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the ciphcr. If she conducts the attack 220 times, what is the total work, the storage requirement and the success probability b. Suppose Trudy pre-computes E ( P , K ) for a selected plaintext P and every possible kcy K . For cacti attack, Trudy chooses the same plaintext P and obtains the corresponding ciphertext C. Then she simply looks up C in her pre-computed list to obtain the key K . If she again conducts the attack 2 times, what is thc total work, the storage requirement and the success probability c. Suppose Trudy implements Hellman s TMTO attack and, as suggested in the text, she chooses T = m = t = 264/3. If she conducts the attack 220 times, what is the total work, the storage requirement and the success probability

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The 4 4 matrix on the left of the multiplication, of course, is the transformation matrix. (And now that you know the correct terminology, the 4 1 column matrices on the left and right are the post-transformation vector and the original vector.) The addition of the fourth column of the transformation matrix, and the dummy dimension of the vector spaces, allows for the tx , ty , and tz coef cients to effect translation in the x, y, and z directions. The scaling, translation, and even rotation matrices are of a familiar form to those in two dimensions. You should be able to guess the forms of the scale and translation matrices. Rotations around the three axes look like this:

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| Rx ( ) = | | | 1 0 0 0 0 cos( ) sin( ) 0 0 sin( ) cos( ) 0 0 0 0 1 0 -sin( ) 1 0 0 cos( ) 0 0 0 0 1 0 0 0 0 1 0 0 0 1 | | | | | | | | | | | |

approx.: 2.0

7. The key size of CMEA is k = 64 bits and the block size n is variable. Suppose the key is restricted to 32 bits by setting all of the first 32 bits of any key equal to 0. Then, effectively, Ic = 32. Choose the block size to be n = 32 bits. Implement Hellman s TMTO attack on this version of the CMEA block cipher. In the TMTO attack, let m = t = T = 215. Empirically determine the probability of success.

Now, because any angle can be represented as a sequence of rotations around the three axes, any angle can be represented as a product of these three matrices, each of which rotate about a single axis. This means that you can represent any orientation with the product of these translation matrices, which is in itself a 4 4 translation matrix. Likewise, any af ne transformation can be composed into a single matrix with matrix multiplication.

ActionScript 3.0, however, actually provides access to the fourth dimension in the 4D vectors used with 4 4 transformation matrices, not to mention access to the bottom row of those matrices which has heretofore been lled with <0, 0, 0, 1>. The fourth dimension is called w, and it is not used as a real, independent dimension. Nor is it used in 3D space or in any transformation that takes place in 3D space (or rather, it is degenerate; the value 1 is always used for w). Where it is used, however, is during projection.

8. Derive the formula in equation (4.3) for the success probability in Hellman s TMTO. Hint: See the occupancy problem in Feller [49].

Almost every screen today, like this piece of paper, is two-dimensional and displays a two-dimensional image. Yet it s easy to look at the page and discern 3D shapes. That s because these 3D shapes are drawn on a 2D surface in much the same way that your eyes focus an image on your retinas, or cameras focus an image on lm or a sensor. Because your brain processes 2D projections of the 3D world all day long from the day you re born, it s adept at recognizing 3D shapes in arti cial projections as well. There are multiple ways you can project a 3D reality onto a 2D image. Artists have been doing this much longer than computers, so the art world is a much better place to look for different examples of

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