A method (and structure) of exchange between two parties interconnected by a device or network. A recipient party (verifier) chooses a secret value x for computing a value X=F1(x), where F1 comprises a first predetermined function having at least one argument, the value x being one of the at least one argument of F1. A signing party (signer) chooses a secret value y for computing a value Y=F2(y), where F2 comprises a second predetermined function having at least one argument, the value y being one of the at least one argument of F2. The signer obtains the value X, and the signer has a private key b and a public key B. The signer computes a value s=F3(y,b,X), where F3 comprises a third predetermined function having at least three arguments: the value y, the private key b, and the value X being three arguments of the at least three arguments of F3. There exists a fourth predetermined function F4(x,Y,B) to calculate a value s', F4 having at least three arguments: the value x, the value Y, and the public key B being three arguments of the at least three arguments of F4, but the value s is not an argument of F4. There exists no secret shared between the verifier and the signer that serves as a basis for any argument in any of the functions F1, F2, F3, and F4. The verifier can consider the values s and s' as valid authenticators if value s' is determined to be related in a predetermined manner to value s.