cryptographic methods and apparatus for forming , checking , blinding , and unblinding of undeniable signatures are disclosed . the validity of such signatures is based on public keys and they are formed by a signing party with access to a corresponding private key , much as with public key digital signatures . a difference is that whereas public key digital signatures can be checked by anyone using the corresponding public key , the validity of undeniable signatures is in general checked by a protocol conducted between a checking party and the signing party . during such a protocol , the signing party may improperly try to deny the validity of a valid signature , but the checking party will be able to detect this with substantially high probability . in case the signing party is not improperly performing the protocol , the checking party is further able to determine with high probability whether or not the signature validly corresponds to the intended message and public key . blinding can be used while obtaining undeniable signatures , while providing them to other parties , and while checking their validity .