Cryptography Reference
In-Depth Information
5.8. Let
p
= 1249,
q
= 13,
e
= 12,
k
=2,
r
= 55.
5.9. Let
p
= 2251,
q
=5,
e
=4,
k
=3,
r
= 23.
5.10. Let
p
= 6619,
q
= 1103,
e
= 110,
k
= 234,
r
= 233.
5.11. Let
p
= 9377,
q
= 293,
e
= 11,
k
= 43,
r
= 199.
Exercises 5.12-5.15 pertain to the protocol for coin flipping by telephone
using discrete logs that we studied on page 209. With the parameters
given, decide whether the outcome of the coin toss is heads or tails.
5.12.
Let
p
= 3011,
α
=2,
β
=7,
x
= 9, and
y
≡
1 (mod
p
), which Alice
guesses is a function of
α
.
5.13. Let
p
= 4129,
α
= 13,
β
= 14,
x
= 5, and
y
≡
3812 (mod
p
), which Alice
guesses is a function of
β
.
5.14. Let
p
= 4561,
α
= 11,
β
= 13,
x
= 7, and
y
≡
2840 (mod
p
), which Alice
guesses is a function of
β
.
5.15. Let
p
= 7481,
α
=6,
β
=7,
x
= 13, and
y
≡
5128 (mod
p
), which Alice
guesses is a function of
α
.
In Exercises 5.16-5.19, we are going to play poker by telephone in a fashion
similar to that described on page 210. However, instead of five cards dealt,
we assume that only one card is dealt by Alice to Bob. In each exercise,
the values of the cards are given specific numerical values. You will need
computing power. The hint to Exercise 4.23 applies here.
5.16. Let
n
= 26904167, (
e
A
,d
A
)=(5
,
21515021); (
e
B
,d
B
) = (11
,
9779555),
c
j
≡
5
j
(mod
n
) for
j
=2
,
3
,
4
,...,
14 where
c
j
=
j
for
j
=2
,
3
,...,
10,
whereas
c
j
for
j
=11
,
12
,
13
,
14 represent the Jack, Queen, King, and
Ace, respectively.
Bob is dealt
h
j
≡
14473275 (mod
n
).
Determine the
card dealt to Bob.
(
Hint: Compute
c
j
≡
h
d
B
(mod
n
)
.
)
5.17.
In Exercise 5.16, assume that
g
j
≡
22526833 (mod
n
).
Determine the
card that was initially enciphered by Alice.
(
Hint: Compute
c
j
≡
g
d
A
·
d
B
j
(mod
n
)
.
)
5.18.
24498353 (mod
n
) with the parameters in Exercise 5.16,
determine the card enciphered by Alice.
(
Hint: Calculate
c
j
≡
Given
f
j
≡
f
d
j
(mod
n
)
.
)
5.19. With reference to the analysis of Jacobi symbols concerning poker playing
by telephone, given on pages 210 and 211, determine which of the values
−
1 or +1 is more advantageous to Alice given the parameters in Exercise
5.16. How may we alter the numbering of the cards to eliminate this
advantage?
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