Cryptography Reference
In-Depth Information
6.4.
p
= 4799, (
r
1
,r
2
,r
3
) = (49
,
492
,
585), (
u
A
,u
B
,u
C
) = (479
,
1078
,
3003).
6.5.
p
= 5009, (
r
1
,r
2
,r
3
) = (59
,
581
,
609), (
u
A
,u
B
,u
C
) = (3232
,
1137
,
1194).
6.6.
p
= 7321, (
r
1
,r
2
,r
3
) = (78
,
777
,
832), (
u
A
,u
B
,u
C
) = (2590
,
2495
,
1979).
6.7.
p
= 8293, (
r
1
,r
2
,r
3
) = (87
,
888
,
929), (
u
A
,u
B
,u
C
) = (186
,
4047
,
152).
6.8.
p
= 9497, (
r
1
,r
2
,r
3
) = (96
,
999
,
9000), (
u
A
,u
B
,u
C
) = (2152
,
1601
,
6133).
6.9.
Explain what would be necessary to embed a PKI into Kerberos (see
Section 6.2).
6.10. Explore the ramifications of eliminating the dual signature in the SET
protocol described in Section 6.3 in favour of a standard signature.
6.11. How do SSL/TLS and SET differ with respect to suitability of e-
commerce applications?
G.7 Chapter 7 Exercises
7.1 This exercise pertains to the birthday attack and related issues described
on pages 252-255. Suppose that we want to solve the DLP, namely, given
a large prime
p
, a generator
m
of
F
p
and an element
c
∈
F
p
, we want to
m
e
(mod
p
). How does the following aid in solving
the problem using the birthday attack?
Alice compiles two lists,
A
and
B
of length
≡
find
e
such that
c
≈
√
p
, satisfying the two
properties:
√
p
List
A
consists of all numbers
m
x
(mod
p
) for approximately
1.
randomly selected values of
x
.
√
p
List
B
contains the values
cm
−
y
(mod
p
) for approximately
2.
randomly chosen values of
y
.
7.2 Suppose that there are 150 students in a class. What is the probability
that at least two of them have the same birthday?
7.3. What is the minimum number of people who should be in a room to ensure
a probability of 99% that at least two of them have the same birthday?
7.4. What is the probability that we have a collision from a randomly chosen
pair of 16-bit numbers, given a random hash function applied to them?
(
Hint: See the formula for
P
2
(
n,m
)
on page 253 and set
n
=2
16
,
m
=1
.
)
7.5. Suppose that you choose a pair of 16-bit numbers as in Exercise 7.4, but
if you do not get a collision, you keep trying until you do. If you do this
n
times, how big must
n
be in order to have a 50% chance of success?
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