#### CSCI 7000 - Cryptanalysis - Spring 2005

### Problem Set #4

#### Due: Mar 29th, 2005 at 11am

**Remember, your homework MUST be submitted in LaTeX!**
Do not hand in homework electronically; print it out and bring it to class.

**Problem 1. (Difficulty: 5) **

Please produce a collision in MD4. You may not submit any collision that
is already known (eg, from Dobbertin's paper, from the Wang report).

Details were given in class on how you should do this.

**Problem 2. (Difficulty: 2) **

Factor the following composites (they're products of two primes):

7176413
1023799717
608501948407
881174755670089
1034763347438959
3554822081115323
6477520741950277
6543766558063627

Use two methods: (1) Trial division (use any speedups you like for this
step, but it must be trial division at its heart). (2) Pollard's Rho
method. Give timings for each method and conclude which method seems
faster. Can you factor 9681919867970786881 with Pollard's Rho method?
How long does it take? Compute how long you think it would take to
factor this number with trial division (ie, extrapolate from the
stuff you did above; don't try and actually factor this thing with
trial division).
**Problem 3. (Difficulty: 1) **

Factor the following composite, which is the product of two primes,
*p* and *q*.

10064294524865371791576617755742533411584130899146514841

(Hint: *p*-1 has no factors larger than 1000.)