpIdentity 

func (q *Rationals) OpIdentity() *Rat

OpIdentity returns the additive identity element of Q.

func (*Rationals) Order 

func (*Rationals) Order() algebra.Cardinal

Order returns the order of the field Q, which is infinite.

func (*Rationals) Random 

func (*Rationals) Random(lowInclusive, highExclusive *Rat, prng io.Reader) (*Rat, error)

Random samples a random *Rat element in the interval [lowInclusive, highExclusive).

func (*Rationals) RandomInt 

func (*Rationals) RandomInt(lowInclusive, highExclusive *Rat, prng io.Reader) (*Int, error)

RandomInt samples a random integer *Int element in the interval [lowInclusive, highExclusive).

The valid integers are those n satisfying lowInclusive <= n < highExclusive, which is equivalent to the half-open integer interval [ceil(lowInclusive), ceil(highExclusive)).

Returns ErrOutOfRange if the interval contains no integers.

func (*Rationals) Zero 

func (*Rationals) Zero() *Rat

Zero returns the zero element of Q.

type Uint 

type Uint struct {
	// contains filtered or unexported fields
}

Uint represents an integer modulo n.

func NewUintGivenModulus 

func NewUintGivenModulus(value *numct.Nat, m *numct.Modulus) (*Uint, error)

NewUintGivenModulus creates a new Uint element given a value Nat and a modulus Modulus.

func (*Uint) Abs 

func (u *Uint) Abs() *Nat

Abs returns the absolute value of the Uint element as a Nat.

func (*Uint) Add 

func (u *Uint) Add(other *Uint) *Uint

Add performs addition of two Uint elements.

func (*Uint) AnnouncedLen 

func (u *Uint) AnnouncedLen() int

AnnouncedLen returns the announced length in bytes of the Uint element.

func (*Uint) Big 

func (u *Uint) Big() *big.Int

Big returns the big.Int representation of the Uint element.

func (*Uint) Bit 

func (u *Uint) Bit(i uint) byte

Bit returns the i-th bit of the Uint element.

func (*Uint) Bytes 

func (u *Uint) Bytes() []byte

Bytes returns the byte slice representation of the Uint element.

func (*Uint) BytesBE 

func (u *Uint) BytesBE() []byte

BytesBE returns the big-endian byte slice representation of the Uint element.

func (*Uint) Cardinal 

func (u *Uint) Cardinal() cardinal.Cardinal

Cardinal returns the cardinality of the Uint element.

func (*Uint) Clone 

func (u *Uint) Clone() *Uint

Clone creates a copy of the Uint element.

func (*Uint) Compare 

func (u *Uint) Compare(other *Uint) base.Ordering

Compare performs a total comparison between two Uint elements.

func (*Uint) CondAssign 

func (u *Uint) CondAssign(choice ct.Choice, x *Uint)

CondAssign conditionally assigns the value of x to the Uint element if choice is true.

func (*Uint) Coprime 

func (u *Uint) Coprime(other *Uint) bool

Coprime checks if the Uint element is coprime to another Uint element.

func (*Uint) Decrement 

func (u *Uint) Decrement() *Uint

Decrement decrements the Uint element by one.

func (*Uint) Double 

func (u *Uint) Double() *Uint

Double returns the result of adding the Uint element to itself.

func (*Uint) Equal 

func (u *Uint) Equal(other *Uint) bool

Equal checks if two Uint elements are equal.

func (*Uint) EqualModulus 

func (u *Uint) EqualModulus(other *Uint) bool

EqualModulus checks if two Uint elements have the same modulus.

func (*Uint) EuclideanDiv 

func (u *Uint) EuclideanDiv(other *Uint) (quot, rem *Uint, err error)

EuclideanDiv performs Euclidean division of the Uint element by another Uint element.

func (*Uint) EuclideanValuation 

func (u *Uint) EuclideanValuation() algebra.Cardinal

EuclideanValuation returns the Euclidean valuation of the Uint element.

func (*Uint) Exp 

func (u *Uint) Exp(exponent *Nat) *Uint

Exp performs exponentiation of the Uint element by a Nat exponent.

func (*Uint) ExpBounded 

func (u *Uint) ExpBounded(exponent *Nat, bits uint) *Uint

ExpBounded performs exponentiation of the Uint element by a Nat exponent, using only the lower 'bits' bits of the exponent.

func (*Uint) ExpI 

func (u *Uint) ExpI(exponent *Int) *Uint

ExpI performs exponentiation of the Uint element by an Int exponent.

func (*Uint) ExpIBounded 

func (u *Uint) ExpIBounded(exponent *Int, bits uint) *Uint

ExpIBounded performs exponentiation of the Uint element by an Int exponent, using only the lower 'bits' bits of the exponent.

func (*Uint) Group 

func (u *Uint) Group() *ZMod

Group returns the ZMod structure that this Uint belongs to.

func (*Uint) HashCode 

func (u *Uint) HashCode() base.HashCode

HashCode returns a hash code for the Uint element.

func (*Uint) Increment 

func (u *Uint) Increment() *Uint

Increment increments the Uint element by one.

func (*Uint) IsBottom 

func (u *Uint) IsBottom() bool

IsBottom checks if the Uint element is the bottom element.

func (*Uint) IsEven 

func (u *Uint) IsEven() bool

IsEven checks if the Uint element is even.

func (*Uint) IsLessThanOrEqual 

func (u *Uint) IsLessThanOrEqual(other *Uint) bool

IsLessThanOrEqual checks if the Uint element is less than or equal to another Uint element.

func (*Uint) IsNegative 

func (u *Uint) IsNegative() bool

IsNegative checks the Uint would have been wrapped around if interpreted as an element of in [-n/2, n/2).

func (*Uint) IsOdd 

func (u *Uint) IsOdd() bool

IsOdd checks if the Uint element is odd.

func (*Uint) IsOne 

func (u *Uint) IsOne() bool

IsOne checks if the Uint element is one.

func (*Uint) IsOpIdentity 

func (u *Uint) IsOpIdentity() bool

IsOpIdentity checks if the Uint element is the additive identity.

func (*Uint) IsPositive 

func (u *Uint) IsPositive() bool

IsPositive checks if the Uint is non-zero.

func (*Uint) IsProbablyPrime 

func (u *Uint) IsProbablyPrime() bool

IsProbablyPrime checks if the Uint element is probably prime.

func (*Uint) IsQuadraticResidue 

func (u *Uint) IsQuadraticResidue() bool

IsQuadraticResidue checks if the Uint element is a quadratic residue modulo the modulus.

func (*Uint) IsTop 

func (u *Uint) IsTop() bool

IsTop checks if the Uint element is the top element.

func (*Uint) IsTorsionFree 

func (*Uint) IsTorsionFree() bool

IsTorsionFree checks if the Uint element is torsion-free.

func (*Uint) IsUnit 

func (u *Uint) IsUnit() bool

IsUnit checks if the Uint element is a unit (i.e., has a multiplicative inverse).

func (*Uint) IsZero 

func (u *Uint) IsZero() bool

IsZero checks if the Uint element is zero.

func (*Uint) Lift 

func (u *Uint) Lift() *Int

Lift lifts the Uint element to an Int element.

func (*Uint) Lsh 

func (u *Uint) Lsh(shift uint) *Uint

Lsh performs left shift on the Uint element. Lsh is equivalent to multiplying by 2^shift mod modulus.

func (*Uint) MarshalCBOR 

func (u *Uint) MarshalCBOR() ([]byte, error)

func (*Uint) Modulus 

func (u *Uint) Modulus() *NatPlus

Modulus returns the modulus NatPlus of the Uint element.

func (*Uint) ModulusCT 

func (u *Uint) ModulusCT() *numct.Modulus

ModulusCT returns the modulus Modulus of the Uint element.

func (*Uint) Mul 

func (u *Uint) Mul(other *Uint) *Uint

Mul performs multiplication of two Uint elements.

func (*Uint) Nat 

func (u *Uint) Nat() *Nat

Nat returns the Nat representation of the Uint element.

func (*Uint) Neg 

func (u *Uint) Neg() *Uint

Neg returns the additive inverse of the Uint element.

func (*Uint) Op 

func (u *Uint) Op(other *Uint) *Uint

Op performs the group operation (addition) on two Uint elements.

func (*Uint) OpInv 

func (u *Uint) OpInv() *Uint

OpInv returns the additive inverse of the Uint element.

func (*Uint) OtherOp 

func (u *Uint) OtherOp(other *Uint) *Uint

OtherOp performs the other group operation (multiplication) on two Uint elements.

func (*Uint) PartialCompare 

func (u *Uint) PartialCompare(other *Uint) base.PartialOrdering

PartialCompare performs a partial comparison between two Uint elements.

func (*Uint) Rsh 

func (u *Uint) Rsh(shift uint) *Uint

Rsh performs right shift on the Uint element. Rsh is equivalent to floor division by 2^shift, then mod modulus.

func (*Uint) ScalarExp 

func (u *Uint) ScalarExp(other *Nat) *Uint

ScalarExp performs exponentiation of the Uint element by a Nat scalar.

func (*Uint) ScalarMul 

func (u *Uint) ScalarMul(other *Nat) *Uint

ScalarMul performs scalar multiplication of the Uint element by a Nat scalar.

func (*Uint) ScalarOp 

func (u *Uint) ScalarOp(other *Nat) *Uint

ScalarOp performs scalar multiplication of the Uint element by a Nat scalar.

func (*Uint) Select 

func (u *Uint) Select(choice ct.Choice, x0, x1 *Uint)

Select sets the Uint element to x0 if choice is true, and to x1 if choice is false.

func (*Uint) Sqrt 

func (u *Uint) Sqrt() (*Uint, error)

Sqrt computes the square root of the Uint element if it exists.

func (*Uint) Square 

func (u *Uint) Square() *Uint

Square returns the result of multiplying the Uint element by itself.

func (*Uint) String 

func (u *Uint) String() string

String returns the string representation of the Uint element.

func (*Uint) Structure 

func (u *Uint) Structure() algebra.Structure[*Uint]

Structure returns the algebraic structure of the Uint.

func (*Uint) Sub 

func (u *Uint) Sub(other *Uint) *Uint

Sub performs subtraction of two Uint elements.

func (*Uint) TrueLen 

func (u *Uint) TrueLen() int

TrueLen returns the true length in bytes of the Uint element.

func (*Uint) TryDiv 

func (u *Uint) TryDiv(other *Uint) (*Uint, error)

TryDiv performs division of the Uint element by another Uint element.

func (*Uint) TryInv 

func (u *Uint) TryInv() (*Uint, error)

TryInv returns the multiplicative inverse of the Uint element.

func (*Uint) TryNeg 

func (u *Uint) TryNeg() (*Uint, error)

TryNeg returns the additive inverse of the Uint element.

func (*Uint) TryOpInv 

func (u *Uint) TryOpInv() (*Uint, error)

TryOpInv returns the additive inverse of the Uint element.

func (*Uint) TrySub 

func (u *Uint) TrySub(other *Uint) (*Uint, error)

TrySub performs subtraction of two Uint elements.

func (*Uint) UnmarshalCBOR 

func (u *Uint) UnmarshalCBOR(data []byte) error

func (*Uint) Value 

func (u *Uint) Value() *numct.Nat

Value returns the underlying numct.Nat value of the Uint.

type ZMod 

type ZMod struct {
	// contains filtered or unexported fields
}

ZMod represents the integers modulo n.

func NewZMod 

func NewZMod(modulus *NatPlus) (*ZMod, error)

NewZMod creates a new ZMod structure given a modulus NatPlus.

func NewZModFromCardinal 

func NewZModFromCardinal(n cardinal.Cardinal) (*ZMod, error)

NewZModFromCardinal creates a new ZMod structure given a cardinal.

func NewZModFromModulus 

func NewZModFromModulus(m *numct.Modulus) (*ZMod, error)

NewZModFromModulus creates a new ZMod structure given a modulus Modulus.

func (*ZMod) AmbientStructure 

func (*ZMod) AmbientStructure() algebra.Structure[*Int]

AmbientStructure returns the ambient structure of quotient group ie. Z.

func (*ZMod) Bottom 

func (zn *ZMod) Bottom() *Uint

Bottom returns the bottom element of the group.

func (*ZMod) Characteristic 

func (zn *ZMod) Characteristic() cardinal.Cardinal

Characteristic returns the characteristic of the group.

func (*ZMod) ElementSize 

func (zn *ZMod) ElementSize() int

ElementSize returns the size in bytes of an element.

func (*ZMod) FromBig 

func (zn *ZMod) FromBig(v *big.Int) (*Uint, error)

FromBig creates a Uint element from a big.Int value. It will reduce the value modulo the modulus of the ZMod.

func (*ZMod) FromBytes 

func (zn *ZMod) FromBytes(input []byte) (*Uint, error)

FromBytes creates a Uint element from a byte slice. It will NOT reduce the value modulo the modulus, and will return an error if the value is out of range.

func (*ZMod) FromBytesBE 

func (zn *ZMod) FromBytesBE(input []byte) (*Uint, error)

FromBytesBE creates a Uint element from a big-endian byte slice. It will NOT reduce the value modulo the modulus, and will return an error if the value is out of range.

func (*ZMod) FromBytesBEReduce 

func (zn *ZMod) FromBytesBEReduce(input []byte) (*Uint, error)

FromBytesBEReduce creates a Uint element from a big-endian byte slice, reducing it modulo the modulus. It will reduce the value modulo the modulus of the ZMod.

func (*ZMod) FromCardinal 

func (zn *ZMod) FromCardinal(v cardinal.Cardinal) (*Uint, error)

FromCardinal creates a Uint element from a cardinal. It will NOT reduce the value modulo the modulus, and will return an error if the value is out of range.

func (*ZMod) FromInt 

func (zn *ZMod) FromInt(v *Int) (*Uint, error)

FromInt creates a Uint element from an Int value. It will reduce the Int modulo the modulus of the ZMod.

func (*ZMod) FromInt64 

func (zn *ZMod) FromInt64(value int64) (*Uint, error)

FromInt64 creates a Uint element from an int64 value.

func (*ZMod) FromNat 

func (zn *ZMod) FromNat(v *Nat) (*Uint, error)

FromNat creates a Uint element from a Nat value. It will reduce the value modulo the modulus of the ZMod.

func (*ZMod) FromNatCT 

func (zn *ZMod) FromNatCT(v *numct.Nat) (*Uint, error)

FromNatCT creates a Uint element from a numct.Nat value. It will reduce the value modulo the modulus.

func (*ZMod) FromNatCTReduced 

func (zn *ZMod) FromNatCTReduced(reducedV *numct.Nat) (*Uint, error)

FromNatCTReduced creates a Uint element from a reduced numct.Nat value. It will NOT reduce the value modulo the modulus, and will return an error if the value is out of range.

func (*ZMod) FromNatPlus 

func (zn *ZMod) FromNatPlus(v *NatPlus) (*Uint, error)

FromNatPlus creates a Uint element from a NatPlus value. It will reduce the value modulo the modulus of the ZMod.

func (*ZMod) FromRat 

func (zn *ZMod) FromRat(v *Rat) (*Uint, error)

FromRat creates a Uint element from a Rat value. It will reduce the Rat modulo the modulus of the ZMod.

func (*ZMod) FromUint64 

func (zn *ZMod) FromUint64(value uint64) *Uint

FromUint64 creates a Uint element from a uint64 value.

func (*ZMod) Hash 

func (zn *ZMod) Hash(input []byte) (*Uint, error)

Hash hashes the input byte slice to an element of the group.

func (*ZMod) IsDomain 

func (zn *ZMod) IsDomain() bool

IsDomain checks if the group is a domain (i.e., if the modulus is probably prime).

func (*ZMod) IsInRange 

func (zn *ZMod) IsInRange(v *Nat) bool

IsInRange checks if a Nat value is in the range of the group.

func (*ZMod) MarshalCBOR 

func (z *ZMod) MarshalCBOR() ([]byte, error)

func (*ZMod) Modulus 

func (zn *ZMod) Modulus() *NatPlus

Modulus returns the modulus NatPlus of the group.

func (*ZMod) ModulusCT 

func (zn *ZMod) ModulusCT() *numct.Modulus

func (*ZMod) Name 

func (zn *ZMod) Name() string

Name returns the name of the structure.

func (*ZMod) One 

func (zn *ZMod) One() *Uint

One returns the one element of the group.

func (*ZMod) OpIdentity 

func (zn *ZMod) OpIdentity() *Uint

OpIdentity returns the additive identity element of the group.

func (*ZMod) Order 

func (zn *ZMod) Order() cardinal.Cardinal

Order returns the order of the group.

func (*ZMod) Random 

func (zn *ZMod) Random(prng io.Reader) (*Uint, error)

Random samples a random element from the group using the provided PRNG.

func (*ZMod) ScalarStructure 

func (*ZMod) ScalarStructure() algebra.Structure[*Nat]

ScalarStructure returns the scalar structure of the group.

func (*ZMod) Top 

func (zn *ZMod) Top() *Uint

Top returns the top element of the group.

func (*ZMod) UnmarshalCBOR 

func (z *ZMod) UnmarshalCBOR(data []byte) error

func (*ZMod) WideElementSize 

func (zn *ZMod) WideElementSize() int

WideElementSize returns the size in bytes of a wide element.

func (*ZMod) Zero 

func (zn *ZMod) Zero() *Uint

Zero returns the zero element of the group.

Source Files

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