### 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 x2 = 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 * n1/4 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 M3 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

```