package ff

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Signature

package ff

ff
- Ff
  - MakeFp
    
    S
  - MakeFp2
    
    Fp
    
    Intf

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Parameters

module S : sig ... end

Signature

include Ff_sig.PRIME

include Ff_sig.BASE

exception Not_in_field of Bytes.t

type t

val order : Z.t

The order of the finite field

val size_in_bytes : int

minimal number of bytes required to encode a value of the field.

val check_bytes : Bytes.t -> bool

check_bytes bs returns true if bs is a correct byte representation of a field element

val zero : t

The neutral element for the addition

val one : t

The neutral element for the multiplication

val is_zero : t -> bool

is_zero x returns true if x is the neutral element for the addition

val is_one : t -> bool

is_one x returns true if x is the neutral element for the multiplication

val random : ?state:Random.State.t -> unit -> t

Use carefully! random () returns a random element of the field. A state for the PRNG can be given to initialize the PRNG in the requested state. If no state is given, no initialisation is performed

val non_null_random : ?state:Random.State.t -> unit -> t

Use carefully! non_null_random () returns a non null random element of the field. A state for the PRNG can be given to initialize the PRNG in the requested state. If no state is given, no initialisation is performed

val add : t -> t -> t

add a b returns a + b mod order

val (+) : t -> t -> t

Infix operator for add

val sub : t -> t -> t

sub a b returns a - b mod order

val mul : t -> t -> t

mul a b returns a * b mod order

val (*) : t -> t -> t

Infix operator for mul

val eq : t -> t -> bool

eq a b returns true if a = b mod order, else false

val (=) : t -> t -> bool

Infix operator for eq

val negate : t -> t

negate x returns -x mod order. Equivalently, negate x returns the unique y such that x + y mod order = 0

val (-) : t -> t

Infix operator for negate

val inverse_exn : t -> t

inverse_exn x returns x^-1 if x is not 0, else raise Division_by_zero

val inverse_opt : t -> t option

inverse_opt x returns x^-1 if x is not 0 as an option, else None

val div_exn : t -> t -> t

div_exn a b returns a * b^-1. Raise Division_by_zero if b = zero

val div_opt : t -> t -> t option

div_opt a b returns a * b^-1 as an option. Return None if b = zero

val (/) : t -> t -> t

Infix operator for div_exn

val square : t -> t

square x returns x^2

val double : t -> t

double x returns 2x

val pow : t -> Z.t -> t

pow x n returns x^n

val (**) : t -> Z.t -> t

Infix operator for pow

val of_bytes_exn : Bytes.t -> t

Construct a value of type t from the bytes representation in little endian of the field element. For non prime fields, the encoding starts with the coefficient of the constant monomial. Raise Not_in_field if the bytes do not represent an element in the field.

val of_bytes_opt : Bytes.t -> t option

From a predefined little endian bytes representation, construct a value of type t. The same representation than of_bytes_exn is used. Return None if the bytes do not represent an element in the field.

val to_bytes : t -> Bytes.t

Convert the value t to a bytes representation. The number of bytes is size_in_bytes and the encoding must be in little endian. For instance, the encoding of 1 in prime fields is always a bytes sequence of size size_in_bytes starting with the byte 0b00000001. For non prime fields, the encoding starts with the coefficient of the constant monomial.

val factor_power_of_two : int * Z.t

Returns s, q such that order - 1 = 2^s * q

val of_string : string -> t

Create a value t from a predefined string representation. It is not required that to_string of_string t = t. By default, decimal representation of the number is used, modulo the order of the field

val to_string : t -> string

String representation of a value t. It is not required that to_string of_string t = t. By default, decimal representation of the number is used

val of_z : Z.t -> t

of_z x builds an element t from the Zarith element x. mod order is applied if x >= order

val to_z : t -> Z.t

to_z x builds a Zarith element, using the decimal representation. Arithmetic on the result can be done using the modular functions on integers

val legendre_symbol : t -> Z.t

Returns the Legendre symbol of the parameter. Note it does not work for p = 2

val is_quadratic_residue : t -> bool

is_quadratic_residue x returns true if x is a quadratic residue i.e. if there exists n such that n^2 mod p = 1

val sqrt_opt : t -> t option

sqrt_opt x returns a square root of x

val get_nth_root_of_unity : Z.t -> t

Returns a nth root of unity

val is_nth_root_of_unity : Z.t -> t -> bool

is_nth_root_of_unity n x returns true if x is a nth-root of unity