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What is the probability that a wafer contains three or more particles in the inspected location Let E denote the event that a wafer contains three or more particles in the inspected location. Then, E consists of the three outcomes {3, 4, 5 or more}. Therefore, P1E2 EXAMPLE 2-12 0.10 0.05 0.10 0.25

Suppose that a batch contains six parts with part numbers {a, b, c, d, e, f }. Suppose that two parts are selected without replacement. Let E denote the event that the part number of the rst part selected is a. Then E can be written as E {ab, ac, ad, ae, af }. The sample space can be enumerated. It has 30 outcomes. If each outcome is equally likely, P1E2 5 30 1 6. Also, if E2 denotes the event that the second part selected is a, E2 {ba, ca, da, ea, fa} and with equally likely outcomes, P1E2 2 5 30 1 6.

(5.42) (5.43) (5.44) (5.45)

2-2.2

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Now that the probability of an event has been de ned, we can collect the assumptions that we have made concerning probabilities into a set of axioms that the probabilities in any random experiment must satisfy. The axioms ensure that the probabilities assigned in an experiment can be interpreted as relative frequencies and that the assignments are consistent with our intuitive understanding of relationships between relative frequencies. For example, if event A is contained in event B, we should have P1A2 P1B2 . The axioms do not determine probabilities; the probabilities are assigned based on our knowledge of the system under study. However, the axioms enable us to easily calculate the probabilities of some events from knowledge of the probabilities of other events.

where A , B and C are 32-bit words, A is the AND operation, V is the OR operation, @ is the XOR, and 1A is the complement of A. The MD5 algorithm pads the message using the same rnethod as MD4. As in the MD4 attack, the padding is not important for the attack we discuss here. And, as with MD4, the MD5 hash operates on 512-bit blocks of data, with the 128-bit output of one block being used as the initial value for the next block. The IV used in MD5 is the same as that used in MD4, which appears in (5.7) in Section 5 . 3 . The MD5 algorithm is given in Table 5.7 and the four MD5 round functions are given in Table 5.8. Here, we number the steps consecutively from 0 to 63, with the i t h output denoted by Q i . These Q values correspond t o the A , B , C and D values in [122]. Each step of MD5 has its own additive constant. We denote the constant for step i as Ki. Although these constants play no role in the attack, for completeness, the I, are given in the Appendix in Table A-1. The shift for ( step i is denoted si and the values of si are listed in Table 5.9. In Table 5.7, the permutation applied to the input blocks is denoted by a, that is, Wi = X,,(i). The values of ~ ( i for i = 0 , 1 , 2 , . . . 63, are given in ), Table 5.10. The significant differences between MD4 and MD5 are the following [122]:

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