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# In multi-factor authentication, a system may require that a user authenticates using more than one method. For example, it can require both a...

In multi-factor authentication, a system may require that a user authenticates using more than one method. For example, it can require both a password and a fingerprint scan. Let us assume a two-factor authentication system that uses methods A and B in such a way that the evidence associated with both A and B must be presented to the system at the same time for it to determine if login will be allowed. Furthermore, we define the two methods to be independent if compromise of method A provides no advantage to an adversary for compromising method B. Answer the following questions with this definition of independent authentication factors. Assume that the entropy for method A is e1 and it is e2 for method B.

The hardening password paper updates the instruction table for feature i based on the value of | µI - ksI |. Explain how password hardening entropy and false negative rate will change as k is increased.