#### CSCI 7000 - Cryptography - Spring 2010

### Problem Set #4

#### Due: May 3rd, 2010

1. 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*.
2. 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}.)

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

4. 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