CSci 5471: Modern Cryptography, Fall 2010 Homework #4
Yongdae Kim
–
Due: Mar. 23 9:00 AM
–
Show all steps. Please ask any questions, if the problems are not clear.
– Send me your solution by email.
–
Total 100 + 40 points.
1. Number Theory (45 points): Try to use theorems you learned. Use of calculator for this homework won’t help
your preparation for upcoming quiz.
(a) (5 points) Solve for
x
using Chinese remainder theorem.
5
·
x
≡
2
(mod 8)
7
·
x
≡
2
(mod 9)
x
≡
0
(mod 11)
(b) (5 points) Let
m, n
be the integers such that
gcd(
m, n
) = 1
. When does
gcd(
m, n
) = gcd(
m
+
n, m

n
)
hold? Justify your claim.
(c) (5 points) Find the remainder of
10
10
+ 10
10
2
+
· · ·
+ 10
10
10
upon division by 7.
(d) (6 points) Compute
2
28224
(mod 113
·
127)
.
(e) (7 points) Compute
2
2
64
(mod 180)
.
(f) (7 points) Find all generators in
Z
*
19
.
(g) (10 points) Let
p
be an odd prime. Show that if
a
h
= 1
mod
p
then
a
ph
= 1
mod
p
2
.
2.
Performance Degradation when Key Size Increases (10 points)
We consider number of bit operations instead
of number of multiplications for this problem. Suppose modular multiplication (i.e.
a
·
b
(mod
n
)
with

a

=

b

=

n

) requires
4 log
2
2
(
n
)
bit operations in the worst case. Furthermore, we assume that squareandmultiply
algorithm is used for modular exponentiation. Compared with 512 bit modular exponentiation, how many times is
1024bit modular exponentiation slower?
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 Spring '08
 Staff
 Cryptography, Database Security, Alice, Modular exponentiation, public key cryptosystem

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