ActiveState Code

Recipe 576737: Inverse modulo p

Very rarely it is necessary to find the multiplicative inverse of a number in the ring of integers modulo p. Thie recipe handles those rare cases. That is, given x, an integer, and p the modulus, we seek a integer x^-1 such that x * x^-1 = 1 mod p. For example 38 is the inverse of 8 modulo 101 since 38 * 8 = 304 = 1 mod 101. The inverse only exists when a and p are relatively prime.

Python 
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def invmodp(a, p):
    '''
    The multiplicitive inverse of a in the integers modulo p.
    Return b s.t.
    a * b == 1 mod p
    '''
    
    r = a
    d = 1
    for count in xrange(p):
        d = ((p // r + 1) * d) % p
        r = (d * a) % p
        if r == 1:
            break
    else:
        raise ValueError('%d has no inverse mod %d' % (a, p))
    return d
    
def __invmodp__test__():
    p = 101
    for i in range(1, p):
        iinv = invmodp(i, p)
        assert (iinv * i) % p == 1
__invmodp__test__()

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