Types and classification of ciphers

Classification of ciphers is presented on a picture 2.

Picture 2 - Classification of ciphers

Pointers on a picture mean the most meaningful subclasses of ciphers. The dotted lines mean that ciphers can be examined and as a block, and as ciphers of replacement.

Symmetric ciphers utillize the same key for encryption and decryption.

Asymmetrical ciphers are utillized the different keys.

In line ciphers every character is ciphered on a separateness.

At the use of block ciphers, a plaintext is divided by the blocks of the fixed length, each of which is ciphered on a separateness. A size of block to date is 64, 128 or 256 bit.

The ciphers of gamming form the subclass of multialphabetical ciphers. They behave to line ciphers and they are symmetric.

One-way functions

Public-key systems utillize ireversible or one-way functions. A concept of one-way function is the base concept of cryptography with the opened key. One-way function possess such features:

- on the set argument х   Х it is easily to calculate the value of this function F(x);

- at the same time, determination х from F(x) is an infeasible task.

In theory х by known value F(x) it is possible to find always, checking up in turn all of possible values of х until the proper value F(x) will not coincide with set. However practically such approach is unrealizable at the considerable dimension of set X.

One-way functions can be compared to the one-sided streets. Easily to reach on such street from a point A to a point B, while it is practically impossible to reach from a point B to the point A. Cipherment examined as direction from A to B. Although we can move in this direction, we are unable to move in retrograde direction – direction of decryption.

easy

x f(x)

with difficultly

Figure 2 – Illustration of one-way function

There is the more strong approach to determination of one-way function:

An one-way function is name the function F(х): X  Y, х   Х, possessing two properties:

- there is a polynomical algorithm of calculation of values y = F(x);

- there is not a polynomical algorithm of inverting of function F(x) = y.

The algorithm will name polynomical algorithm when an implementation of it is closed no more than after p(n) steps, where n is a size of entrance task, measureable, as a rule, by the amount of characters of text describing this task.

The multitude of classes of one-way functions is generated variety of the systems with the opened key.

Two important requirements are using for an order to guarantee reliable work of the systems with the opened key:

- transformation of the opened message must be irreversible and to eliminate his restoration on the base of the opened key;

- to define the closed key by analysing opened it must be impossible at modern technological level (thus desirably have exact lower estimation of complication of opening of cipher, i.e. amount of operations).

We will notice that until now is not well-proven existence of one-way functions. Use them as basis of asymmetric algorithms of encipherement possibly only until effective algorithms, executing finding of one-way functions for polinomical time, are not found.

The example of candidate on the rank of one-way function is module involution, i.e. function of F(x)  а^x mod р, where а is a primitive element of the field of GF(p); р is a large prime number. That this function can be effectively calculated even at the bit of parameters in a few hundred signs, it is possible to show on a next example.

Example. а²⁵ can be calculated by six operations of multiplication (a multiplication will consider operation of raise to square). Number 25 in the binary notation written down as 11001, so that 25 = 2⁴ + 2³ + 2⁰.

Therefore: а²⁵ mod р  (а¹⁶а⁸а) mod р  ((((а² а)² )² )² а) mod р.

The task of calculation of function, reverse module involution, is named the task of the discrete taking the logarithm. To date unknown not a single effective algorithm of calculation of discrete logarithms of large numbers.

An one-wave function as a function of encryption is inapplicable, because, if F(x) is a crypted message of х, nobody, including legal recipient, not able to recover х. Going round this problem is possible by an one-way function with a secret (one-way trapdoor function). Sometimes a term is yet used function with trap.

For example, function E_k: X Y, has a reverse function D_k: Y X, however it is impossible to know a reverse function only on E_k without knowledge of secret k.

Function E_k: X Y, depending on a parameter k and possessing next three properties is named by an one-way function with a secret. There are following properties:

1) at any k there is a polinomical algorithm of calculation of values of Ek(x);

2) at unknown k there is not a polinomical algorithm of inverting of E_k;

3) at known k there is a polinomical algorithm of inverting of E_k;

The function of E_k can be utillized for encrypting of information, and reverse by it function of D_k - for decrypting, because at all х   Х justly D_k (E_k(x)) = x.

Implied thus, that, who knows, how information to encrypt, quite not necessarily must know how to decrypt it. Similarly as well as in case with an one-way function, a question about existence of one-way trapdoor function is opened. For practical cryptography a few functions - candidates on the rank of one-way trapdoor function are found. For them the second property is not well-proven, however known it is, that the task of inverting is equivalent the decision of difficult mathematical task.

Application of one-way trapdoor function in cryptography allows:

- to organize an exchange the encrypted messages with the use of the only opened channels of connection, i.e. to turn down the secret channels of connection for a preliminary exchange by the keys;

- to include at dissection of cipher an complicated mathematical problem and the same to increase cipher firmness;

- to decide new cryptographic tasks, different from an encipherement (electronic digital signature and other).

Firmness of most modern asymmetric algorithms is based on mathematical problems which on this stage are an infeasible task:

1) factorization of large numbers (decomposition of large numbers on simple multipliers);

2) the discrete taking the logarithm in the eventual fields (a search of logarithm in the eventual fields);

3) search of roots of algebraic equalizations.

As to date there are not effective algorithms of decision of these tasks or their

decision requires bringing in of large calculable resources or temporal expenses, these mathematical tasks found a wideuse in the construction of asymmetric algorithms.