Just as with symmetric and public-key encryption, we can group attacks on hash functions and MACs into two categories: brute-force attacks and cryptanalysis.

**Security of Hash Functions and
Macs**

Just as
with symmetric and public-key encryption, we can group attacks on hash
functions and MACs into two categories: brute-force attacks and cryptanalysis.

**Brute-Force Attacks**

The
nature of brute-force attacks differs somewhat for hash functions and MACs.

**Hash Functions**

The
strength of a hash function against brute-force attacks depends solely on the
length of the hash code produced by the algorithm. Recall from our discussion
of hash functions that there are three desirable properties:

·
One-way: For any given code h, it is
computationally infeasible to find x such that H(x) = h.

·
Weak collision resistance: For any given block x,
it is computationally infeasible to find y x with H(y) = H(x).

·
Strong collision resistance: It is computationally
infeasible to find any pair (x, y) such that H(x) = H(y).

·
For a hash code of length n, the level of effort
required, as we have seen is proportional to the following:

**Message Authentication Codes**

A
brute-force attack on a MAC is a more difficult undertaking because it requires
known message-MAC pairs.. To attack a hash code, we can proceed in the
following way. Given a fixed message x with n-bit hash code h = H(x), a
brute-force method of finding a collision is to pick a random bit string y and
check if H(y) = H(x). The attacker can do this repeatedly off line. To proceed,
we need to state the desired security property of a MAC algorithm, which can be
expressed as follows:

Computation
resistance: Given one or more text-MAC pairs (x_{i}, C_{K}[x_{i}]),
it is computationally infeasible to compute any text-MAC pair (x, C_{K}(
x)) for any new input x ≠x_{i}.

In other
words, the attacker would like to come up with the valid MAC code for a given
message x. There are two lines of attack possible: Attack the key space and
attack the MAC value. We examine each of these in turn.

To summarize,
the level of effort for brute-force attack on a MAC algorithm can be expressed
as min(2^{k}, 2^{n}). The assessment of strength is similar to
that for symmetric encryption algorithms. It would appear reasonable to require
that the key length and MAC length satisfy a relationship such as min(k, n) ≥N,
where N is perhaps in the range of 128 bits.

**Cryptanalysis**

As with
encryption algorithms, cryptanalytic attacks on hash functions and MAC
algorithms seek to exploit some property of the algorithm to perform some
attack other than an exhaustive search.

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Cryptography and Network Security : Security of Hash Functions and Macs |

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