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Instead of generating all of the putative 2-lists, then all of the Y-lists, followed by all of the X-lists, we generate a single 2-list, then all of the Ylists that are consistent with this 2-list, then the X-lists that are consistent with the Y-lists. The resulting X-lists are then tested to determine whether we have found the correct list, and the entire process is repeated for each putative 2-list until the solution is found. In this way, we only need to store a relatively small set of lists at any given time--not the entire set of 238 lists. This simplified attack is outlined in Table 3.15. Table 3.15: Outline of PKZIP Attack

into the computer. I set my cameras to Continuous to avoid duplicate lenames for my images. Filenames begin with the designation IMG_ followed by the four-digit image number and file extension: .jpg for JPEG files, or .cr2 for RAW images.

(12-11)

In general, y is an (n 1) vector of the observations, X is an (n p) matrix of the levels of the independent variables, is a ( p 1) vector of the regression coefficients, and is a (n 1) vector of random errors. We wish to find the vector of least squares estimators, , that minimizes

f o r i = 0,1,. . . , 1 2

We will not give the details of taking the derivatives above; however, the resulting equations that must be solved are

(12-12)

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Equations 12-12 are the least squares normal equations in matrix form. They are identical to the scalar form of the normal equations given earlier in Equations 12-10. To solve the normal equations, multiply both sides of Equations 12-12 by the inverse of X X. Therefore, the least squares estimate of is

next i f o r each ( 2 1 2 ) 1 6 ...29 // expect 64 f o r each (212)0...15 // 216 choices f o r i = 11,10,. . . , O

(X X)

(12-13)

Find (Zi)16...29consistent with (Zi+l)o 29 ... Extend (Zz)l6 ...29 to ( 2 i ) O ...29 next i Complete to 2-list: 2, = (Zi)O,,,31, = 1 , 2 , . . . , 1 2 i Solve for (K)o i = 2,3,. . . , 12 ...7, Solve for Y-lists: i = 3 , 4 , . . . , 1 2 // expect 2" lists f o r each Y-list // expect 2" Solve for (Xi)24 1, i = 4 , 5 , . . . , 1 2 ...3 Solve for Xg using ( X 9 ) 2 4 ...31, ( x 1 0 ) 2 4 ...31, (xll) 4 . 3 and (x12) ...31 2 24 Solve for X-list: X i , i = 4 , 5 , . . . , 1 2 i f (Xi)24,..31 o r i = 8 , 7 , 6 , 5 , 4 verified t h e n f r e t u r n X-list, Y-list, 2-list end i f n e x t Y-list next ( 2 1 2 ) O ...15 n e x t (212)16...29

Note that there are p k 1 normal equations in p k 1 unknowns (the values of , , p , 2. Furthermore, the matrix X X is always nonsingular, as was assumed above, 0 1 k so the methods described in textbooks on determinants and matrices for inverting these matrices can be used to find 1X X2 1. In practice, multiple regression calculations are almost always performed using a computer. It is easy to see that the matrix form of the normal equations is identical to the scalar form. Writing out Equation 12-12 in detail, we obtain

If the indicated matrix multiplication is performed, the scalar form of the normal equations (that is, Equation 12-10) will result. In this form it is easy to see that X X is a ( p p) symmetric matrix and X y is a ( p 1) column vector. Note the special structure of the X X matrix. The diagonal elements of X X are the sums of squares of the elements in the columns of X, and the off-diagonal elements are the sums of cross-products of the elements in the columns of X. Furthermore, note that the elements of X y are the sums of cross-products of the columns of X and the observations 5 yi 6. The fitted regression model is

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