The objective of the Cryptographie is to build sure systems of coding. That being, it is advisable to define this criterion of safety rigorously.

Primarily, one has two concepts:

• the semantic, or unconditional Safety
• the calculative Safety

The first concept is introduced by Claude Shannon, in his famous article Communication theory off secrecy systems published in 1949. Currently, only the system of the disposable Masque is proven unconditionally sure. Shannon itself showed it in the article quoted above. This concept formalizes the idea that, if one has only the only coded message, it is impossible to deduce least information on the message in light.

The second concept is less strong, it supposes that if one has only one capacity of limited calculation, one will not be able to deduce the message.

Distinctions between symmetrical and asymmetrical cryptography

It is now necessary to distinguish the symmetrical Cryptographie from the asymmetrical Cryptographie.

In the symmetrical field, only semantic safety can be proven, which is a very important problem of the field. Indeed, except this safety, the only thing which one manages to prove on symmetrical systems is their resistance to known techniques of Cryptanalyse, such as for example the Cryptanalyse linear differential or . One cannot prove resistance to still unknown attacks.

In the asymmetrical field, the problem arises differently and it is besides in the latter that one finds more the concept of proof of safety. The asymmetrical systems rest on calculative problems of the Théorie of the numbers or discrete Algèbre. For example, an algorithm as ElGamal rests on the problem of the discrete Logarithme. The general outline of a proof of safety is then to prove that to break the system is reduced, by a number of polynomial operations (in certain quantities depending on the system), with another presumedly difficult problem . One retouve thus, with the help of a overcost considered as negligible, to solve a problem (supposed) difficult.

The theory of complexity

But what does one have to call difficult ? An answer to this question is brought by the Théorie of the complexity from which one borrows inter alia the concept of reduction between problems. This theory seeks to classify the problems according to the computing time necessary to solve them, and defines classes of “difficulty”. In fact, the class which interests us is that known as not polynomial determinist (NP). These are the problems whose solution, data, are “easy” with vérifer (checks itself in polynomial time), but risk on the other hand to be difficult (potentially in nonpolynomial time) with to find .

The membership of one problem to the class NP does not mean not that this one is not resolvable in polynomial time. Indeed, all the problems of P are in NP, and the fact of knowing if on the contrary there exist NP problems which are not in P is one of the great open-ended questions in mathematics.

In practice

The problems used by cryptography are all in NP: it is “easy” to code a message, or to decode a message when one has the key of it. On the other hand, in the actual position of knowledge , all the existing methods to break these codes are exponential in the size of the key. It is practical disproportion between the time of coding or decoding with key on the one hand, and of breaking on the other hand, which make the methods useful.

There is for the moment no theoretical objection with the existence of polynomial algorithms of breaking of the codes used currently, but just the report practices that these problems resist the sustained efforts of the community since sufficiently a long time. Let us note in addition that the quantum computers, if one sometimes happens to build some of “size” (number of qbits) sufficient, would make it possible to break systems like RSA.

Lastly, it is important to specify that the evidence of safety is to be taken with precaution. For example, a system which one owes in Ajtai and Dwork, accompanied by a proof of theoretical safety supposing a certain difficult problem, is found broken in practice by Phong Nguyen and Jacques Stern.

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