## Math 5410 Take-Home Midterm Spring 2004

This exam is due in class on Thursday, March 11th. Do all problems. The exam totals 100 points. Please show all work as answers alone are not sufficient. Submit partial solutions for partial credit. All work submitted must be your own.

1. Construct the finite field GF(16).

1. The following ciphertext was obtained by XORing (adding mod 2) an LFSR output with the plaintext.

0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1

Suppose you know that the plaintext starts: 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0

Find the complete plaintext.

1. Find the number of bases, b, such that 837 is an Euler psuedo prime to the base b. (Note that b is relatively prime to 837)

2. A plaintext x is said to be fixed if an encryption algorithm encrypts x to itself. In the RSA cryptosystem, 0 is clearly a fixed plaintext. If n = pq is the public modulus and b is the public encryption exponent of an RSA cryptosystem, show that the number of non-zero fixed plaintexts in Zn is (1 + gcd(b-1, p-1)) ´(1 + gcd(b-1, q-1)) - 1. (Hint: You will need to use the fact that if a º b mod (pq) then a º b mod p and a º b mod q.)

3. Here is an example of RSA ciphertext:

```    6340    8309    14010   8936    27358   25023  16481   25809
23614   7135    24996   30590   27570   26486  30388   9395
27584   14999   4517    12146   29421   26439  1606    17881
25774   7647    23901   7372    25774   18436  12056   13547
7908    8635    2149    1908    22076   7372   8686    1304
4082    11803   5314    107     7359    22470  7372    22827
15698   30317   4685    14696   30388   8671   29956   15705
1417    26905   25809   28347   26277   7897   20240   21519
12437   1108    27106   18743   24144   10685  25234   30155
23005   8267    9917    7994    9694    2149   10042   27705
15930   29748   8635    23645   11738   24591  20240   27212
27486   9741    2149    29329   2149    5501   14015   30155
18154   22319   27705   20321   23254   13624  3249    5443
2149    16975   16087   14600   27705   19386  7325    26277
19554   23614   7553    4734    8091    23973  14015   107
3183    17347   25234   4595    21498   6360   19837   8463
6000    31280   29413   2066    369     23204  8425    7792
25973   4477    30989```

The public modulus is n = 31313 and the public encryption exponent is e = 4913. Perform the following computations:

a) Factor n by trial and error.

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

c) Decrypt the ciphertext, obtaining the plaintext as a sequence of elements of Z31313 . Use the square-and-multiply algorithm for modular exponentiation.

d) Translate the plaintext into English text, given that each element of Z31313 represents three alphabetic characters, as in the following examples:

DOG ® 3 × 262 + 14 × 26 + 6 = 2398
CAT ® 2 × 262 + 0 × 26 + 19 = 1371
ZZZ ® 25 × 262 + 25 × 26 + 25 = 17575