Chinese Remainder Theorem

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How do you solve Chinese Remainder Theorem? How do you solve a system of congruence? How can CRT be used to speed up RSA decryption?

Chinese Remainder Theorem (C.R.T), given by Sun Zi, is one of the brilliant concepts in mathematics, it represents Horace's quote beautifully, omne tulit punctum qui miscuit utile dulci (he gains everyone’s approval who mixes the pleasant with the useful).

 

In number theory, the C.R.T. gives us the unique remainder from the simultaneous linear congruences by solving the congruences using the Euclidean division where the divisors are coprime.


Statement:

If we take t number of integers a1,a2,a3,…,at and m1,m2,m3,…,mt be coprime (i.e gcd of two numbers is 1) then there is a number x with the property that, 

x º ai (mod mi)    1 £ i £ t    .......(1)
The number x is unique in the following sense: 
Let M be the product of all the mi¢s, 
M = m1.m2.....mt 
and y satisfies the system of congruences (1). Then, 

y º x ( mod M)        

                                            

Proof:


Examples of C.R.T:


Applications of C.R.T: 


C.R.T has many applications and it will continue to show its use in various fields one of its major application is in computing it is also used in cryptography for encrypting data, GÖdel numbering for sequences (it is used to represent each finite sequence of natural numbers as one natural number.). The theory of codes and algorithms are its recent applications, it is also used in range ambiguity resolution and fast Fourier transform.

                                                

                  

 

 

 

 


 

  
Hey there, This is Hifza Siddiqui, an aspiring Mathematician pursuing masters from the University of Mumbai. I'm writing this blog to share the knowledge and the way I understand mathematics. I hope it helps you to get a better insight. My blog's frequency will be twice per week.

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