1. (4 pt) Increasing the key length is one way to increase security of an encryption
algorithm against the bruteforce type of attack. DES uses 56bit key, which is not secure,
given the modern computing power. Assume that 56bit key was just sufficient in 1979
when DES was standardized, and assume that the hardware performance improves about
40% per year, then how many bits of a DES key would just suffice this year? Until what
year would a 112bit DES key be sufficient?
Answer:
40% improvement per year doubles the performance every two years (1.4×1.4
≈
2). So
keys must grow by about 1 bit every two years. (20081979)/2+56=70.5, so we need 71
bit keys which would just suffice this year. (x1979)/2+56=112,
x=2091; 112bit key
would suffice till year 2091.
2. (6 pt) Suppose you encrypt message m by computing
. How do you decrypt?
13
mod 77
c
m
≡
Answer:
(77)
(7
11)
(7
1)(11
1)
60
ϕ
ϕ
=
×
=
−
−
=
13 is relatively prime to 60,
1
13
mod 60
d
−
=
Show steps of extended Euclidean algorithm to find
d=
23 mod 60 or 37 mod 60 = 37
13
13
0
60
8
60
4
13
5
13
1
8
3
8
1
5
2
5
1
3
1
3
1
2
=
−
×
=
−
×
=
− ×
=
− ×
=
− ×
=
− ×
՜
1
3
1
2
3
1
(5
1
3)
2
3
1
5
2
(8
1
5)
1
5
2
8
3
5
2
8
3
(13
1
8)
5
8
3
13
5
(60
4
13)
3 13
5
60
23 13
=
− ×
=
− ×
− ×
=
×
− ×
=
×
− ×
− ×
=
×
−
×
=
×
−
×
− ×
=
×
−
×
=
×
−
×
−
×
=
×
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 Spring '10
 attkin
 Cryptography, Encryption, Alice Bob, plaintext block, different ciphertext block

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