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Rating 2.9

Almost everything in ActionScript is an object. Functions are no exception. Every function in ActionScript 3.0 is an instance of Function, with properties and methods just like any other object. In this section you ll look at how this relationship can be exploited for your bene t. The word function, written in lowercase, is the generic term that I ve been using so far in this chapter to describe an executable block of code. Function, written capitalized, is the Function class that all functions are instances of. Normally when you create a new instance of a class you use the new keyword followed by the class name. However, because functions are such an integral and frequently used part of ActionScript, there are commands (like the function keyword) that are used instead of the new Function() statement.



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distance measure to compute the distance of a feature vector from a given class. Thus, the distance di y is computed as follows: di y = y

As you have doubtless realized by now, the kernel language is not like ActionScript 3.0. But it s not entirely foreign either. The Pixel Bender kernel language is akin to GLSL, OpenGL s shader language. Some key things to note: Types are placed before the variable name, without a colon, as in C.





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which is much shorter than a typical vector in C. Furthermore, W has a very special form, since its first nl entries consist of 0s and 1s with its last m cntries I-xing all 0. Is it possible to take advantage of this special structure t o find W In 1982, Lenstra, Lenstra and Lov&sz [91] discovered the so-called LLL Algorithm, which provides an efficient method t o find short vectors in a lattice. In Table 6.1, we give a n outline of their algorithm in pseudo-code, where G S ( M ) refers t o the Grarn--Schmidt process, which returns an orthonormal basis for the subspace spanned by the columns of M . The GramSchmidt process appears in Table 6.2. Note that a small number of lines of pseudo-code suffices t o specify the entire LLL Algorithm. With clever insight, Shaniir [I321 realized that lattice reduction could be used t o attack the Merkle-Hellman knapsack cryptosystern. Suppose that Bob s public knapsack is given by T = (to,t l , . . . , t r - l ) , and Alice sends Bob a ciphertext block C, encrypted with Bob s public knapsack. Since the attacker, Trudy, knows the public knapsack T and C, she can break the system if she is able t o solve the matrix equation TU = C , where U is a n T x 1 column matrix consisting of 0s and 1s. Trudy can rewrite the matrix equation T U = C in block matrix form as

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float radius = 1.0;

In function declarations, the return value comes before the name of the function, and the parameters place the types before the parameter names. However, Pixel Bender in Flash Player does not currently allow you to call functions you de ne yourself.

arid apply the LLL Algorithm t o the matrix A f . The resulting short vectors which are obtained can be checked t o see if they have the special form required of W ,which is a column vector where the first T entries are all 0 or 1 and last entry is 0. The LLL Algorithm will not always produce the desired vector and therefore, the attack is not always successful. However, in practirp, the lattice reduction attack is highly effectivc against the original Merkle-Hellman knapsack. To illustrate the lattice reduction attack, suppose Alice constructs her knapsack key pair from the superincreasing knapsack

void evaluatePixel() {} float2 customFunction(float a, float b) {} //not in Flash Player!

Functions are overloaded that is, many functions may exist by the same name, with different arguments. The correct version will be called depending on its arguments. This makes for exible handling of different dimensions. The length() function, for instance, calculates the length of a one, two, three, or four-dimensional vector, depending on the type of its argument:

(16.39)

1 . .. , ~ 7 = )

float float float float length(float x) {} length(float2 x) {} length(float3 x) {} length(float4 x) {}

Literals are not automatically converted between numeric types. Include a decimal somewhere in your literal to indicate it is a float (a oating-point number like Number). Leave out the decimal to interpret it as an int.

(2,3.7,14,30,57,120,251),

float float float float int e a b c d = = 2.0; //Preferred = .4; = 5.; = 6; //ERROR 7;

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