The theoretical foundation

Computational vs unconditional security

Not all security is equal. There is a sharp boundary between what resists any amount of computing power and what resists only reasonably available resources.

Two levels of security

Unconditional security (also called perfect): the cipher is unbreakable even against an adversary with infinite computing power. No amount of computation reveals the message.
Computational security: the cipher is breakable in theory, but doing so would require time or resources out of reach in practice.

The one-time pad: unconditional security

The one-time pad combines each bit of the message with a key bit that is truly random, as long as the message, and never reused. Shannon proved that the ciphertext reveals no information about the plaintext: all plaintexts are equally possible.

This security is proven and absolute — but its price is prohibitive: a key as long as everything one wishes to exchange, transmitted securely. That is why it remains reserved for rare and critical uses.

RSA, AES: computational security

Nearly all deployed cryptography — RSA, AES, elliptic curves — offers only computational security. In principle, an attacker who tried every key, or who factored the RSA modulus, would eventually succeed.

But "eventually" here means astronomical durations. Breaking AES-128 by brute force, or factoring a 2048-bit RSA modulus, far exceeds what humanity can muster.

   +------------------+---------------------+----------------------+
   |                  | INCONDITIONNELLE    | CALCULATOIRE         |
   +------------------+---------------------+----------------------+
   | Exemple          | masque jetable      | RSA, AES, ECC        |
   | Cassable ?       | non, jamais         | oui, en theorie      |
   | Puissance infinie| resiste             | tombe                |
   | En pratique      | incassable          | hors de portee       |
   | Repose sur       | l'information       | la difficulte calcul |
   | Cout             | cle enorme          | cles courtes         |
   +------------------+---------------------+----------------------+

The notion of reduction

How can computational security be justified without absolute proof? Through a reduction. We prove the implication:

   casser le schema  ===>  resoudre le probleme dur

In other words: if there existed an efficient way to break RSA, then there would exist an efficient way to factor. Since we firmly believe factoring is hard, we deduce that breaking RSA is too.

Security thus rests not on a direct proof, but on a transfer of trust to a well-studied problem.

	Unconditional	Computational
Guarantee	mathematical, absolute	depends on the hardness of a problem
Facing `P = NP`	always holds	may collapse
Examples	one-time pad	RSA, AES, ECC
Real-world adoption	marginal	almost universal

In summary