Method for authenticating an entity with a verifier, the entity possessing an identifier, the verifier possessing a private key (x) / public key (y) pair, comprising: - a sending to the entity of a first random (r) chosen by the verifier, - a step of encrypting a value (v) by the entity by means of the public key of the verifier, said value comprising the first random and an authentication datum comprising the identifier of the entity and an authentication message computed by means of a secret and of a message to be protected, said secret depending on the identifier and said message to be protected comprising at least one part of the first random (r), said encryption step complying with a public-key encryption scheme, called the modified El Gamal scheme, in which: - the step of encrypting the value (v) produces two elements T 1 ′ = v ⊕ H y w, and T 2 ′ = g w, where - H is the identity function or a hash function and ⊕ an exclusive or operation, - g is a generator of a group used by said scheme, - y is the public key with which the private key x of the verifier is associated, said keys being such that y = g x, - w is a second random chosen by the entity, - the step of decryption by the verifier of the two elements T 1 and T 2 computes T 1 ′ ⊕ H ′ T 2 ′ x so as to provide said value (v), - a sending in response, by the entity, of a message comprising said encrypted value, for authentication of said entity. - a sending to the entity of a first random (r) chosen by the verifier, - a step of encrypting a value (v) by the entity by means of the public key of the verifier, said value comprising the first random and an authentication datum comprising the identifier of the entity and an authentication message computed by means of a secret and of a message to be protected, said secret depending on the identifier and said message to be protected comprising at least one part of the first random (r), said encryption step complying with a public-key encryption scheme, called the modified El Gamal scheme, in which: - the step of encrypting the value (v) produces two elements T 1 ′ = v ⊕ H y w, and T 2 ′ = g w, where - H is the identity function or a hash function and ⊕ an exclusive or operation, - g is a generator of a group used by said scheme, - y is the public key with which the private key x of the verifier is associated, said keys being such that y = g x, - w is a second random chosen by the entity, - the step of decryption by the verifier of the two elements T 1 and T 2 computes T 1 ′ ⊕ H ′ T 2 ′ x so as to provide said value (v), - a sending in response, by the entity, of a message comprising said encrypted value, for authentication of said entity. - the step of encrypting the value (v) produces two elements T 1 ′ = v ⊕ H y w, and T 2 ′ = g w, where - H is the identity function or a hash function and ⊕ an exclusive or operation, - g is a generator of a group used by said scheme, - y is the public key with which the private key x of the verifier is associated, said keys being such that y = g x, - w is a second random chosen by the entity, - H is the identity function or a hash function and ⊕ an exclusive or operation, - g is a generator of a group used by said scheme, - y is the public key with which the private key x of the verifier is associated, said keys being such that y = g x, - w is a second random chosen by the entity, - the step of decryption by the verifier of the two elements T 1 and T 2 computes T 1 ′ ⊕ H ′ T 2 ′ x so as to provide said value (v), - a sending in response, by the entity, of a message comprising said encrypted value, for authentication of said entity.