Module `Z`

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 : 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 : Buffer.t -> t -> unit

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

val pp_print : 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: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: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

package zarith

Module Z