Does some standard Python module contain a function to compute modular multiplicative inverse of a number, i.e. a number y = invmod(x, p) such that x*y == 1 (mod p)? Google doesn't seem to give any good hints on this.
Of course, one can come up with home-brewed 10-liner of extended Euclidean algorithm, but why reinvent the wheel.
For example, Java's BigInteger has modInverse method. Doesn't Python have something similar?
13 Answers
Python 3.8+
y = pow(x, -1, p)Python 3.7 and earlier
Maybe someone will find this useful (from wikibooks):
def egcd(a, b): if a == 0: return (b, 0, 1) else: g, y, x = egcd(b % a, a) return (g, x - (b // a) * y, y)
def modinv(a, m): g, x, y = egcd(a, m) if g != 1: raise Exception('modular inverse does not exist') else: return x % mIf your modulus is prime (you call it p) then you may simply compute:
y = x**(p-2) mod p # PseudocodeOr in Python proper:
y = pow(x, p-2, p)Here is someone who has implemented some number theory capabilities in Python:
Here is an example done at the prompt:
m = 1000000007
x = 1234567
y = pow(x,m-2,m)
y
989145189L
x*y
1221166008548163L
x*y % m
1LYou might also want to look at the gmpy module. It is an interface between Python and the GMP multiple-precision library. gmpy provides an invert function that does exactly what you need:
>>> import gmpy
>>> gmpy.invert(1234567, 1000000007)
mpz(989145189)Updated answer
As noted by @hyh , the gmpy.invert() returns 0 if the inverse does not exist. That matches the behavior of GMP's mpz_invert() function. gmpy.divm(a, b, m) provides a general solution to a=bx (mod m).
>>> gmpy.divm(1, 1234567, 1000000007)
mpz(989145189)
>>> gmpy.divm(1, 0, 5)
Traceback (most recent call last): File "<stdin>", line 1, in <module>
ZeroDivisionError: not invertible
>>> gmpy.divm(1, 4, 8)
Traceback (most recent call last): File "<stdin>", line 1, in <module>
ZeroDivisionError: not invertible
>>> gmpy.divm(1, 4, 9)
mpz(7)divm() will return a solution when gcd(b,m) == 1 and raises an exception when the multiplicative inverse does not exist.
Disclaimer: I'm the current maintainer of the gmpy library.
Updated answer 2
gmpy2 now properly raises an exception when the inverse does not exists:
>>> import gmpy2
>>> gmpy2.invert(0,5)
Traceback (most recent call last): File "<stdin>", line 1, in <module>
ZeroDivisionError: invert() no inverse existsAs of 3.8 pythons pow() function can take a modulus and a negative integer. See here. Their case for how to use it is
>>> pow(38, -1, 97)
23
>>> 23 * 38 % 97 == 1
TrueHere is a one-liner for CodeFights; it is one of the shortest solutions:
MMI = lambda A, n,s=1,t=0,N=0: (n < 2 and t%N or MMI(n, A%n, t, s-A//n*t, N or n),-1)[n<1]It will return -1 if A has no multiplicative inverse in n.
Usage:
MMI(23, 99) # returns 56
MMI(18, 24) # return -1The solution uses the Extended Euclidean Algorithm.
Sympy, a python module for symbolic mathematics, has a built-in modular inverse function if you don't want to implement your own (or if you're using Sympy already):
from sympy import mod_inverse
mod_inverse(11, 35) # returns 16
mod_inverse(15, 35) # raises ValueError: 'inverse of 15 (mod 35) does not exist'This doesn't seem to be documented on the Sympy website, but here's the docstring: Sympy mod_inverse docstring on Github
1Here is a concise 1-liner that does it, without using any external libraries.
# Given 0<a<b, returns the unique c such that 0<c<b and a*c == gcd(a,b) (mod b).
# In particular, if a,b are relatively prime, returns the inverse of a modulo b.
def invmod(a,b): return 0 if a==0 else 1 if b%a==0 else b - invmod(b%a,a)*b//aNote that this is really just egcd, streamlined to return only the single coefficient of interest.
I try different solutions from this thread and in the end I use this one:
def egcd(a, b): lastremainder, remainder = abs(a), abs(b) x, lastx, y, lasty = 0, 1, 1, 0 while remainder: lastremainder, (quotient, remainder) = remainder, divmod(lastremainder, remainder) x, lastx = lastx - quotient*x, x y, lasty = lasty - quotient*y, y return lastremainder, lastx * (-1 if a < 0 else 1), lasty * (-1 if b < 0 else 1)
def modinv(a, m): g, x, y = self.egcd(a, m) if g != 1: raise ValueError('modinv for {} does not exist'.format(a)) return x % mHere is my code, it might be sloppy but it seems to work for me anyway.
# a is the number you want the inverse for
# b is the modulus
def mod_inverse(a, b): r = -1 B = b A = a eq_set = [] full_set = [] mod_set = [] #euclid's algorithm while r!=1 and r!=0: r = b%a q = b//a eq_set = [r, b, a, q*-1] b = a a = r full_set.append(eq_set) for i in range(0, 4): mod_set.append(full_set[-1][i]) mod_set.insert(2, 1) counter = 0 #extended euclid's algorithm for i in range(1, len(full_set)): if counter%2 == 0: mod_set[2] = full_set[-1*(i+1)][3]*mod_set[4]+mod_set[2] mod_set[3] = full_set[-1*(i+1)][1] elif counter%2 != 0: mod_set[4] = full_set[-1*(i+1)][3]*mod_set[2]+mod_set[4] mod_set[1] = full_set[-1*(i+1)][1] counter += 1 if mod_set[3] == B: return mod_set[2]%B return mod_set[4]%BThe code above will not run in python3 and is less efficient compared to the GCD variants. However, this code is very transparent. It triggered me to create a more compact version:
def imod(a, n): c = 1 while (c % a > 0): c += n return c // afrom the cpython implementation source code:
def invmod(a, n): b, c = 1, 0 while n: q, r = divmod(a, n) a, b, c, n = n, c, b - q*c, r # at this point a is the gcd of the original inputs if a == 1: return b raise ValueError("Not invertible")according to the comment above this code, it can return small negative values, so you could potentially check if negative and add n when negative before returning b.
2To figure out the modular multiplicative inverse I recommend using the Extended Euclidean Algorithm like this:
def multiplicative_inverse(a, b): origA = a X = 0 prevX = 1 Y = 1 prevY = 0 while b != 0: temp = b quotient = a/b b = a%b a = temp temp = X a = prevX - quotient * X prevX = temp temp = Y Y = prevY - quotient * Y prevY = temp return origA + prevYWell, I don't have a function in python but I have a function in C which you can easily convert to python, in the below c function extended euclidian algorithm is used to calculate inverse mod.
int imod(int a,int n){
int c,i=1;
while(1){ c = n * i + 1; if(c%a==0){ c = c/a; break; } i++;
}
return c;}Python Function
def imod(a,n): i=1 while True: c = n * i + 1; if(c%a==0): c = c/a break; i = i+1 return cReference to the above C function is taken from the following link C program to find Modular Multiplicative Inverse of two Relatively Prime Numbers
