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Animating using a Timer object gives you some distinct bene ts. When you are using a Timer object, you can set the frequency at which new frames are drawn completely independently of the published frame rate of the SWF. You can employ this to change the frame rate of programmed animations without interfering with timeline animations created by the Flash authoring tool or running two animations at different frame rates. For example, you can animate the main characters of a game at 30 fps but save some CPU time by animating the background at 15 fps. It s possible for events from a Timer object to re more frequently than the published frame rate of a movie, and you can ensure that the results of these events are made visible by calling



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updateAfterEvent() on the event object. This can actually make Flash Player play faster than the published frame rate. However, updating more frequently than the frame rate without showing the results visually is a waste.

Example 39-1 animates a DisplayObject moving the object to the right at whatever frame rate is assigned to the fps variable (here, it is 60), regardless of the published frame rate (set to 1 here).

M(l)k(ppl) M (mod q ) , =





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Pattern-Recognition Transforms, John Wiley & Sons, Inc, New York, 1993 [15] Oirrak, A, Daoudi, M and Aboutajdine, D Estimation of general 2D affine motion using Fourier descriptors, Pattern Recognition, 35, pp 223 228, 2002 [16] Oppenheim, A and Lim, J The importance of phase in signals, Proceedings of the IEEE, 69, pp 529 541, 1981 [17] Ozaktas, H, Kutay, M A and Zalevsky, Z Fractional Fourier Transform: With Applications in Optics and Signal Processing, John Wiley & Sons, Ltd, Chichester, 2001 [18] Persoon, E and Fu, K Shape discrimination using Fourier descriptors, IEEE Transactions on Systems, Man and Cybernetics, SMC-7 (3), pp 170 179, 1977 [19] Persoon, E and Fu, K Shape discrimination using Fourier descriptors, IEEE Transactions on Pattern Analysis and Machine Intelligence, 8 (8), pp 388 397, 1986 [20] Pinkowski, B Principal component analysis of speech spectrogram images, Pattern Recognition, 30, pp 777 787, 1997.

EXAMPLE 39-1

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package { import flash.display.*; import flash.events.TimerEvent; import flash.utils.Timer; [SWF(frameRate="1")] public class ch39ex1 extends Sprite { protected var ball:DisplayObject; public function ch39ex1() { ball = new Ball(); ball.y = stage.stageHeight/2; addChild(ball); // fps = frames/second var fps:int = 60; // 1/fps = seconds/frame, 1/fps*1000 = milliseconds/frame var mspf:int = Math.round(1 / fps * 1000); var t:Timer = new Timer(mspf); t.addEventListener(TimerEvent.TIMER, onTimer); t.start(); } protected function onTimer(event:TimerEvent):void { ball.x += 0.5; event.updateAfterEvent(); } } } import flash.display.Shape; class Ball extends Shape { public function Ball(color:uint = 0xff0000, size:Number = 50) { graphics.beginFill(color); graphics.drawCircle(0, 0, size); } }

again: by Euler's Theorem. Also note that in this case, M = 0 (mod p ) , which implies ( M e ) d 0 (mod p ) . It follows that ( M e ) d M (mod N ) ; see = = Problem 14. In the next section, we consider some mathematical and implernentation issues related to RSA. Then in 7 we discuss implementation-related attacks on RSA.

The onTimer() method is called every 17 milliseconds, or about 60 times a second, or as close to that as Flash Player can manage. After the method modi es the x property of the display object, it asks Flash Player to redraw the screen. You ll see that even a SWF published at 1 fps runs the animation at the requested 60 fps.

You can tie the frame rate of programmatic animation to the frame rate of the movie by using the Event.ENTER_FRAME event broadcast by display objects when Flash Player advances to the next frame. This is a less invasive way of scheduling animations: it lets the host SWF determine the frame rate, and it doesn t force frames to be drawn when they wouldn t normally be drawn.

There is large body of literature focusing on the cryptanalysis of RSA. An excellent source of information is Boneh's survey [19]. In this section, we mention a few mathematical issues related to the security of RSA. Our intent is to illustrate some of the issues that can lead to attacks on flawed implementations of RSA. First, we mention a generic attack that applies to public key systems (and hash functions in some situations) but not to symmetric ciphers. We state the

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