Quantum key distribution theory and practice

1.4 Diffie-Hellman Key Exchange

Diffie-Hellman is the first asymmetric encryption algorithm, invented in 1976, using discrete logarithms in a finite field. It allows two users to exchange a secret key over an insecure medium without any prior secrets. This Public-Key is based on the discrete logarithm in finite field which is hard to solve. In general, we consider a Diffie-Hillman protocol secure when an appropriate mathematical group is used.

Diffie-hellman key-exchange is a way that two persons agree about a number without the third person knows the number. This method is the same quite than Elgamal public-key encryption because the security is based on discrete logarithme and Diffie-Hellman problem [Mer07]. This key-Exchange methode use a group notion denoted by G. Diffie-Hellman the choose number by Alice and Bob still secret Eve in classical scheme can be in the middle understand the communication, she couldn't know the secrete number. The Diffie-Hellman is define also as (DHP) given a prime p, a generator g of Z* p and element ga mod (p) and g^b mod (p),find g^abmod(p). We need to find a generator

1. Alice and Bob will choose the finite group where they will play the security game and generate a generator

2. Alice will choose randomly a number and compute ga

3. Bob must do the same thing than Alice but will choose b as a natural number and compute gb

4. Alice will compute (gb)a

Section 1.5. Information Theory Page 7

5. Bob, he will compute (g^a)^b

For the Diffie-Hellmann protocole, the secret values of Alice and Bob, a and b, must be big numbers. There are steps for encryption which Alice and Bob will follow if they want to share secret messages without Eavesdropper knows.

Note: The computation must be difficult from Alice to solve Bob's private key and from Bob to solve Alice's private key. If the computation is easy, that allows Eve simply to substitute her own private, public key pair, plug Bob's public key into her private key, produce a fake shared secret key, and share it in both sides.

1.5 Information Theory

In our century, the information is still in order the most important in all part of life (societies, entreprises, millitaries) and there are some assumption which expect that if you know to manipulate an information you can do many things. So, people developped a concept in information named Information theory. The principal actor for this theory is nammed Claude Shannon who in 1949 published a article with a title »Communication Theory of Secrecy Systems» in the bell System newspaper that made a big influence in cryptography science. With Shannon theory, we are now able to make a quantification of our information.

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2. Fundamentals of Quantum Mechanics

Classical physics failed to explain phenomenon such as the black body radiation and presence of spectral lines in the spectrum of absorption and emission of some atoms. A new theory known as quantum mechanics arised in the 20^th century and gave a satisfactory explanation of those phenomenon. A conceptual difference between this new theory and classical mechanics is the desapperences of the full description of the trajectory of a particle. Indeed, in quantum mechanics, one speaks of the probability of the particle to be at a certain position at a given time. Quantum mechanics is a physical theory that discribes some systems where h is not negligible anymore. It is especially efficient to describe physical phenomena at molecular scales and above (except at high energies such that solar energy. We need to take into account relativistic effects). We will recall the five postulat of quantum mechanics

1. The knowledge of state of quantum system is completely known in normalizable vector of a hilbert space H usually denoted |?(t)i;

2. The correspondance principle: to any physical observable corresponds an hermitian operator which acts on the vectors of a hilbert space;

3. If we have a initial state ái, the probability that it is at the final state áf is given by

Páf = |háf|áii|², (2.0.1)

If the system is in the state á, the mean value of the measure of an observable is given by

h ^àAi = há| ^àA|ái. (2.0.2)

4. Projection of quantum states Let a_n be an eigenvalue of an operator A and is the output of measurement at time t and let ö be its associated eigenstate. The state of the system after the measure is projected on the eigenspace associated to a_n.

5. Time evolution of state öt is given by the schrödinger equation

in ? ?t|ö, ti = ^àH|ö,ti. (2.0.3)

One notices that H is the generator of time translation and is called the hamiltonian is the

operator is associated to the energy of the system and the hermitian have a dimension tw.