Information Assurance: Homework 5 – Answer key
Due October 13, 2006.
1. The strength of the RSA algorithm is based on the difficulty of factoring large prime
numbers. Assume you are given the factorization for the modulus of a public key (n
= p x q). Show now this breaks the RSA key pair.
If we know n = p*q, we can easily compute Φ(n) = (p-1)(q-1). The public key exponent e
is relatively prime to n and the private key exponent d is the inverse of e modulo Φ(n),
i.e., e*d mod Φ(n) = 1. If we know the toitent and e, we can compute d. Once we know d
the key pair is broken.
2. Suppose Alice and Bob have RSA public keys in a file on a server. They
communicate regularly using authenticated, confidential messages. Eve wants to read
the messages but is unable to crack the RSA private keys of Alice and Bob. However,
she is able to break into the server and alter the file containing Alice's and Bob's
public keys.
a. How should Eve alter that file so that she can read confidential messages sent
between Alice an Bob, and forge messages from either?
Assume Alice has a public key (eA, nA) and Bob has a public key (eB, nB) registered. Eve
could create two new key pairs (eEA, nEA) and (eEB, nEB) and register them for Alice and
Bob. Then if Eve can intercept all communication exchanged between Alice and Bob,
she can undo their computation and replace it with a signature/encryption using her
version of the key. E.g., suppose Bob sends a signed encrypted message to Alice
{m {h(m)}dB}eEA. Eve would intercept it, decrypt it using her version of Alice's private
key, recalculate the hash using her version of Bob's private key, and re-encrypt using her
version of Alice's public key to compute {m {h(m)}dEB}eA.
b. How might Alice and/or Bob detect Eve's subversion of the public keys?
If Eve fails to intercept a message the deception is found, because the messages destined
to Alice and Bob will not correctly decrypt, since Alice and Bob will use their real
private keys to decrypt rather than Eve's fake private keys.
Alternatively, Alice and Bob should periodically verify that the public key they have
posted on the key server is correct. They could also post a fingerprint of their public key
at a separate location (e.g. On their personal web site or at the end of emails) to
encourage people they communicate to validate what they receive from the certificate
server.
3. Thanks to the birthday paradox one can find collisions using the DES-MAC
cryptographic hash function in 232 messages. Alice wants to take advantage of that
fact to make it swindle Bob. She has two contracts. One that Bob is willing to sign
