| 16 | bgcolor="#F2F2F2" | 64 | bgcolor="#F2F2F2" | | 6.1 | | | | | | | | | |- align="center" | bgcolor="#F2F2F2" | (24) | bgcolor="#F2F2F2" | (96) | bgcolor="#F2F2F2" | () | | | | | | | | | | |- align="center" | bgcolor="#F2F2F2" | 32 | bgcolor="#F2F2F2" | 128 | bgcolor="#F2F2F2" | | | | | | | | | | | |- align="center" | bgcolor="#F2F2F2" | (48) | bgcolor="#F2F2F2" | (192) | bgcolor="#F2F2F2" | () | | | | | | | | | | |- align="center" | bgcolor="#F2F2F2" | 64 | bgcolor="#F2F2F2" | 256 | bgcolor="#F2F2F2" | | | | | | | | | | | |- align="center" | bgcolor="#F2F2F2" | (96) | bgcolor="#F2F2F2" | (384) | bgcolor="#F2F2F2" | () | | | | | | | | | | |- align="center" | bgcolor="#F2F2F2" | 128 | bgcolor="#F2F2F2" | 512 | bgcolor="#F2F2F2" | | | | | | | | | | | |} The lighter fields in this table show the number of hashes needed to achieve the given probability of collision (column) given a hash space of a certain size in bits (row). Using the birthday analogy: the "hash space size" resembles the "available days", the "probability of collision" resembles the "probability of shared birthday", and the "required number of hashed elements" resembles the "required number of people in a group". One could also use this chart to determine the minimum hash size required (given upper bounds on the hashes and probability of error), or the probability of collision (for fixed number of hashes and probability of error). For comparison, to is the uncorrectable bit error rate of a typical hard disk. In theory, 128-bit hash functions, such as MD5, should stay within that range until about documents, even if its possible outputs are many more. An upper bound on the probability and a lower bound on the number of people The argument below is adapted from an argument of Paul Halmos. As stated above, the probability that no two birthdays coincide is As in earlier paragraphs, interest lies in the smallest such that ; or equivalently, the smallest such that . Using the inequality in the above expression we replace with . This yields Therefore, the expression above is not only an approximation, but also an upper bound of . The inequality implies . Solving for gives Now, is