An authenticatable device for use with an authentication server by processing a cryptographic operation on auxiliary data ( Aux ) communicated by the authentication server to the authenticable device, the authenticatable device comprising: one or more physically-unclonable function, 'PUF', devices, constructed to generate, in response to the input of a challenge (c), an output value that is characteristic to the PUF and the challenge; and a processor connected to the one or more PUF devices, the processor configured to: issue a first challenge ( c 1 ) via the processor output to a first PUF device of the one or more PUF devices and receive a first output ( O' 1 ) from the first PUF device in response to issuing the first challenge ( c 1 ); use the first output ( O' 1 ) to recover, internally to the authenticatable device, a first share ( r 1 ) of a private key or secret associated with the first output ( O' 1 ); and perform, internally to the authenticatable device, a first threshold cryptographic operation with the first share ( r 1 ) and the auxiliary data ( Aux ); issue a second challenge (c 2 ) to the first PUF device or a second PUF device of the one or more PUF devices and receive, from the first PUF device or the second PUF device, a second output ( O' 2 ) in response to issuing the second challenge( c 2 ); use the second output ( O' 2 ) to recover, internally to the authenticatable device, a second share ( r 2 ) of the private key or secret associated with the second output ( O' 2 ), wherein the second share ( O' 2 ) is different from the first share ( O' 1 ); and perform, internally to the authenticatable device, a second threshold cryptographic operation with the second share ( r 2 ) and the auxiliary data ( Aux ); and enable processing of the cryptographic operation with a combined output of at least the first and second threshold cryptographic operations. issue a first challenge ( c 1 ) via the processor output to a first PUF device of the one or more PUF devices and receive a first output ( O' 1 ) from the first PUF device in response to issuing the first challenge ( c 1 ); use the first output ( O' 1 ) to recover, internally to the authenticatable device, a first share ( r 1 ) of a private key or secret associated with the first output ( O' 1 ); and perform, internally to the authenticatable device, a first threshold cryptographic operation with the first share ( r 1 ) and the auxiliary data ( Aux ); issue a second challenge (c 2 ) to the first PUF device or a second PUF device of the one or more PUF devices and receive, from the first PUF device or the second PUF device, a second output ( O' 2 ) in response to issuing the second challenge( c 2 ); use the second output ( O' 2 ) to recover, internally to the authenticatable device, a second share ( r 2 ) of the private key or secret associated with the second output ( O' 2 ), wherein the second share ( O' 2 ) is different from the first share ( O' 1 ); and perform, internally to the authenticatable device, a second threshold cryptographic operation with the second share ( r 2 ) and the auxiliary data ( Aux ); and enable processing of the cryptographic operation with a combined output of at least the first and second threshold cryptographic operations. <EOS>