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One of the design problems when using a SOM is deciding on the size of the map. Too many neurons may cause overfitting of the training patterns, with small clusters containing a few patterns. Alternatively, the final SOM may have succeeded in forming good clusters of similar patterns, but with many neurons with a zero, or close to zero frequency. The frequency of a neuron refers to the number of patterns for which that neuron is the winner, referred to as the best matching neuron (BMN). Too many neurons also cause a substantial increase in computational complexity. Too few neurons, on the other hand, will result in clusters with a high variance among the cluster members. An approach to find near optimal SOM architectures is to start training with a small architecture, and to grow the map when more neurons are needed. One such SOM growing algorithm is given below, assuming a square map structure. Note that the map-growing process coexists with the training process. 1. Initialize the codebook vectors for a small, undersized SOM. 2. Grow the map: (a) Train the SOM for pattern presentations, using any SOM training method. (b) Find the neuron kj with the largest quantization error. (c) Find the furthest immediate neighbor mn in the row-dimension of the map, and the furthest neuron op in the column-dimension. (d) Insert a column between neurons kj and op and a row between neurons kj and mn (this step preserves the square structure of the map).

7. What is the advantage of a cipher (such as the Enigma) that is its own inverse, as compared to a cipher that is not (such as Purple and Sigaba)

b. Explain how this restriction gives the cryptanalyst an advantage when searching for a suitable crib. 9. Consider the Enigma attack discussed in the text and suppose that only cycles of S ( E ) are used to recover the correct rotor settings. Then, after the attack is completed, only the stecker value of S ( E ) is known. Using only the matched plaintext and ciphertext in Table 2.2, how many additional stecker values can be recovered

8.28 The Web dialog allows you to set how often Aperture checks for updates to a MobileMe Gallery album, among other things.

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10. Write a program to simulate the Enigma cipher. Use your program to answer the following questions, where the rotor and reflector perrnutations are known to be

(e) For each neuron ab in the new column, initialize the corresponding codebook vectors wab using wab = a(wa,b-i + wa,b+i) and for each neuron in the new row,

where Re is the left rotor, R ,is the middle rotor, R, is the right rotor, , and T is the reflector. Thc notch that causes the odometer effect is at position Q for Re, V for R,, and J for R,. For example, the middle rotor steps when the right rotor steps from V to W.

8.29 The MobileMe Gallery Album Settings sheet lets you adjust your album s settings after you create your album.

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Plaintext A D H 0 C A D L 0 C 9 U I D P R 0 9 U 0 S 0 Ciphertext S W Z S 0 F C J M D C V U G E L H S M B G G

where a 6 (0, 1) (f) Stop growing when any one of the following criteria is satisfied: the maximum map size has been reached; the largest neuron quantization error is less than a user specified threshold, A; the map has converged to the specified quantization error 3 Refine the map: Refine the weights of the final SOM architecture with additional training steps until convergence has been reached A few aspects of the growing algorithm above need some explanation These are the constants A, 7 and the maximum map size A good choice for a is 05 The idea of the interpolation step is to assign a weight vector to the new neuron ab such that it removes patterns from the largest quantization error neuron kj in order to reduce the quantization error of neuron kj A value less than 0.

To remove a MobileMe Gallery Album, select it in the Library Inspector and choose File Delete MobileMe Album.

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Plaintext L I T T L E T I M E S 0 M U C H T 0 K N 0 W Ciphertext N B S M q T 9 Z I Y D D X K Y N E W J K Z R

In the past few years, Facebook has exploded from a private social-networking site with its membership limited to select colleges into one of the most trafficked sites on the Web. Everyone and his or her Uncle Bob seem to have a profile. Aperture 3 provides direct Facebook integration, which lets you directly publish your images to Facebook rather than using Facebook s clunky image uploader.

11. Suppose that the same Enigma rotors (in the same order) and reflector are used as in Problem 10, and the stecker has no cables connected. Solve for the initial rotor settings and recover the plaintext given the following ciphertext.

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