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Introduction to cryptographic notions

What is cryptography ?

Cryptography can be defined essentially as data treatment techniques that render certain actions by some people impossible, very difficult, or very improbable, while allowing these actions to others without problems.  Cryptography consists mainly in setting up data operations that render certain actions or operations a practically untraceable puzzle, while nevertheless allowing other operations to be performed relatively easily.

The original application of cryptography was of course "secret communication".  This is still a very important part of cryptography, but in the mean time, the cryptographic applications have exploded, and "secret communication" has become just one, although still important, aspect of it.  Secret communication consists in making public a source of information, that conveys a message that is unintelligible to most people, but that is perfectly intelligible to others, because these others possess an extra amount of information that renders the message visible.   In the standard secret communication application, one considers "friends" and "enemies".   The friends are the people that are supposed to understand the message, and hence are in possession of the extra amount of information (often called the key) that allows them to render the message intelligible ; the enemies are all the others, who may eavesdrop on the source, are not supposed to possess the key. We want them to be in the impossibility of reading the message. A friend is a person possessing keys, and is hence able to do operations that others are not able to do.  People who are targeted not to be able to do something, are called enemies.  One shouldn't consider this too much in the military sense, although of course, there are similarities and originally, cryptography was mostly a military affair.

We will see that modern cryptography makes these concepts much more involved: some keys are distributed publicly, sometimes agents are at the same time enemies and friends.  In as much as "allowing friends to do certain actions on data, which are rendered impossible, hard to do, or very improbable to succeed, for enemies" becomes an essential part of economy in our increasingly data-driven world, one can imagine the importance that cryptography has taken the last few decades. Nevertheless, cryptography is still a very poorly understood domain by most users who depend nevertheless very strongly on it for their economic activity, their privacy and even their security.  This is a worrisome situation.  You could think of it as equivalent to most merchants not mastering  arithmetic sufficiently to do the bookkeeping of their trades !

Cryptography is a technology, mainly based upon mathematics, computing and entropy, as we will see.  Let us return to the standard application: secret communication.  If the goal were simply to send out information to be understood by everybody, we would be in the normal communication engineers' domain.  Everybody would be a friend.  If the goal were to make information unintelligible, that wouldn't be too much of a hassle too: adding sufficient noise to it would swamp it mostly and render it unintelligible to everybody.  Everybody would be an enemy.  The subtlety in cryptography is of course that one needs techniques which make the operation of rendering the information readable to friends much more easy by using a key, than any operation on that same information by enemies, not possessing the key.   The intelligible message to be shared with friends is called the clear text.  The unintelligible message is called the cypher text.  The extra amount of information shared amongst friends is called the key.   The cypher text should be produced with the help of the key, from the clear text.   The clear text should be produced again from the cypher text, with the key.  Without the key, it should be impossible, very difficult or highly improbable to produce the clear text from the cypher text.  As such, a mathematical operation should be found which does exactly that.   However, mathematics is usually not involved with not finding solutions.  This is what makes cryptography special: one should have mathematical puzzles which are hard or impossible to solve !  This is also what makes "doing cryptography in your basement" dangerous.  Indeed, in as much as it is always possible to show that a mathematical puzzle can be solved, it is much harder, or even essentially impossible, to show that a mathematical puzzle cannot be solved.  It is not because you cannot solve it, or your friends cannot solve it, that nobody can solve it.  As such, there is in fact no way to test the reliability of a cryptographic technique individually. The only thing one can do, is to publish the technique, hope it will get a certain attention by sufficient cryptographers in the world who try to "break" it, where breaking a cryptographic technique consists essentially in finding a way to do with the data what an enemy wasn't supposed to be able to do with it.

This is why the golden rule in cryptography is:  for important cryptographic applications, only use well-known, published techniques, and don't invent your own in your basement.

If there is one thing to remember in cryptography, it is the above statement.

Cryptographic applications and systems

As we said before, the original application of cryptography was "secret communication", but in the mean time, the applications have exploded, and "secret communication" has become only one of many applications.

 As in any engineering, there is a bottom-up structure.  At the basis, there are fundamental mathematical cryptographic techniques.  One step up, there are the application blocks such as listed above: there is still a strong link between a certain technique, and a specific application.  Next, these application blocks can be combined in different ways to make up a cryptographic system with a certain use.  Even though the blocks themselves are already simple systems by themselves, much more complex systems can be constructed on top of that.

Safe web browsing is one such system.  Anonymous communication another system.  Crypto currencies are another recent development.