An implicit certificate is based on a ring learning with errors (“RLWE”) public keys that are, in some examples, resistant to quantum-based computing attacks. Various methods are described that request, generate, verify, and use the implicit certificates. In some examples, the system provides an implicit certificate that enables communication between two parties that are identified at the time of certificate generation. In another example, the system provides a certificate that may be used to communicate with a variety of different parties. The implicit certificate generation algorithm yields a public key purportedly bound to U. Confirmation that the public key is bound to U is obtained after use of the corresponding private key. Binding of an entity to its associated public key and accessibility to the private key, are verified as a result of successful key use.