package ff

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Module type Ff.PRIME_WITH_ROOT_OF_UNITYSource

Module type for prime field with additional functions to manipulate roots of unity

include PRIME
include BASE
Sourceexception Not_in_field of Bytes.t
Sourcetype t
Sourceval order : Z.t

The order of the finite field

Sourceval size_in_bytes : int

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

Sourceval zero : t

The neutral element for the addition

Sourceval one : t

The neutral element for the multiplication

Sourceval is_zero : t -> bool

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

Sourceval is_one : t -> bool

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

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

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

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

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

Sourceval add : t -> t -> t

add a b returns a + b mod order

Sourceval (+) : t -> t -> t

Infix operator for add

Sourceval mul : t -> t -> t

mul a b returns a * b mod order

Sourceval (*) : t -> t -> t

Infix operator for mul

Sourceval eq : t -> t -> bool

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

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

Infix operator for eq

Sourceval negate : t -> t

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

Sourceval (-) : t -> t

Infix operator for negate

Sourceval inverse_exn : t -> t

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

Sourceval inverse_opt : t -> t option

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

Sourceval div_exn : t -> t -> t

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

Sourceval div_opt : t -> t -> t option

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

Sourceval (/) : t -> t -> t

Infix operator for div_exn

Sourceval square : t -> t

square x returns x^2

Sourceval double : t -> t

double x returns 2x

Sourceval pow : t -> Z.t -> t

pow x n returns x^n

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

Infix operator for pow

Sourceval of_bytes_exn : Bytes.t -> t

From a predefined bytes representation, construct a value t. It is not required that to_bytes of_bytes_exn t = t. Raise Not_in_field if the bytes do not represent an element in the field.

Sourceval of_bytes_opt : Bytes.t -> t option

From a predefined bytes representation, construct a value t. It is not required that to_bytes (Option.get (of_bytes_opt t)) = t. By default, little endian encoding is used and the given element is modulo the prime order

Sourceval to_bytes : t -> Bytes.t

Convert the value t to a bytes representation which can be used for hashing for instance. It is not required that to_bytes of_bytes_exn t = t. By default, little endian encoding is used, and length of the resulting bytes may vary depending on the order.

Sourceval 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

Sourceval 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

Sourceval of_z : Z.t -> t

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

Sourceval 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 integer

Sourceval get_nth_root_of_unity : Z.t -> t

Returns a nth root of unity

Sourceval 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