ICS 555 - Data Security and Cryptography TERM 142 Sultan Almuhammadi

Homework Assignment - HW1 Due: Wednesday, Feb. 18, by 6:00 pm

Problem Set: [60 points] 1. [5 pts] In the base 26, with digits A Z representing 0 25, add YOU + ME.

2. [5 pts] Compute: (30 32 34 + 936494 3099865) mod 31

3. [10 pts] Refer to the fundamental theory of arithmetic. a) Find the prime factorization of 360 and 294. b) How many divisors does 360 and 294 have?

4. [5 pts] What is the smallest positive integer that has exactly 12 divisors.

5. [5 pts] Prove or disprove: if pn || a and pn || b then pn || a + b

6. [5 pts] Using the Euclidean algorithm, find the gcd(2152, 764), and express the gcd(2152, 764) as a linear combination of 2152 and 764.

7. [5 pts] Find the multiplicative inverse of 550 mod 1769.

8. [10 pts] Assume multiplying k-bit by l-bit integers takes O(kl) bit-operations. Using the big-O notation in terms of n, estimate the number of bit-operations required to compute:

a) 3n by multiplying 3 to itself n times. b) nn by multiplying n to itself (n 1) times

9. [10 pts] Assume division of a k-bit number by an l-bit number takes O(kl) bit-operations. If we want to test if a large odd number n is a prime or not, estimate the number of bit operations that each of the following procedures takes:

a) Test by trial division by all odd numbers (n) b) Suppose you have a list of prime numbers up to (n) and you test by

trial division by those primes in the list (i.e. no longer running through all odd numbers). Hint: pi(n) = (n/log n) by Prime Number Theorem.

Bonus Problems: [10 points] 10. [5 pts] In Problem 8, we assume that multiplying two k-bit numbers takes

O(k2). Search the internet for a better time estimate of multiplying two k-bit large integers. Briefly explain the algorithm (cite the reference for details).

11. [5 pts] Use your result in Problem 9, if any, to improve the time estimate for computing nn you have obtained in Problem 3(b).