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One-way function

One-way functions are easy to compute but it is very difficult to compute their inverse functions. Thus, having data x it is easy to calculate f(x) but, on the other hand, knowing the value of f(x) it is quite difficult to calculate the value of x.

There is not a mathematical prove that one-way functions exist. In practical applications functions that behave similarly as real one-way functions are used.

One-way functions are key elements of various tools useful in modern cryptography. They are used in pseudorandom generators, authentication of messages and digital signatures.

Trapdoor one-way function

Trapdoor one-way functions are types of one-way functions that contain a kind of "back door" (trapdoor). As in the case of ordinary one-way functions it is easy to compute their values for given data but it is very difficult to compute their inverse functions. However, if one has some additional secret information, he can easily compute the inverse function as well.

An example of such trapdoor one-way functions may be finding the prime factors of large numbers. Nowadays, this task is practically infeasible. On the other hand, knowing one of the factors, it is easy to compute the other ones.

One-way hash function

One-way hash functions (there are a lot of other names of functions of this type) transform input messages of various length into output sequences of fixed length (usually shorter). The output sequence is often called a hash value. Hash values are often used to mark input sequences, that is to assign to them some unique values that characterize them.

One-way hash functions fulfil all conditions of one-way functions. It is easy to compute their values based on input data but having only a hash value one can't determine the original input sequence.

A one-way hash function should be collision-free. This means that it should be very difficult to find two different sequences that produce the same hash value.

Algorithms of one-way hash functions are often known to the public. They provide security thanks to their properties as one-way functions. Usually, a change of one bit of input data, causes changing about half of the output bits. Data generated by hash functions should be pseudorandom (it cannot be possible to distinguish output data from ordinary random data).

One-way hash functions are used to protect data against intentional or unintentional modifications. Having some data, one can calculate a checksum that may be attached to the message and checked by other recipients (by computing the same checksum and compare it with the received checksum value). They are used for example in an message authentication algorithm HMAC.

Moreover, hash functions are used for storing data efficiently, in so-called hash tables. Data can be accessed by finding hash values, which are stored in computer memory.

Hash function expanding

Having a hash function that operates on small blocks of data, one can expand this function and create a new function that operates on larger input data of various sizes. Is such a way it is possible to compute a hash value for any messages. To achieve this, one can use so called Merkle-Damgard construction.

A scheme below presents a way of using multiple hash functions (h: T x X -> T, where T is a set of all possible hash values) that allows to compute a hash value for the whole message (H: XL -> T). Each hash function operates on one data block.

Scheme of Merkle-Damgard construction

The first h function is initialized by a previously determined vector IV. The last data block should be fulfilled to the full-length using previously agreed bits. A popular method of aligning the length of the last block is about appending an additional bit equals to 1 and then filling the rest of the block with bits equal to 0. It allows to determine precisely the end of the real message.

It may be proved that if the function h is collision-free, then the function H (which is created based on functions h) is also collision-free. Every collision in the function H would automatically result in a collision in the function h.

Hash functions based on block ciphers

There are a few popular ways of creating one-way hash functions, that operate on input data of various lengths, using algorithms of block ciphers.

The Davies-Meyer hash function (denoted h) uses the encryption algorithm E that operates on subsequent data blocks:
    h(H, m) = E(m, H) XOR H

A scheme of Davies-Meyer function is presented below:

Scheme of Davies-Meyer function

It can by proved that if E is a secure algorithm of a block cipher, then an intruder would have to perform about O(2n/2) encryption operations to find a collision (thus, to find a message with the same hash value as the hash value that he would like to find).

Another kind of hash functions based on block ciphers are Miyaguchi-Preneel functions. There are 12 kinds of those functions, for example:
    h(H, m) = E(m, H) XOR H XOR m
    h(H, m) = E(H XOR m, m) XOR m