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We introduce a new class of public-key functions involving a number n = pq having two large prime factors. As usual, the key n is public, while p and q are the private key used by the issuer for production of signatures and function inversion. These functions can be used for all the applications involving public-key functions proposed by Diffie and Hellman, including digitalized signatures. We prove that for any given n, if we can invert the function y = E (x1) for even a small percentage of the values y then we can factor n. Thus, as long as factorization of large numbers remains practically intractable, for appropriate chosen keys not even a small percentage of signatures are forgeable. Breaking the RSA function is at most hard as factorization, but is not known to be equivalent to factorization even in the weak sense that ability to invert all function values entails ability to factor the key. Computation time for these functions, i.e. signature verification, is several hundred times faster than for the RSA scheme. Inversion time, using the private key, is comparable. The almost-everywhere intractability of signature-forgery for our functions (on the assumption that factoring is intractable) is of great practical significance and seems to be the first proved result of this kind.