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In functional link neural networks (FLNN) input units do implement activation functions. A FLNN is simply a FFNN with the input layer expanded into a layer of functional higher-order units [Ghosh and Shin 1992, Hussain et al. 1997]. The input layer, with dimension /, is therefore expanded to functional units h 1 , h 2 , - ,hL, where L is the total number of functional units, and each functional unit hi is a function of the input parameter vector (z 1 , , z I ), i.e. h l (z 1 , , zI) (see Figure 3.2). The weight matrix U between the input layer and the layer of functional units is defined as 1 if functional unit hl is dependent of Zi , , 0 otherwise For FLNNs, Vji is the weight between hidden unit yj and functional link hi. Calculation of the activation of each output unit ok occurs in the same manner as for FFNNs, except that the additional layer of functional units is taken into account:

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The famous Project VENONA [151] details this impressive cryptanalytic success The VENONA decrypts provide tremendous insight into Soviet spying in general, and nuclear espionage in particular Modern stream ciphers are a generalization of the one-time pad, where provable security is traded for practicality In a stream cipher a short secret key is stretched into a long pseudo-random string of bits, which is then used just like a one-time pad The provable security is lost since the number of possible keys is much smaller than the number of possible messages Stream ciphers are discussed in 3..

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The use of higher-order combinations of input units may result in faster training times and improved accuracy (see, for example, [Ghosh and Shin 1992, Hussain et al. 1997]).

At the price of $276,495 Alf Blowfish sells his house to Ann Bonidea ... Figure 5.4: MD4 collision [42].

MD5 attack, we give an example that shows how a meaningless collision can be used to break security in a very meaningful way.

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Product unit neural networks (PUNN) have neurons which compute the weighted product of input signals, instead of a weighted sum [Durbin and Rumelhart 1989, Janson and Frenzel 1993, Leerink et al. 1995]. For product units, the net input is computed as given in equation (2.2). Different PUNNs have been suggested. In one type each input unit is connected to SUs, and to a dedicated group of PUs. Another PUNN type has alternating layers of product and summation units. Due to the mathematical complexity of having PUs in more than one hidden layer, this section only illustrates the case for which just the hidden layer has PUs, and no SUs. The output layer has only SUs, and linear activation functions are assumed for all neurons in the network. Then, for each hidden unit yj, the net input to that hidden unit is (note that no bias is included)

Message Digest 5, or MD5, is a strengthened version of MD4. Like MD4, the MD5 hash was invented by Rivest [122]. Also, MD5 was obviously used as the model for SHA-1, since they share many common features. It is undoubtedly the case that MD5 and SHA-1 are the two most widely used hash algorithms today, but use of MD5 will certainly decline over time, since it is now considered broken. At the time of this writing, it appears certain that SHA-1 will soon share the same fate as MD5.

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MD5 is similar to MD4 and in this section, we often refer to our previous discussion of MD4. It is important to read Section 5.3 carefully before attempting this section, since both the operation of MD4 and various aspects of the MD4 attack will be referenced here.

F ( A ,B, ) = ( AA B ) V ( 1 AA C ) C G ( A ,B, ) = ( AA C ) V ( B A -C) C H(A,B,C) A @ B @ C = I ( A ,B, C) = B @ ( AV -C)

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