lecture12b

lecture12b - CS 70 Spring 2005 Discrete Mathematics for CS...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
CS 70 Discrete Mathematics for CS Spring 2005 Clancy/Wagner Notes 12 RSA and the Chinese remainder theorem The Chinese remainder theorem Suppose we have a system of simultaneous equations, like maybe this one: x 2 ( mod 5 ) x 5 ( mod 7 ) What can we say about x ? Well, notice that one solution is x = 12; x = 12 satisfies both equations. This is not the only solution: for instance, x = 12 + 35 also works, as does x = 12 + 70, x = 12 + 105, and so on. Evidently adding any multiple of 35 to any solution gives another valid solution, so we might as well summarize this state of affairs by saying that x 12 ( mod 35 ) is one solution of the above system of equations. What about other solutions? For this example, there are no other solutions; every solution is of the form x 12 ( mod 35 ) . Why not? Well, suppose x and x 0 are two valid solutions. From the first equation, we know that x 2 ( mod 5 ) and x 0 2 ( mod 5 ) , so we must have x x 0 ( mod 5 ) . Similarly x x 0 ( mod 7 ) . But the former means that 5 is a divisor of x - x 0 , and the latter means that 7 is a divisor of x - x 0 , so x - x 0 must be a multiple of 35 (here we have used that gcd ( 5 , 7 ) = 1), which in turn means that x x 0 ( mod 35 ) . In other words, all solutions are the same modulo 35: or, equivalently, if all we care about is x mod 35, the solution is unique. You can check that the same would be true if we replaced the numbers 5 , 7 , 2 , 5 above by any others. The only thing we used is that gcd ( 5 , 7 ) = 1. Here is the generalization:
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 09/03/2011 for the course CS 70 taught by Professor Papadimitrou during the Fall '08 term at Berkeley.

Page1 / 3

lecture12b - CS 70 Spring 2005 Discrete Mathematics for CS...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online