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rj+n/2 = sj j sj from the FFT algorithm (Figure 29), via a pattern of wires known as a butter y: Unraveling the FFT circuit completely for n = 8 elements, we get Figure 104 Notice the following 1 For n inputs there are log 2 n levels, each with n nodes, for a total of n log n operations 2 The inputs are arranged in a peculiar order: 0, 4, 2, 6, 1, 5, 3, 7 Why Recall that at the top level of recursion, we rst bring up the even coef cients of the input and then move on to the odd ones Then at the next level, the even coef cients of this rst group (which therefore are multiples of 4, or equivalently, have zero as their two least signi cant bits) are brought up, and so on To put it otherwise, the inputs are arranged by increasing last bit of the binary representation of their index, resolving ties by looking at the next more signi cant bit(s) The resulting order in binary, 000, 100, 010, 110, 001, 101, 011, 111, is the same as the natural one, 000, 001, 010, 011, 100, 101, 110, 111 except the bits are mirrored! 3 There is a unique path between each input a j and each output A( k ) This path is most easily described using the binary representations of j and k (shown in Figure 104 for convenience) There are two edges out of each node, one going up (the 0-edge) and one going down (the 1-edge) To get to A( k ) from any input node, simply follow the edges speci ed in the bit representation of k, starting from the rightmost bit (Can you similarly specify the path in the reverse direction ) 4 On the path between aj and A( k ), the labels add up to jk mod 8 Since 8 = 1, this means that the contribution of input a j to output A( k ) is aj jk , and therefore the circuit computes correctly the values of polynomial A(x) 5 And nally, notice that the FFT circuit is a natural for parallel computation and direct implementation in hardware 72.

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Security and access controls today and tomorrow will largely be based on authorization models that use roles, group memberships, and data attributes This is because the users in many situations are unknown not only to the database, but sometimes even to the application! Therefore, no user entry will exist in an application s USERS table (for example), nor will an entry exist in the Oracle Database USER$ table and perhaps not even in the local LDAP directory With no user entry, you have no access control list (ACL) Without a way to capture users ahead of time, identity is meaningless for security enforcement purposes To help you understand a bit better, consider an example that consists of a Web services architecture that federates or makes many calls as part of a single business transaction Each of these services may be on separate servers, using separate databases, providing separate information for separate and potentially shared purposes As such, the ability to execute a multi-call transaction requires some way of conveying to all the services, and subsequently the programs that access the databases, that the user is authorized or unauthorized to perform an action Ideally, this model needs to support an infinite number of users and needs to be as adaptable as the standards on which it relies The actual user identities will therefore not be stored locally Authorization information and other aspects about the data and how the information is being accessed or used will be employed to ensure that proper access is controlled User identities, if properly propagated, will be used only for auditing and accountability reasons Hopefully, you now are starting to see the new thinking of today These highly distributed architectures supporting vast numbers of unknown users are forcing radical changes in architectures and implementations In addition, another paradigm shift concerns how you address security concerns: What are these concerns How do you know if your data is secure What are you protecting and why You can address all these questions by looking at what motivates the security end of businesses today.

To add multiple recipients of the same type, use a make new statement for each recipient You can also use a repeat loop

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