We disclose methods and apparatuses for securing cryptographic devices against attacks involving external monitoring and analysis. A "self-healing" property is introduced, enabling security to be continually re-established following partial compromises. In addition to producing useful cryptographic results, a typical leak-resistant cryptographic operation modifies or updates secret key material in a manner designed to render useless any information about the secrets that may have previously leaked from the system. Exemplary leak-proof and leak-resistant implementations are shown for symmetric authentication, certified Diffie-Hellman (when either one or both users have certificates), RSA, ElGamal public key decryption.