# Module Z

    module Z: sig .. end

Integers.

This modules provides arbitrary-precision integers. Small integers internally use a regular OCaml `int`. When numbers grow too large, we switch transparently to GMP numbers (`mpn` numbers fully allocated on the OCaml heap).

This interface is rather similar to that of `Int32` and `Int64`, with some additional functions provided natively by GMP (GCD, square root, pop-count, etc.).

This file is part of the Zarith library http://forge.ocamlcore.org/projects/zarith . It is distributed under LGPL 2 licensing, with static linking exception. See the LICENSE file included in the distribution.

Copyright (c) 2010-2011 Antoine Miné, Abstraction project. Abstraction is part of the LIENS (Laboratoire d'Informatique de l'ENS), a joint laboratory by: CNRS (Centre national de la recherche scientifique, France), ENS (École normale supérieure, Paris, France), INRIA Rocquencourt (Institut national de recherche en informatique, France).

## Toplevel

For an optimal experience with the `ocaml` interactive toplevel, the magic commands are:

        #load "zarith.cma";;
        #install_printer Z.pp_print;;

Alternatively, using the new `Zarith_top` toplevel module, simply:

        #require "zarith.top";;

## Types

    type t

Type of integers of arbitrary length.

    exception Overflow

Raised by conversion functions when the value cannot be represented in the destination type.

## Construction

    val zero : t

The number 0.

    val one : t

The number 1.

    val minus_one : t

The number -1.

    val of_int : int -> t

Converts from a base integer.

    val of_int32 : int32 -> t

Converts from a 32-bit (signed) integer.

    val of_int64 : int64 -> t

Converts from a 64-bit (signed) integer.

    val of_nativeint : nativeint -> t

Converts from a native (signed) integer.

    val of_int32_unsigned : int32 -> t

Converts from a 32-bit integer, interpreted as an unsigned integer.

- Since 1.13

    val of_int64_unsigned : int64 -> t

Converts from a 64-bit integer, interpreted as an unsigned integer.

- Since 1.13

    val of_nativeint_unsigned : nativeint -> t

Converts from a native integer, interpreted as an unsigned integer..

- Since 1.13

    val of_float : float -> t

Converts from a floating-point value. The value is truncated (rounded towards zero). Raises `Overflow` on infinity and NaN arguments.

    val of_string : string -> t

Converts a string to an integer. An optional `-` prefix indicates a negative number, while a `+` prefix is ignored. An optional prefix `0x`, `0o`, or `0b` (following the optional `-` or `+` prefix) indicates that the number is, represented, in hexadecimal, octal, or binary, respectively. Otherwise, base 10 is assumed. (Unlike C, a lone `0` prefix does not denote octal.) Raises an `Invalid_argument` exception if the string is not a syntactically correct representation of an integer.

    val of_substring : string -> pos:int -> len:int -> t

`of_substring s ~pos ~len` is the same as `of_string (String.sub s pos len)`

- Since 1.4

    val of_string_base : int -> string -> t

Parses a number represented as a string in the specified base, with optional `-` or `+` prefix. The base must be between 2 and 16.

    val of_substring_base : int -> string -> pos:int -> len:int -> t

`of_substring_base base s ~pos ~len` is the same as `of_string_base base (String.sub s pos len)`

- Since 1.4

## Basic arithmetic operations

    val succ : t -> t

Returns its argument plus one.

    val pred : t -> t

Returns its argument minus one.

    val abs : t -> t

Absolute value.

    val neg : t -> t

Unary negation.

    val add : t -> t -> t

Addition.

    val sub : t -> t -> t

Subtraction.

    val mul : t -> t -> t

Multiplication.

    val div : t -> t -> t

Integer division. The result is truncated towards zero and obeys the rule of signs. Raises `Division_by_zero` if the divisor (second argument) is 0.

    val rem : t -> t -> t

Integer remainder. Can raise a `Division_by_zero`. The result of `rem a b` has the sign of `a`, and its absolute value is strictly smaller than the absolute value of `b`. The result satisfies the equality `a = b * div a b + rem a b`.

    val div_rem : t -> t -> t * t

Computes both the integer quotient and the remainder. `div_rem a b` is equal to `(div a b, rem a b)`. Raises `Division_by_zero` if `b = 0`.

    val cdiv : t -> t -> t

Integer division with rounding towards +oo (ceiling). Can raise a `Division_by_zero`.

    val fdiv : t -> t -> t

Integer division with rounding towards -oo (floor). Can raise a `Division_by_zero`.

    val ediv_rem : t -> t -> t * t

Euclidean division and remainder. `ediv_rem a b` returns a pair `(q, r)` such that `a = b * q + r` and `0 <= r < |b|`. Raises `Division_by_zero` if `b = 0`.

    val ediv : t -> t -> t

Euclidean division. `ediv a b` is equal to `fst (ediv_rem a b)`. The result satisfies `0 <= a - b * ediv a b < |b|`. Raises `Division_by_zero` if `b = 0`.

    val erem : t -> t -> t

Euclidean remainder. `erem a b` is equal to `snd (ediv_rem a b)`. The result satisfies `0 <= erem a b < |b|` and `a = b * ediv a b + erem a b`. Raises `Division_by_zero` if `b = 0`.

    val divexact : t -> t -> t

`divexact a b` divides `a` by `b`, only producing correct result when the division is exact, i.e., when `b` evenly divides `a`. It should be faster than general division. Can raise a `Division_by_zero`.

    val divisible : t -> t -> bool

`divisible a b` returns `true` if `a` is exactly divisible by `b`. Unlike the other division functions, `b = 0` is accepted (only 0 is considered divisible by 0).

- Since 1.10

    val congruent : t -> t -> t -> bool

`congruent a b c` returns `true` if `a` is congruent to `b` modulo `c`. Unlike the other division functions, `c = 0` is accepted (only equal numbers are considered equal congruent 0).

- Since 1.10

## Bit-level operations

For all bit-level operations, negative numbers are considered in 2's complement representation, starting with a virtual infinite number of 1s.

    val logand : t -> t -> t

Bitwise logical and.

    val logor : t -> t -> t

Bitwise logical or.

    val logxor : t -> t -> t

Bitwise logical exclusive or.

    val lognot : t -> t

Bitwise logical negation. The identity `lognot a`=`-a-1` always hold.

    val shift_left : t -> int -> t

Shifts to the left. Equivalent to a multiplication by a power of 2. The second argument must be nonnegative.

    val shift_right : t -> int -> t

Shifts to the right. This is an arithmetic shift, equivalent to a division by a power of 2 with rounding towards -oo. The second argument must be nonnegative.

    val shift_right_trunc : t -> int -> t

Shifts to the right, rounding towards 0. This is equivalent to a division by a power of 2, with truncation. The second argument must be nonnegative.

    val numbits : t -> int

Returns the number of significant bits in the given number. If `x` is zero, `numbits x` returns 0. Otherwise, `numbits x` returns a positive integer `n` such that `2^{n-1} <= |x| < 2^n`. Note that `numbits` is defined for negative arguments, and that `numbits (-x) = numbits x`.

- Since 1.4

    val trailing_zeros : t -> int

Returns the number of trailing 0 bits in the given number. If `x` is zero, `trailing_zeros x` returns `max_int`. Otherwise, `trailing_zeros x` returns a nonnegative integer `n` which is the largest `n` such that `2^n` divides `x` evenly. Note that `trailing_zeros` is defined for negative arguments, and that `trailing_zeros (-x) = trailing_zeros x`.

- Since 1.4

    val testbit : t -> int -> bool

`testbit x n` return the value of bit number `n` in `x`: `true` if the bit is 1, `false` if the bit is 0. Bits are numbered from 0. Raise `Invalid_argument` if `n` is negative.

- Since 1.4

    val popcount : t -> int

Counts the number of bits set. Raises `Overflow` for negative arguments, as those have an infinite number of bits set.

    val hamdist : t -> t -> int

Counts the number of different bits. Raises `Overflow` if the arguments have different signs (in which case the distance is infinite).

## Conversions

Note that, when converting to an integer type that cannot represent the converted value, an `Overflow` exception is raised.

    val to_int : t -> int

Converts to a signed OCaml `int`. Raises an `Overflow` if the value does not fit in a signed OCaml `int`.

    val to_int32 : t -> int32

Converts to a signed 32-bit integer `int32`. Raises an `Overflow` if the value does not fit in a signed `int32`.

    val to_int64 : t -> int64

Converts to a signed 64-bit integer `int64`. Raises an `Overflow` if the value does not fit in a signed `int64`.

    val to_nativeint : t -> nativeint

Converts to a native signed integer `nativeint`. Raises an `Overflow` if the value does not fit in a signed `nativeint`.

    val to_int32_unsigned : t -> int32

Converts to an unsigned 32-bit integer. The result is stored into an OCaml `int32`. Beware that most `Int32` operations consider `int32` to a signed type, not unsigned. Raises an `Overflow` if the value is negative or does not fit in an unsigned 32-bit integer.

- Since 1.13

    val to_int64_unsigned : t -> int64

Converts to an unsigned 64-bit integer. The result is stored into an OCaml `int64`. Beware that most `Int64` operations consider `int64` to a signed type, not unsigned. Raises an `Overflow` if the value is negative or does not fit in an unsigned 64-bit integer.

- Since 1.13

    val to_nativeint_unsigned : t -> nativeint

Converts to a native unsigned integer. The result is stored into an OCaml `nativeint`. Beware that most `Nativeint` operations consider `nativeint` to a signed type, not unsigned. Raises an `Overflow` if the value is negative or does not fit in an unsigned native integer.

- Since 1.13

    val to_float : t -> float

Converts to a floating-point value. This function rounds the given integer according to the current rounding mode of the processor. In default mode, it returns the floating-point number nearest to the given integer, breaking ties by rounding to even.

    val to_string : t -> string

Gives a human-readable, decimal string representation of the argument.

    val format : string -> t -> string

Gives a string representation of the argument in the specified printf-like format. The general specification has the following form:

`% [flags] [width] type`

Where the type actually indicates the base:

- `i`, `d`, `u`: decimal
- `b`: binary
- `o`: octal
- `x`: lowercase hexadecimal
- `X`: uppercase hexadecimal

Supported flags are:

- `+`: prefix positive numbers with a `+` sign
- space: prefix positive numbers with a space
- `-`: left-justify (default is right justification)
- `0`: pad with zeroes (instead of spaces)
- `#`: alternate formatting (actually, simply output a literal-like prefix: `0x`, `0b`, `0o`)

Unlike the classic `printf`, all numbers are signed (even hexadecimal ones), there is no precision field, and characters that are not part of the format are simply ignored (and not copied in the output).

    val fits_int : t -> bool

Whether the argument fits in an OCaml signed `int`.

    val fits_int32 : t -> bool

Whether the argument fits in a signed `int32`.

    val fits_int64 : t -> bool

Whether the argument fits in a signed `int64`.

    val fits_nativeint : t -> bool

Whether the argument fits in a signed `nativeint`.

    val fits_int32_unsigned : t -> bool

Whether the argument is non-negative and fits in an unsigned `int32`.

- Since 1.13

    val fits_int64_unsigned : t -> bool

Whether the argument is non-negative and fits in an unsigned `int64`.

- Since 1.13

    val fits_nativeint_unsigned : t -> bool

Whether the argument is non-negative fits in an unsigned `nativeint`.

- Since 1.13

## Printing

    val print : t -> unit

Prints the argument on the standard output.

    val output : Stdlib.out_channel -> t -> unit

Prints the argument on the specified channel. Also intended to be used as `%a` format printer in `Printf.printf`.

    val sprint : unit -> t -> string

To be used as `%a` format printer in `Printf.sprintf`.

    val bprint : Stdlib.Buffer.t -> t -> unit

To be used as `%a` format printer in `Printf.bprintf`.

    val pp_print : Stdlib.Format.formatter -> t -> unit

Prints the argument on the specified formatter. Can be used as `%a` format printer in `Format.printf` and as argument to `#install_printer` in the top-level.

## Ordering

    val compare : t -> t -> int

Comparison. `compare x y` returns 0 if `x` equals `y`, -1 if `x` is smaller than `y`, and 1 if `x` is greater than `y`.

Note that Pervasive.compare can be used to compare reliably two integers only on OCaml 3.12.1 and later versions.

    val equal : t -> t -> bool

Equality test.

    val leq : t -> t -> bool

Less than or equal.

    val geq : t -> t -> bool

Greater than or equal.

    val lt : t -> t -> bool

Less than (and not equal).

    val gt : t -> t -> bool

Greater than (and not equal).

    val sign : t -> int

Returns -1, 0, or 1 when the argument is respectively negative, null, or positive.

    val min : t -> t -> t

Returns the minimum of its arguments.

    val max : t -> t -> t

Returns the maximum of its arguments.

    val is_even : t -> bool

Returns true if the argument is even (divisible by 2), false if odd.

- Since 1.4

    val is_odd : t -> bool

Returns true if the argument is odd, false if even.

- Since 1.4

    val hash : t -> int

Hashes a number, producing a small integer. The result is consistent with equality: if `a` = `b`, then `hash a` = `hash b`. The result is the same as produced by OCaml's generic hash function, `Hashtbl.hash`. Together with type `Z.t`, the function `Z.hash` makes it possible to pass module `Z` as argument to the functor `Hashtbl.Make`.

- Before 1.14 a different hash algorithm was used.

    val seeded_hash : int -> t -> int

Like `Z.hash`, but takes a seed as extra argument for diversification. The result is the same as produced by OCaml's generic seeded hash function, `Hashtbl.seeded_hash`. Together with type `Z.t`, the function `Z.hash` makes it possible to pass module `Z` as argument to the functor `Hashtbl.MakeSeeded`.

- Since 1.14

## Elementary number theory

    val gcd : t -> t -> t

Greatest common divisor. The result is always nonnegative. We have `gcd(a,0) = gcd(0,a) = abs(a)`, including `gcd(0,0) = 0`.

    val gcdext : t -> t -> t * t * t

`gcdext u v` returns `(g,s,t)` where `g` is the greatest common divisor and `g=us+vt`. `g` is always nonnegative.

Note: the function is based on the GMP `mpn_gcdext` function. The exact choice of `s` and `t` such that `g=us+vt` is not specified, as it may vary from a version of GMP to another (it has changed notably in GMP 4.3.0 and 4.3.1).

    val lcm : t -> t -> t

Least common multiple. The result is always nonnegative. We have `lcm(a,0) = lcm(0,a) = 0`.

    val powm : t -> t -> t -> t

`powm base exp mod` computes `base`^`exp` modulo `mod`. Negative `exp` are supported, in which case (`base`^-1)^(-`exp`) modulo `mod` is computed. However, if `exp` is negative but `base` has no inverse modulo `mod`, then a `Division_by_zero` is raised.

    val powm_sec : t -> t -> t -> t

`powm_sec base exp mod` computes `base`^`exp` modulo `mod`. Unlike `Z.powm`, this function is designed to take the same time and have the same cache access patterns for any two same-size arguments. Used in cryptographic applications, it provides better resistance to side-channel attacks than `Z.powm`. The exponent `exp` must be positive, and the modulus `mod` must be odd. Otherwise, `Invalid_arg` is raised.

- Since 1.4

    val invert : t -> t -> t

`invert base mod` returns the inverse of `base` modulo `mod`. Raises a `Division_by_zero` if `base` is not invertible modulo `mod`.

    val probab_prime : t -> int -> int

`probab_prime x r` returns 0 if `x` is definitely composite, 1 if `x` is probably prime, and 2 if `x` is definitely prime. The `r` argument controls how many Miller-Rabin probabilistic primality tests are performed (5 to 10 is a reasonable value).

    val nextprime : t -> t

Returns the next prime greater than the argument. The result is only prime with very high probability.

    val jacobi : t -> t -> int

`jacobi a b` returns the Jacobi symbol `(a/b)`.

- Since 1.10

    val legendre : t -> t -> int

`legendre a b` returns the Legendre symbol `(a/b)`.

- Since 1.10

    val kronecker : t -> t -> int

`kronecker a b` returns the Kronecker symbol `(a/b)`.

- Since 1.10

    val remove : t -> t -> t * int

`remove a b` returns `a` after removing all the occurences of the factor `b`. Also returns how many occurrences were removed.

- Since 1.10

    val fac : int -> t

`fac n` returns the factorial of `n` (`n!`). Raises an `Invaid_argument` if `n` is non-positive.

- Since 1.10

    val fac2 : int -> t

`fac2 n` returns the double factorial of `n` (`n!!`). Raises an `Invaid_argument` if `n` is non-positive.

- Since 1.10

    val facM : int -> int -> t

`facM n m` returns the `m`-th factorial of `n`. Raises an `Invaid_argument` if `n` or `m` is non-positive.

- Since 1.10

    val primorial : int -> t

`primorial n` returns the product of all positive prime numbers less than or equal to `n`. Raises an `Invaid_argument` if `n` is non-positive.

- Since 1.10

    val bin : t -> int -> t

`bin n k` returns the binomial coefficient `n` over `k`. Raises an `Invaid_argument` if `k` is non-positive.

- Since 1.10

    val fib : int -> t

`fib n` returns the `n`-th Fibonacci number. Raises an `Invaid_argument` if `n` is non-positive.

- Since 1.10

    val lucnum : int -> t

`lucnum n` returns the `n`-th Lucas number. Raises an `Invaid_argument` if `n` is non-positive.

- Since 1.10

## Powers

    val pow : t -> int -> t

`pow base exp` raises `base` to the `exp` power. `exp` must be nonnegative. Note that only exponents fitting in a machine integer are supported, as larger exponents would surely make the result's size overflow the address space.

    val sqrt : t -> t

Returns the square root. The result is truncated (rounded down to an integer). Raises an `Invalid_argument` on negative arguments.

    val sqrt_rem : t -> t * t

Returns the square root truncated, and the remainder. Raises an `Invalid_argument` on negative arguments.

    val root : t -> int -> t

`root x n` computes the `n`-th root of `x`. `n` must be positive and, if `n` is even, then `x` must be nonnegative. Otherwise, an `Invalid_argument` is raised.

    val rootrem : t -> int -> t * t

`rootrem x n` computes the `n`-th root of `x` and the remainder `x-root**n`. `n` must be positive and, if `n` is even, then `x` must be nonnegative. Otherwise, an `Invalid_argument` is raised.

- Since 1.10

    val perfect_power : t -> bool

True if the argument has the form `a^b`, with `b>1`

    val perfect_square : t -> bool

True if the argument has the form `a^2`.

    val log2 : t -> int

Returns the base-2 logarithm of its argument, rounded down to an integer. If `x` is positive, `log2 x` returns the largest `n` such that `2^n <= x`. If `x` is negative or zero, `log2 x` raise the `Invalid_argument` exception.

- Since 1.4

    val log2up : t -> int

Returns the base-2 logarithm of its argument, rounded up to an integer. If `x` is positive, `log2up x` returns the smallest `n` such that `x <= 2^n`. If `x` is negative or zero, `log2up x` raise the `Invalid_argument` exception.

- Since 1.4

## Representation

    val size : t -> int

Returns the number of machine words used to represent the number.

    val extract : t -> int -> int -> t

`extract a off len` returns a nonnegative number corresponding to bits `off` to `off`+`len`-1 of `a`. Negative `a` are considered in infinite-length 2's complement representation. Raises an `Invalid_argument` if `off` is strictly negative, or if `len` is negative or null.

    val signed_extract : t -> int -> int -> t

`signed_extract a off len` extracts bits `off` to `off`+`len`-1 of `b`, as `extract` does, then sign-extends bit `len-1` of the result (that is, bit `off + len - 1` of `a`). The result is between `- 2{^[len]-1}` (included) and `2{^[len]-1}` (excluded), and equal to `extract a off len` modulo `2{^len}`. Raises an `Invalid_argument` if `off` is strictly negative, or if `len` is negative or null.

    val to_bits : t -> string

Returns a binary representation of the argument. The string result should be interpreted as a sequence of bytes, corresponding to the binary representation of the absolute value of the argument in little endian ordering. The sign is not stored in the string.

    val of_bits : string -> t

Constructs a number from a binary string representation. The string is interpreted as a sequence of bytes in little endian order, and the result is always positive. We have the identity: `of_bits (to_bits x) = abs x`. However, we can have `to_bits (of_bits s) <> s` due to the presence of trailing zeros in s.

## Pseudo-random number generation

    val random_int : ?rng:Stdlib.Random.State.t -> t -> t

`random_int bound` returns a random integer between 0 (inclusive) and `bound` (exclusive). `bound` must be greater than 0.

The source of randomness is the `Random` module from the OCaml standard library. The optional `rng` argument specifies which random state to use. If omitted, the default random state for the `Random` module is used.

Random numbers produced by this function are not cryptographically strong and must not be used in cryptographic or high-security contexts. See `Z.random_int_gen` for an alternative.

- Since 1.13

    val random_bits : ?rng:Stdlib.Random.State.t -> int -> t

`random_bits nbits` returns a random integer between 0 (inclusive) and `2{^nbits}` (exclusive). `nbits` must be nonnegative. This is a more efficient special case of `Z.random_int` when the bound is a power of two.

The source of randomness and the `rng` optional argument are as described in `Z.random_int`.

Random numbers produced by this function are not cryptographically strong and must not be used in cryptographic or high-security contexts. See `Z.random_bits_gen` for an alternative.

- Since 1.13

    val random_int_gen : fill:(bytes -> int -> int -> unit) -> t -> t

`random_int_gen ~fill bound` returns a random integer between 0 (inclusive) and `bound` (exclusive). `bound` must be greater than 0.

The `fill` parameter is the source of randomness. It is called as `fill buf pos len`, and is responsible for drawing `len` random bytes and writing them to offsets `pos` to `pos + len - 1` of the byte array `buf`.

Example of use where `/dev/random` provides the random bytes: << In_channel.with_open_bin "/dev/random" (fun ic -> Z.random_int_gen ~fill:(really_input ic) bound) >> Example of use where the Cryptokit library provides the random bytes: << Z.random_int_gen ~fill:Cryptokit.Random.secure_rng#bytes bound >>

- Since 1.13

    val random_bits_gen : fill:(bytes -> int -> int -> unit) -> int -> t

`random_bits_gen ~fill nbits` returns a random integer between 0 (inclusive) and `2{^nbits}` (exclusive). `nbits` must be nonnegative. This is a more efficient special case of `Z.random_int_gen` when the bound is a power of two. The `fill` parameter is as described in `Z.random_int_gen`.

- Since 1.13

## Prefix and infix operators

Classic (and less classic) prefix and infix `int` operators are redefined on `t`.

This makes it easy to typeset expressions. Using OCaml 3.12's local open, you can simply write `Z.(~$2 + ~$5 * ~$10)`.

    val (~-) : t -> t

Negation `neg`.

    val (~+) : t -> t

Identity.

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

Addition `add`.

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

Subtraction `sub`.

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

Multiplication `mul`.

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

Truncated division `div`.

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

Ceiling division `cdiv`.

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

Flooring division `fdiv`.

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

Exact division `divexact`.

    val (mod) : t -> t -> t

Remainder `rem`.

    val (land) : t -> t -> t

Bit-wise logical and `logand`.

    val (lor) : t -> t -> t

Bit-wise logical inclusive or `logor`.

    val (lxor) : t -> t -> t

Bit-wise logical exclusive or `logxor`.

    val (~!) : t -> t

Bit-wise logical negation `lognot`.

    val (lsl) : t -> int -> t

Bit-wise shift to the left `shift_left`.

    val (asr) : t -> int -> t

Bit-wise shift to the right `shift_right`.

    val (~$) : int -> t

Conversion from `int` `of_int`.

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

Power `pow`.

    module Compare: sig .. end

## Miscellaneous

    val version : string

Library version.

- Since 1.1