#### CSCI 7000 - Cryptanalysis - Spring 2005

### Problem Set #5

#### Due: Apr 12th, 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: 1)**

Suppose you are in possession of a device such that if you give it a
positive integer n and an integer a in [1,n], you are told one solution
to x^{2} = a (mod n) if a solution exists, or told that no
solution exists if such is the case. If the equation has more than one
solution, the device picks one of these by some method unknown to you.
Assume that the device takes polynomial time; that is, the time it takes
to present its answer is bounded by a constant times a fixed power of
log(n). Show how, armed with such a device, one can factor via a
probabilistic algorithm with expected running time being polynomial.

**Problem 2. (Difficulty: 1)**
Let *n* be the product of distinct primes and let *e,d* be
inverses mod phi(*n*). Factor *n* for the following values:

n = 12167743697078823982285244561960404776234566754991007
e = 28258238181581995919207
d = 9711423862090244315080011600854726586677192251800743

**Problem 3. (Difficulty: 1)**
Let *n* be the product of distinct primes *p,q*
and let *e,d* be
inverses mod phi(*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

**Problem 4. (Difficulty: 1)**
Let *n1*, *n2*, and
*n3*, all be RSA moduli. Assume they are pairwise
coprime.
Let M be a number coprime to *n1*, *n2*, and
*n3*. Here is M^{3} modulo each of the above
three moduli; recover M.

n1 = 495686857566818933643731261008796623698621036637084896532377802272962445660829
M cubed mod n1 = 415187664033726855915723321090424502301717752950859170859603113265082093429509
n2 = 7002863119690523625149150663368289022346584130230947342910706423731579283453
M cubed mod n2 = 6398774707225696589164222858440863055200201126165875935672268746441800383699
n3 = 80635254989105072933971955953143348542680821191821297820451940193722465944913
M cubed mod n3 = 71519715094104667253654598558184719595919913945972035102241985006191784298462