Digital signatures are just like hand-written ones, which guarantee integrity, authenticity, and nonrepudiation. On the basis of an intractable mathematical assumption, that is, bilinear square Diffie–Hellman problem, we give a concrete construction of the first provably secure probabilistic signature scheme and its extension to a universal designated verifier signature scheme. A universal designated verifier signature scheme allows any signature holder to designate a publicly verifiable signature to a specific verifier such that only the designated verifier can validate the signature for protecting the privacy of signature holder. In addition, the designated verifier is unable to convince any third party of the fact. Such schemes can be applied to privacy-preserving electronic applications such as the certificate for medical records and income summary. Our scheme is especially suitable for computation constrained devices because the signature generation and designation are pairing free. Copyright © 2012 John Wiley & Sons, Ltd.