Cryptography Reference
In-Depth Information
4.10. Let
p
= 401,
e
= 21 and
c
= (256
,
232
,
127
,
0
,
10).
Exercises 4.11-4.15 pertain to the Di
7
e-Hellman key exchange protocol de-
scribed on page 166. In Exercises 4.11-4.14, with the given parameters,
find the shared secret key in each case.
4.11.
p
= 397,
α
=5,
x
= 295, and
y
= 301.
4.12.
p
= 643,
α
= 11,
x
= 540, and
y
= 603.
4.13.
p
= 907,
α
=2,
x
= 101, and
y
=2.
4.14.
p
= 1181,
α
=7,
x
= 1000, and
y
=5.
4.15. Explain why we choose the generator
α
, for the DiGe-Hellman protocol,
in the range 2
≤
α
≤
p
−
2. In other words, why would it be a very bad
idea to choose
α
=
p
−
1?
4.16. Verify the statement made on page 167, namely, if Eve can solve the
DLP, she can solve the DHP.
4.17. Suppose that
n
is an RSA modulus and you know both the enciphering
exponent
e
as well as the deciphering exponent
d
. Show how this allows
you to factor
n
.
In Exercises 4.18-4.21, use the repeated squaring method highlighted on page
171 to find the given modular power residue.
4.18. 5
72
(mod 103).
4.19. 3
81
(mod 303).
4.20. 7
92
(mod 97).
4.21. 2
51
(mod 101).
4.22. Suppose we alter the modulus in Exercise 4.20 so that we have
7
92
(mod 105)
.
Explain how this may be easily done
without
using the repeated squaring
method.
(
Hint: Use Euler's theorem A.14 on page 479 once you factor out the
gcd
of the modulus and the base.
)
Exercises 4.23-4.26 pertain to the RSA public-key cryptosystem described
on page 173. Find the plaintext numerical value of
m
from the param-
eters given. You will first have to determine the private key
d
from the
given data via the methodology illustrated in Example 4.5 on page 173. If
repeated squaring is not employed
(
see page 171
)
, then a computer will be
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