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bigint-mod-arith - v3.3.0

Some common functions for modular arithmetic using native JS implementation of BigInt

Table of contents

Interfaces

Type Aliases

Functions

Type Aliases

PrimeFactor

Ƭ PrimeFactor: number | bigint | PrimePower

Defined in

modPow.ts:8

PrimeFactorization

Ƭ PrimeFactorization: [bigint, bigint][]

Defined in

phi.ts:1

PrimePower

Ƭ PrimePower: [number | bigint, number | bigint]

Defined in

modPow.ts:7

Functions

abs

▸ abs(a): number | bigint

Absolute value. abs(a)==a if a>=0. abs(a)==-a if a<0

Parameters

Name Type
a number | bigint

Returns

number | bigint

The absolute value of a

Defined in

abs.ts:8

bitLength

▸ bitLength(a): number

Returns the (minimum) length of a number expressed in bits.

Parameters

Name Type
a number | bigint

Returns

number

The bit length

Defined in

bitLength.ts:7

crt

▸ crt(remainders, modulos, modulo?): bigint

Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer n by several integers, then one can determine uniquely the remainder of the division of n by the product of these integers, under the condition that the divisors are pairwise coprime (no two divisors share a common factor other than 1). Provided that n_i are pairwise coprime, and a_i any integers, this function returns a solution for the following system of equations: x ≡ a_1 mod n_1 x ≡ a_2 mod n_2 ⋮ x ≡ a_k mod n_k

Parameters

Name Type Description
remainders bigint[] the array of remainders a_i. For example [17n, 243n, 344n]
modulos bigint[] the array of modulos n_i. For example [769n, 2017n, 47701n]
modulo? bigint the product of all modulos. Provided here just to save some operations if it is already known

Returns

bigint

x

Defined in

crt.ts:16

eGcd

▸ eGcd(a, b): Egcd

An iterative implementation of the extended euclidean algorithm or extended greatest common divisor algorithm. Take positive integers a, b as input, and return a triple (g, x, y), such that ax + by = g = gcd(a, b).

Throws

RangeError if a or b are <= 0

Parameters

Name Type
a number | bigint
b number | bigint

Returns

Egcd

A triple (g, x, y), such that ax + by = g = gcd(a, b).

Defined in

egcd.ts:17

gcd

▸ gcd(a, b): bigint

Greatest common divisor of two integers based on the iterative binary algorithm.

Parameters

Name Type
a number | bigint
b number | bigint

Returns

bigint

The greatest common divisor of a and b

Defined in

gcd.ts:11

lcm

▸ lcm(a, b): bigint

The least common multiple computed as abs(a*b)/gcd(a,b)

Parameters

Name Type
a number | bigint
b number | bigint

Returns

bigint

The least common multiple of a and b

Defined in

lcm.ts:11

max

▸ max(a, b): number | bigint

Maximum. max(a,b)==a if a>=b. max(a,b)==b if a<b

Parameters

Name Type
a number | bigint
b number | bigint

Returns

number | bigint

Maximum of numbers a and b

Defined in

max.ts:9

min

▸ min(a, b): number | bigint

Minimum. min(a,b)==b if a>=b. min(a,b)==a if a<b

Parameters

Name Type
a number | bigint
b number | bigint

Returns

number | bigint

Minimum of numbers a and b

Defined in

min.ts:9

modAdd

▸ modAdd(addends, n): bigint

Modular addition of (a_1 + ... + a_r) mod n

Parameters

Name Type Description
addends (number | bigint)[] an array of the numbers a_i to add. For example [3, 12353251235n, 1243, -12341232545990n]
n number | bigint the modulo

Returns

bigint

The smallest positive integer that is congruent with (a_1 + ... + a_r) mod n

Defined in

modAdd.ts:9

modInv

▸ modInv(a, n): bigint

Modular inverse.

Throws

RangeError if a does not have inverse modulo n

Parameters

Name Type Description
a number | bigint The number to find an inverse for
n number | bigint The modulo

Returns

bigint

The inverse modulo n

Defined in

modInv.ts:14

modMultiply

▸ modMultiply(factors, n): bigint

Modular addition of (a_1 * ... * a_r) mod n *

Parameters

Name Type Description
factors (number | bigint)[] an array of the numbers a_i to multiply. For example [3, 12353251235n, 1243, -12341232545990n] *
n number | bigint the modulo *

Returns

bigint

The smallest positive integer that is congruent with (a_1 * ... * a_r) mod n

Defined in

modMultiply.ts:9

modPow

▸ modPow(b, e, n, primeFactorization?): bigint

Modular exponentiation b**e mod n. Currently using the right-to-left binary method if the prime factorization is not provided, or the chinese remainder theorem otherwise.

Throws

RangeError if n <= 0

Parameters

Name Type Description
b number | bigint base
e number | bigint exponent
n number | bigint modulo
primeFactorization? PrimeFactor[] an array of the prime factors, for example [5n, 5n, 13n, 27n], or prime powers as [p, k], for instance 5, 2], [13, 1], [27, 1. If the prime factorization is provided the chinese remainder theorem is used to greatly speed up the exponentiation.

Returns

bigint

b**e mod n

Defined in

modPow.ts:22

phi

▸ phi(primeFactorization): bigint

A function that computes the Euler's totien function of a number n, whose prime power factorization is known

Parameters

Name Type Description
primeFactorization PrimeFactorization an array of arrays containing the prime power factorization of a number n. For example, for n = (p1k1)*(p2k2)...(pr**kr), one should provide p1, k1], [p2, k2], ... , [pr, kr

Returns

bigint

phi((p1k1)*(p2k2)...(pr**kr))

Defined in

phi.ts:9

toZn

▸ toZn(a, n): bigint

Finds the smallest positive element that is congruent to a in modulo n

Remarks

a and b must be the same type, either number or bigint

Throws

RangeError if n <= 0

Parameters

Name Type Description
a number | bigint An integer
n number | bigint The modulo

Returns

bigint

A bigint with the smallest positive representation of a modulo n

Defined in

toZn.ts:14