Algebraic Cryptography
Exam 1
Choose 5 of the following 7 problems to complete.
Problem 1. Recall that in the RSA cryptosystem a user generates primes p, q
and computes the product n. The user then generates an encryption exponent
erelatively prime to (p−1) and (q−1). Then the user computes a decryption
exponent dso that de ≡1 (mod (p−1)(q−1)). The user then publishes the
public key (n, e) and keeps the private key da secret.
(a) Prove that the map x7→ xd(mod n) is the inverse of x7→ xe(mod n).
(b) Suppose Alice and Bob generate public keys (n, e) and (n0, e0) which share
a prime p(i.e., n=pq and n0=pq0), perhaps due to insufficient en-
tropy (randomness) during key generation. Explain how an attacker, with
knowledge only of the public keys can crack both Alice’s and Bob’s private
keys.
Problem 2. Consider a commutative public key encryption decryption scheme
(i.e., the encryption functions eand dcommute, e◦d=d◦e). Let eA, dA
be Alice’s encryption/decryption functions, and let eB, dBbe Bob’s. Let h
denote a hash function and ma message. The functions eA, eB, h are all public
knowledge.
(a) Explain how Alice can use these functions to sign a message min such a
way that everyone (not just Bob) can read the message and everyone can
verify that Alice wrote it.
(b) Explain how Alice can send a secret message to Bob that only he can read,
but that Bob (and only Bob) can know and be sure that Alice wrote it.
Problem 3. Show that at least n
2of the numbers 1,2, . . . , n have (binary)
length equal to or greater than log2n−1. Then show that n! has length at least
equal to n
2(log2n−2), and that for large n, this is greater than C n ln nfor some
positive constant C.
Problem 4. Given a k-bit integer, you want to compute the highest power
of this number that has lor fewer bits (we suppose lk). Estimate (with
big-O) the number of bit operations required to do this. Your answer should be
a very simple expression in terms of kand/or l. Moreover, you should explicitly
describe the algorithm you are using.
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