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For example, assume that the password space is represented by P = {a0,a1,a2,....,a8,a9,a10,a11,a12,...,z13}. In this case, you do not check each possible username, but rather, you check a single username, then a two-character password, then a four-character password, and so on, using a method similar to the one described above. This is faster, but you are no longer able to detect differences in fingerprints. The memory requirement remains the same, however.
Actually, this password space is not completely usable. Instead of having a set of characters used for individual characters, a better option would be to have three sets of characters, A, B, and C. A would correspond to the first half of the password space, B to the second half, and C to the last half.
Because there are already 220 unique fingerprints in the table, we have to remove some of them to make room for the new ones, and that reduces the data space by a factor of about 200. For example, suppose the fingerprint is h(P0), and that the user wants a password starting with the letter “M” and with the first character of the alphabet. This password would produce the fingerprint Yg(x)h(x)h(P0)h(x)h(x). If we remove the space in the middle, we get Yg(x)h(x)h(P0)h(x) and then two hash functions to produce the fingerprint Yg(x)h(x)h(P0)h(x)h(x)h(x)h(x). The fingerprint that starts with a 3-digit number, which is the first number of the alphabet, is removed, and the final fingerprint is Xs,h(P0)h(P0).
A set of numbers is generated by adding sequentially the fingerprints of numbers one through 20. For example, suppose that we have already generated the fingerprints of numbers 1 through 6, and that the fingerprint of your password begins with fingerprint 4. Then, the next fingerprint to be generated is h(x)h(x)h(x)h(x)h(x)h(x) and so on until the 10-digit number. The resulting number is a password.
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