Authentication method

A method of authenticating a first module with a second module, comprising:

• generating with the second module a random datum;
• transmitting the random datum to the first module;
• generating with the first module a first number by starting with the random datum and applying a private key associated with the first module through a bilinear function comprising a Tate pairing;
• at least partially concurrently with the generating of the first number, generating with the second module a second number using public parameters;
• and
• authenticating the first module by operating the second module to directly compare the second number with the first number,
• wherein generating the first and second numbers are performed concurrently, the first module comprises a smart card, a USB data key, a computer or a cell phone, and the second module comprises a cell phone, a set-top-box, a server or a computer.

An authentication method of a first module by a second module, comprising:

• generating with the second module a random datum, wherein the generating of the random datum comprises selecting a random integer and computing a scalar multiplication between the selected random integer and a further public parameter to encrypt the random integer;
• transmitting the random datum to the first module;
• generating with the first module a first number by starting with the random datum and applying a private key associated with the first module;
• wherein the generating with the first module the first number comprises applying a bilinear function to the random datum and to the private key of the first module and wherein the bilinear function comprises a Tate pairing;
• and
• generating with the second module a second number to be compared with the first number to authenticate the first module, wherein the generating of the second number is carried out starting from public parameters and is performed independent of the generating of the first number,
• wherein generating the first and second numbers are performed concurrently, the first module comprises a smart card, a USB data key, a computer or a cell phone, and the second module comprises a cell phone, a set-top-box, a server or a computer.