An authentication method of a first module associated with an identity ID_A by a second module, implemented according to an identity mapping scheme involving a bilinear function (e), the method comprising the steps of: - generating, by the second module, a first random datum to be sent to the first module; - generating by the first module a first number ( v; v') starting from said first datum and by means of a private key s Q _ A; 1 a + s Q 1 of the first module, comprising public key Q_A of the first module obtained from ID_A, number a obtained from ID_A, and master key s of a Trusted Authority associated with the first module, wherein said generating the first number comprises further - applying the bilinear function (e) to the first random datum and to the private key s Q _ A; 1 a + s Q 1 of the first module; - sending by the first module the first number ( v, v ') to the second module; - generating by the second module a second number ( n; n') said step of generating the second number ( n; n ') being carried out starting from public parameters ( Q_A; P Pub; P 1; Q 1) and being independent of the step of generating said first number, wherein at least one of the public parameters includes a first parameter being correlated to the public key (Q_A) of the first module and wherein said public parameters include second, third and fourth parameters being representative of public points ( P Pub), ( P 1), and ( Q 1) associated with the Trusted Authority of the first module; said generating the second number comprising further - applying the bilinear function (e) to first parameter (Q_A) and second parameter ( P Pub); - characterized by further comprising: - comparing by the second module said second number ( n; n ') with said first number ( v; v') such as to authenticate the first module, the steps of generating the first ( v; v') and second ( n; n ') numbers being at least partially simultaneous. - generating, by the second module, a first random datum to be sent to the first module; - generating by the first module a first number ( v; v') starting from said first datum and by means of a private key s Q _ A; 1 a + s Q 1 of the first module, comprising public key Q_A of the first module obtained from ID_A, number a obtained from ID_A, and master key s of a Trusted Authority associated with the first module, wherein said generating the first number comprises further - applying the bilinear function (e) to the first random datum and to the private key s Q _ A; 1 a + s Q 1 of the first module; - sending by the first module the first number ( v, v ') to the second module; - generating by the second module a second number ( n; n') said step of generating the second number ( n; n ') being carried out starting from public parameters ( Q_A; P Pub; P 1; Q 1) and being independent of the step of generating said first number, wherein at least one of the public parameters includes a first parameter being correlated to the public key (Q_A) of the first module and wherein said public parameters include second, third and fourth parameters being representative of public points ( P Pub), ( P 1), and ( Q 1) associated with the Trusted Authority of the first module; said generating the second number comprising further - applying the bilinear function (e) to first parameter (Q_A) and second parameter ( P Pub); - characterized by further comprising: - comparing by the second module said second number ( n; n ') with said first number ( v; v') such as to authenticate the first module, the steps of generating the first ( v; v') and second ( n; n ') numbers being at least partially simultaneous.