authentication methods are provided that allow for superior security , power consumption , and resource utilization over existing authentication methods . by computing only two hashes of a shared secret password for each protocol run , the methods described in this disclosure dramatically reduce the computational power needed to perform authentication . similarly , by exchanging these hashes bitwise or piecewise for verification , rather than performing new hashes including each bit of the password separately , the methods described in this disclosure reveal less information about the password being authenticated than existing methods . the methods described in this disclosure also allow for authentication using fewer messages and with lower latency , reducing the amount of operational power used in the authentication process .