1
Chapter 17: Computing kth roots Modulo m
Practice HW p. 121 # 1, 2, 3a, 4, 5(Additional Web Exercises)
In this chapter, we want to look at ways to solve the congruence
x k  b (mod m)
for x, that is, to find the kth roots modulo m.
Note: If m is larger, than substituting by brute force all possible incongruent solutions
x  0, 1, 2,, m  1 can take a very long time.
Method for Computing kth roots Modulo m
Let b, k, and m be integers where gcd( b, m )  1 and gcd( k ,  ( m))  1 . Then the
following steps will give a solution to the congruence
x k  b (mod m)
1. Find  (m) .
2. Find the multiplicative inverse u of k modulo  (m) , that is, we want to find an
integer u where
ku  1 (mod  ( m ))
(that is, u  k 1 (mod  (m))
To do this, use the Euclidean Algorithm to find integers u and v where
ku   ( m)v  1  gcd( k ,  (m)) .
Note that if u  0 , then u  u . if u  0 , then u  u (mod  ( m)) , that is, we take u to
be a positive value in the congruence class of u .
3. Compute x  bu (mod m) using u from step 2 by successive squaring to get the
solution for x.
2
Proof of Why Method Works:
3
Example 1: Solve x 205  127 (mod 1323)
Solution:
4
5
█
Note: gcd( k ,  ( m))  1 or gcd( b, m )  1 , the method may work only in special
circumstances (See Exercises 4 and 5 on p. 121)