Download Foundations of Cryptography, 23rd of September 2016

Foundations of Cryptography, 23rd of September 2016
Exercises 2
1. Let n be a positive integer. Sieve of Erastothenes is an algorithm for finding all the prime
numbers smaller than or equal to n. The algorithm works as follows: First list all integers
from 2 to n. Remove all proper multiples of 2, i.e., multiples greater than 2, from the list.
The next unremoved integer in the list is 3. Again remove all proper multiples of 3 from
the list. Continue this process of taking the first unremoved integer and then removing
the proper multiples of that integer. Finally, the remaining integers are the prime numbers
smaller than or equal to n. Indeed, no prime number is removed and every composite
number has a factor smaller than or equal to n. Moreover, by Theorem 1.4, every integer
√
smaller than or equal to n has a prime factor p ≤ n. Hence, in the algorithm, we can
√
stop removing the multiples when the considered integer is greater than n.
Using the sieve of Erastothenes, find all the prime numbers smaller than or equal to n = 50.
2. Let a and b integers such that a 6= 0 and b 6= 0. Denote d = gcd(a, b) and
m=
ab
.
d
(a) Show that m is divisible by a and b.
(b) Show that if a|c and b|c for some integer c, then m|c. (Hint: Recall that there exist
integers u and v such that d = ua + vb.)
(c) Conclude that gcd(a, b)lcm(a, b) = ab.
(d) Finally, determine lcm(408, 156). (Recall that gcd(408, 156) has been calculated in the
first set of exercises.)
3. Calculate the remainder when 32016 + 20169 is divided by 11.
4. Prove the following statements.
(a) Integer k 2 + 1 is not divisible by 4 for any integer k.
(b) Integer k 2 − 1 is divisible by 4 for all odd integers k.
5. A natural number a can be written in decimal system as as as−1 · · · a1 a0 , where ai ∈
{0, 1, 2, . . . , 9}. For example, the natural number a = 2016 can be written as a3 = 2,
a2 = 0, a1 = 1 and a0 = 6 with s = 3. Show that a natural number a is divisible by 11 if
and only if
s
X
(−1)n an = a0 − a1 + a2 − a3 + − · · · + (−1)n an
n=0
is divisible by 11. Then determine whether 123456789 is divisible by 11 or not.
6. Consider the RSA cryptosystem with n = 493 and e = 23. Encrypt the message M = 42.
7. Continuing the previous problem, find out the decryption exponent d. Then decrypt the
secret message m = 440. (Hint: You can break the system by finding the prime factors of
n = 493.)