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Then Alice s public key is the polynomial h ( z )= fq(z) *g(z) (mod q ) . The ploynomial h ( z ) ,along with the parameters N , p , and 4 , are made public. Alice s private key consists of t.he polynomials f(x) and f,(z). To summarize; we have Public key: h(z) Private key: ( f ( z )f,(z)) , where h(z)= fq(z) g(z) (mod q ) and f ( z )* f,(z) = 1 (mod q ) . * Bob sends Alice an encrypted as follows. Bob first selects a polynomial M ( z ) E L , that represents the plaintext message. Recall that the coefficients of the message ploynomial M ( z ) are in the range - ( p - 1)/2 and ( p - l ) / 2 and that q is much larger than p . Consequently, the message M ( z ) can be viewed as a small polynomial modulo q , in the sense that the vector of coefficients has small Euclidean length. Bob then chooses a random blinding polynomial r ( z ) E L, and uses Alice s public key to compute the ciphertext message C(z) (also a polynomial) as

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(I ll show how to z-sort faces in the section Z-Sorting, Shading, and Further Topics. ) Another approach is to determine which part of the shell faces away from you and simply elect not to draw it. So in reality only the front half of the baseball would exist, but you d never know. As you rotate the ball, the new parts that face you become visible. You can never see parts of it that face away from you. (If the object isn t convex, this guarantee breaks down.) When you don t draw polygons that face away from you, those polygons are sometimes called single-sided; removing those faces from the list of polygons to draw is called backface culling. Beyond solving the depth sorting issue for convex objects, it s an easy way to speed up rendering, because (to make a rash generalization) around half of the surface is usually facing away from you. To enable backface culling, rst make sure that all your triangles are de ned in the same winding order. Then pass an appropriate culling mode to the culling parameter of drawTriangles(): TriangleCulling.POSITIVE to remove back-facing polygons when you de ne the faces in a counterclockwise winding order, TriangleCulling.NEGATIVE when they re de ned in a clockwise winding order. Or, use the opposite value to cull front-facing polygons for whatever reason. Passing in TriangleCulling.NONE turns off backface culling, rendering all polygons (the default). If you modify the drawTriangles() call in Example 40-3 to read

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The advantage of this method is that in cases where the age information is present in certain features of the face and not the entire face, NMF would be able to capture that information However, it has the same problem as PCA, ie basis vector determination, and was also handled in the same way as PCA In each case, the basis vectors were obtained using the set of training examples From the experiments it was found that all representational schemes, namely: PCA, RPCA and NMF, yielded very low accuracies of around 50 % for the age classification problem On the other hand, classification done using raw image values for the same set of data gave accuracies around 70 % This indicated that when the images were projected along the vector subspace for PCA, RPCA and NMF, data crucial for age classification was lost.

C(z) = r(x) * h(z) M ( z ) (mod q ) ,

graphics.drawTriangles(projectedPoints, model.indices, null, TriangleCulling.POSITIVE);

you notice that the plane disappears for half its revolution. Later in the chapter, you apply backface culling to a more exciting shape and see that it really works.

The coefficients of a(.) are chosen to be in the interval -4/2 to q / 2 (it is crucial that the coefficients be taken in this interval before the next step in the decryption). Then Alice computes b(z)= u ( z ) (mod p ) . Although it is not obvious, Alice recovers the message M ( z ) by computing

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