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Mathematics 336
Prelim 2
April 4, 2008
4 problems, 25 points each
No books, notes or electronic devices may be used. Your proofs may use anything that has
been given in class or in the book, as long as you show clearly what you are using. You must
EXPLAIN ALL ANSWERS!
1. Consider an RSA public key cryptosystem in which your public key is (
e
1
, n
1
) = (27
,
55)
and my public key is (
e
2
, n
2
) = (7
,
22).
(a) Your decoding key is (
d
1
,
55). Determine
d
1
.
Solution:
φ
(
n
1
) =
φ
(5
·
11) = 40
,
3
·
27 = 81
so
d
1
= 3
.
(b) “Sign” the numerical message
02 03 06
so that I know it is from you.
Soln.:
02
3
03
3
06
3
mod
55 = 082751
signed message
(c) Encrypt the result of part (b) for transmission to me. (There is no need to carry
out the arithmetic in this part; just indicate what has to be done.)
Soln.:
08
7
27
7
51
7
mod
22
encrypted signed message
2. Let
H
=
1
0
0
1
0
1
0
1
0
1
1
0
0
0
1
0
1
1
be a parity check matrix for a binary code.
(a) Determine a generator matrix
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 Spring '08
 BILLERA
 Math, Algebra

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