Chinese Reminder Theorem
- 1. Cryptography and Network Security
Jeevanantham Arumugam
Chinese Reminder Theorem
Used to solve a set of congruent equations with one variable but different moduli, which are
relatively prime
A @ a1 (mod m1)
A @ a2 (mod m2)
A @ a3 (mod m3)
.
.
.
A @ ak (mod mk)
Solution :
1. Find M = m1 * m2 * m3 * ……….* mk [This will be the common modulus]
2. Find M1 =
!
"#
, M2 =
!
"$
, ……….. , Mk =
!
"%
3. Find the multiplicative inverse of M1, M2,…….., Mk using the
corresponding moduli ( m1, m2,………, mk)
4. Answer = ( (a1 * M1 * M1
-1
) + (a2 * M2 * M2
-1
) + ….+ (ak * Mk * Mk
-1
) )
mod M
Example :
A @ 2 mod 3
A @ 3 mod 5
A @ 2 mod 7
Solution :
1. M = 3 * 5 * 7 = 105
2. M1 =
#&'
(
= 35
M2 =
#&'
'
= 21
M3 =
#&'
)
= 15
3. M1
-1
= 35-1
mod 3 = 2
M2
-1
= 21-1
mod 5 = 1
M3
-1
= 15-1
mod 7 = 1
4. Answer = (( 2*35*2) + ( 3*21*1) + (2*15*1)) mod 105
= 23 mod 105 = 23