#### CSCI 6454 - Advanced Algorithms - Spring 2014

### Homework #5

#### Due: May 5th, 2014

1. Calculate the following without using a calculator or computer.
(Feel free to use python to *verify* your answers, but you
must show your work to get any credit.)
- 4
^{1536} - 9^{4824} mod 35.
- 2
^{22012} mod 3.
- 5
^{30000}-6^{123456} mod 31.

2. In RSA, we know that factoring *n* allows you to recover
φ(*n*). Show it works the other way as well: given
φ(*n*), give an algorithm to efficiently recover the
factors of *n*. As a test that your method works, provide
the factors of *n* given below. Hand in your python source
along with the factors.

*n* = 1565548969872265465246036414997755958596184387096896694310755704575264720478601522962278179860838506157941

φ(*n*) = 1565548969872265465246036414997755958596184387096896588825919091033285362770995843274094868624380615603584

3. Here is one way in which RSA can be misused: suppose you have
*n = pq* for distinct large primes *p* and *q*,
and distinct encryption exponents e_{1}, e_{2} where
gcd(e_{1}, e_{2}) = 1. You publish *n* along
with e_{1}, e_{2}. Show that if an adversary has
*C*_{1} = M^{ e1} mod n and
*C*_{2} = M^{ e2} mod n,
then she can recover *M*. (Note: *M* is the same for
*C*_{1} and *C*_{2}.)

Use your solution to compute *M* for the parameters given below; hand in any code you use
to aid your computation. (You need not hand in code that was re-used above.)

*n* = 640434271860669796692811836922138143942513719203565769421924022297363333847089887235971007435680486193657059

*e*_{1} = 65537

*e*_{2} = 65539

*M*^{ e1} mod n = 400030256839145194441034228199292487980894977737102147552044462667917219509871638663296814615652770720888715

*M*^{ e2} mod n = 48384876797138828670281479166255073593234801358795810198774095180850824157124747742456773738763877257747936

5. Use Pollard's Rho algorithm to factor the number below. Please include your
source code with your solution.

$N =$ 121932632103337941464563328643500519

6. What irrational is represented by [1,3,3,3,3,...]? Show your work.

7. Let *n* be the product of distinct primes *p,q*
and let *e,d* be
inverses mod φ(*n*). Here *d* is less than
1/3 * *n*^{1/4} and *p* is between *q* and
*2q*. Factor *n* for the following values:

*n* = 151339355784268862864759120910925660361478568759617718849686393875476226860927728820162298798444560267328429
*e* = 47011760371307472336991404805847545822065047648754062545767018099994341894667675275660856278086265516059587