Modular arithmetic (a*b/e)%m?

Question

Is there a modular expression possible for following :

((a*b)/e)%m

For ex :

(a*b*c)%m = ((a%m)*(b%m)*(c%m))%m

@user2040251 m is a prime but a*b may or may not be exactly divisible by e — Akash Rana, Nov 11 '14 at 18:52
@TobiLehman No i am asking if given an expression like above if i can caculate quickly by finding out the remainders obtained from m — Akash Rana, Nov 11 '14 at 18:53
`c = (1/e) (mod m)`, i.e., the multiplicative inverse of `(e)` modulo `(m)` : `c.e (mod m) = 1` — Brett Hale, Nov 11 '14 at 19:03
See http://en.wikipedia.org/wiki/Extended_Euclidean_algorithm#Computing_multiplicative_inverses_in_modular_structures for an algorithm for computing the multiplicative inverse. — rici, Nov 11 '14 at 19:21

user448810 · Accepted Answer · 2014-11-11T19:57:46.867

Instead of performing division when using modular arithmetic, you must multiply by the modular inverse. For instance, to divide by e, you would multiply by the modular inverse c where c × e ≡ 1 (mod m). You can calculate the modular inverse of x (mod m) by the following algorithm:

function inverse(x, m)
    a, b, u = 0, m, 1
    while x > 0
        q = b // x # integer division
        x, a, b, u = b % x, u, x, a - q * u
    if b == 1 return a % m
    error "must be coprime"

Thus, for your expression ((a*b)/e)%m you would compute (a * b * inverse(e,m)) % m.

score 1 · Answer 2 · answered Nov 11 '14 at 20:08

It depends on what you mean for division; for example for

a = 1
b = 1
e = 3
m = 7

what is the result you expect? 1/3 is 0 by conventional arithmetic, if it's that what you're looking for in this case then there is no shortcut in general (you can however trade the division for a multiplication and a shift if a*b is known to be small enough).

Another option for division x/y is however looking for the number that multiplied by y gives x in modular arithmetic (and 1/3 with this meaning is 5 because 5*3 is 1 when thinking modulo 7). If this is what you're looking for instead of dividing by e you can multiply by the modular inverse of e i.e. for the number that multiplied by e gives 1 (mod m).

Modular arithmetic (a*b/e)%m?

2 Answers2