More Number Theory and The RSA Cryptosystem

More Number Theory and The RSA Cryptosystem... Write function gcda,b in Python example... The order of a group element... The order of group elements• Example: Find the order of 3 and 2

More Number Theory

and The RSA Cryptosystem

More number Theory

Write function gcd(a,b)

in Python

example

Exercise

Write

gcdex(a, b, ’s’, ’t’)

in Python

example

Exercise

Multiplicative inverse

• Proof

Exercise

Write

a^(-1) mod b in

Python

Chinese Remainder Theorem

Write function

chrem([a1, …, ar],[m1, …, mr]) in Python

Exercise

The order of a group element

The order of group elements

• Example: Find the order of 3 and 2 in Z7*

• The order of an element g in Z7* must divides 6, ord(g) is in{1, 2, 3, 6}.

• The order of an element g in Z11* must divides 10, ord(g) is in{1, 2, 5, 10}.

Facts

Primitive element

Definition: An element having order p – 1 modulo p is

call a primitive element modulo p

Exercise

Public key cryptosystem

Symmetric key cryptosystem

Public key cryptosystem (PKC)

Remind: One-way function

The RSA cryptosystem

Exercise: show that (xb)a = x (mod n) if x in Zn\ Zn*.

Example

Cryptool it!

The trapdoor in RSA

RSA Parameter generation

• Fast exponentiation

Primality testing

A simple deterministic algorithm

Randomized Algorithm

• They can make random choices during their

execution

• Las Vegas algorithm:

– may fail to give an answer, but if the algorithm does return

an answer, then the answer must be correct.

• Mote Carlo algorithm:

Quadratic residue

Legendre symbol

Jacobi symbol

Example

Compute Jacobi symbol

The Solovay-Strassen Algorithm

The Solovay-Strassen Algorithm properties

• This is a yes-biased Monte Carlo algorithm with

error probability at most ½

• If we run the algorithm m times, the probability that such an integer n is prime is 1 – 2-m

such an integer n is prime is 1 – 2

Write isprime(a) in Python;

Factoring Algorithms

60 = 2 * 2 * 3* 5

The three most effective algorithms

1 The Quadratic Sieve

2 The Elliptic Curve Factoring Algorithm

3 The Number Field Sieve

write ifactor(n) in Python