## Math 5410/4410 Take-Home Midterm Spring 1998

*This exam is due on Thursday, March 12th. The exam totals 100 points. Please show all work
as answers alone are not sufficient. All work submitted must be your own.*

** GOOD LUCK !!**
**1**. Consider the following ciphertext:
EYMHP GZYHH PTIAP QIHPH YIRMQ EYPXQ FIQHI AHYIW ISITK MHXQZ PNMQQ
XFIKJ MKXIJ RIKIU XSSXQ ZEPGS ATIHP PSXZY H

a) Calculate the Index of Coincidence and speculate on whether or not this might be a
polyalphabetic cipher.

b) Supposing that the cipher is either a transposition or a monoalphabetic substitution (and I
am not claiming that it is either of these), examine the frequency distribution and decide
which of these two choices is the most likely.

c) Determine the plaintext if you know that the message starts with "What ought ..." (a
plaintext attack).

d) Explain how the key was formed.

**2**. Suppose that n = 1363 in the **RSA Cryptosystem**, and it has been revealed that (n) =
1288. Use this information to find the factorization of n by solving a quadratic equation.

**3**. Suppose the **Affine Cipher** is implemented in **Z**_{99}.

a) Determine the number of possible keys.

b) If the key is (16,7) (i.e., the encryption function is x (16x + 7) mod 99) determine the
decryption function corresponding to this key.

**4**. Consider the non-linear feedback function f(s_{0}, s_{1}, s_{2}, s_{3}) = s_{0}s_{1} + s_{3} + 1.

a) Determine the repeating portion of the sequence obtained by a FSR with this feedback
function whose starting state is 0 1 0 1.

b) Find the linear equivalence of this FSR and a **LFSR** which will produce the same
repeating portion.

**5**. Here is an example of **RSA** ciphertext:

11437 6198 16611 2405 18636 2679 12205 24142 6375 2134
16611 2405 9529 7260 7834 15094 4667 24027 762 5878
5206 16683 5359 10888 4168 3536 23229 20351 15580 6704
7977 374 6525 4287 14402 527 12887 21628 11884 9402
15470 1339 10420 18051 23125 7747 135 22007 20049 9984
13199 15176 1379 8313 19574 7989 22869 406 10057 21758
3918 23991 14237 7989 3947 19529 15728 5601 3527 7200
7601 13282 21160 6291 15994 7785 8982 3045 6596 16796
4663 2405 20302 11929 17125 14533 21001 8351 11571 22082
11040 8687 6704 3330 5630 19650 13024

The public modulus is n = 24637 and the public encryption exponent is *b* = 3. Perform the
following computations:
a) Factor n by trial and error.

b) Compute the private decryption exponent *a* using the extended Euclidean algorithm.

c) Decrypt the ciphertext, obtaining the plaintext as a sequence of elements of **Z**_{24637} . Use the
square-and-multiply algorithm for modular exponentiation.

d) Translate the plaintext into English text, given that each element of **Z**_{24637} represents three
alphabetic characters, as in the following examples:

DOG 3 × 26^{2} + 14× 26 + 6 = 2398

CAT 2 × 26^{2} + 0× 26 + 19 = 1371

ZZZ 25 × 26^{2} + 25× 26 + 25 = 17575