Hash Functions

H: {0, 1}^* -> {0, 1}^n;
n - hash length;
Input - binary sequence of finite length;
Output - binary sequence of fixed length (n);

Properties

Security Properties:
- It is computationally easy to get H(x) given x;
- It is computationally hard to get x', given x, such that x' != x and H(x') = H(x) -> second pre-image;
- It is computationally hard to get (x, x'), such that x != x' and H(x) = H(x') -> collision;
The hash of m represents m ("digital signature");
Based on boolean and arithmetic operations;

Examples

Data integrity;
Derivation of keys from passwords;
MAC algorithms;
Digital signature (asymmetric scheme);
Multiple cryptographic protocols.

Hash functions with key

It is usual to designate a MAC scheme, with a deterministic algorithm T, as a keyed hash function (Keyed-Hashing for Message Authentication, HMAC);
HMAC is a set of MAC algorithms to use with different hash functions H.