Cryptography Reference
In-Depth Information
the ASIC (manufacturing the printed circuit boards, power supply, cooling, etc.).
How many ASICs can we run in parallel with the given budget? How long does
an average key search take? Relate this time to the age of the Universe, which is
about 10
10
years.
2. We try now to take advances in computer technology into account. Predicting
the future tends to be tricky but the estimate usually applied is Moore's Law,
which states that the computer power doubles every 18 months while the costs
of integrated circuits stay constant. How many years do we have to wait until a
key-search machine can be built for breaking AES with 128 bit with an average
search time of 24 hours? Again, assume a budget of $1 million (do not take
inflation into account).
1.4.
We now consider the relation between passwords and key size. For this purpose
we consider a cryptosystem where the user enters a key in the form of a password.
1. Assume a password consisting of 8 letters, where each letter is encoded by the
ASCII scheme (7 bits per character, i.e., 128 possible characters). What is the
size of the key space which can be constructed by such passwords?
2. What is the corresponding key length in bits?
3. Assume that most users use only the 26 lowercase letters from the alphabet in-
stead of the full 7 bits of the ASCII-encoding. What is the corresponding key
length in bits in this case?
4. At least how many characters are required for a password in order to generate a
key length of 128 bits in case of letters consisting of
a. 7-bit characters?
b. 26 lowercase letters from the alphabet?
1.5.
As we learned in this chapter, modular arithmetic is the basis of many cryp-
tosystems. As a consequence, we will address this topic with several problems in
this and upcoming chapters.
Let's start with an easy one: Compute the result without a calculator.
1. 15
·
29 mod 13
2. 2
·
29 mod 13
·
3. 2
3 mod 13
−
·
3 mod 13
The results should be given in the range from 0
,
1
,...,
modulus-1. Briefly describe
the relation between the different parts of the problem.
4.
11
1.6.
Compute without a calculator:
1. 1
/
5 mod 13
2. 1
/
5mod7
3. 3
·
2
/
5mod7
1.7.
We consider the ring
Z
4
. Construct a table which describes the addition of all
elements in the ring with each other:
