The aim of this diploma thesis is to theoretically explain what are secure communication, message integrity, hash-functions and rainbow tables. The first part of this thesis is concentrated on two most important goals of cryptography, i.e. obtaining private communication and guarantee message integrity. Encryption does not in general provide any integrity, therefore encryption should be used in combination with other techniques for message authentication such as Message Authentication Code (MAC). The second part of this thesis refers to hash-functions that are mapping strings of arbitrary length to strings of some fixed length, called hash-values. Particularly interesting are collision-resistant hash-functions. A collision in a function \(H\) is a pair of distinct inputs \(x\) and \(x'\) such that \(H(x)=H(x')\). Usually, we consider functions that have an infinite domain and a finite range, so a requirement is therefore that such collisions should be ”hard” to find. ”Birthday” attack finds a collision in any hash-function. Then we presented the Merkle-Damgard transform that is used for constructing collision-resistant hash-functions. The Merkle-Damgard transform enables a conversion from any fixed-length hash-function to an arbitrary-length hash-function while maintaining the collision resistance property. The third part of this thesis describes rainbow tables which have been developed for cracking password directly from hash-values. Rainbow tables were invented by P.Oechlin as an application of an earlier algorithm by Martin Hellman. The main restriction of the original method is the fact that when two chains collide in a single table they merge. Rainbow tables efficiently solve this problem by replacing one reduction function with successive reduction functions. Rainbow tables are ineffective against hash-values that include large ”salts”. Also, important techniques that help with prevention for those attacks are key stretching and key strengthening. From the side of the user, it is recommended to use passwords that have more than fourteen characters and to use symbols that are out of the usual range (eg., @, \(\%, \&,\dots\)).