Square and Multiply
Introducing RSA
Prof Hans Georg Schaathun
Høgskolen i Ålesund
Autumn 2013 – Crupto PK 4/4
Recorded: October 14, 2013
Prof Hans Georg Schaathun
Square and Multiply
Crypto PK 4/4
1/8
The RSA Cipher
ee,n (x) = x e
mod n
Problem
What is the algorithmic complexity of the encryption function?
Prof Hans Georg Schaathun
Square and Multiply
Crypto PK 4/4
2/8
The Naïve Approach
Two problems
1
Calculate x e = x · x · . . . · x
2
Reduce x e mod n
Prof Hans Georg Schaathun
Square and Multiply
Crypto PK 4/4
3/8
Square and Multiply
(x a )b = x ab ,
x a x b = x a+b .
Prof Hans Georg Schaathun
Square and Multiply
Crypto PK 4/4
4/8
Reduction after every step
xy mod n = (x mod n)(y mod n) mod n
(1)
To calculate x e mod n
1
Calculate y = x be/2c mod n
2
Square y and reduce mod n
3
If e is odd, then multiply y by x and reduce mod n
4
Return
Prof Hans Georg Schaathun
Square and Multiply
Crypto PK 4/4
5/8
1
2
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4
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6
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Recursive Pseudo-code
Input positive integers x, e, n
Return x e mod n
A l g o r i t h m S q u a r e N m u l t i p l y (x, e, n)
i f e = 1 , return x
y := S q u a r e N m u l t i p l y (x, be/nc , n)
y := y 2 mod n
i f e mod 2 = 1 ,
y := y · x mod n
return y
Prof Hans Georg Schaathun
Square and Multiply
Crypto PK 4/4
6/8
Example
311
Prof Hans Georg Schaathun
mod 13
Square and Multiply
Crypto PK 4/4
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Closure
Exercise
Write pseudo code for an iterative algorithm to calculate x a mod n
using square and multiply.
Prof Hans Georg Schaathun
Square and Multiply
Crypto PK 4/4
8/8