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19. Suppose that in Kocher s timing attack, we obtain the timings T ( C j ) and the emulated timings & . . 2 for dodld2 E (100,101,110, 111}, as given in the table below.

Addition and subtraction are the same in binary as they are in decimal, so when you add two uints, the binary numbers add up just as well. It wouldn t be a very good system if two numbers in binary added up to a different sum than they did in decimal.

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var a:uint = 0xf; var b:uint = 0x2; var c:uint = a + b; trace("0x" + c.toString(16)); //0x11 trace(a.toString(2) + " + " + b.toString(2) + " = " + c.toString(2)); //1111 + 10 = 10001

5 4 1 2 1 1 1 7

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Bit shifting is the process of shifting all the digits in a binary string up or down. This is like when you multiply or divide by factors of 10 in decimal and move the decimal place, adding 0s where necessary. Of course, in binary, shifting the digits multiplies and divides the original value by factors of 2. To shift, use the binary (as in having two arguments ) operators << and >>. These shift the digits to the left and right, or multiply and divide by factors of 2. The operand on the right determines how many places to shift. You can also use the shift-and-assign operators <<= and >>=, which shift the value and assign it back to the same variable. Example 13-1 shows how this works. EXAMPLE 13-1

package { import com.actionscriptbible.Example; public class ch13ex1 extends Example { public function ch13ex1() { var a:uint = 0x4; trace2(a); // 100 trace2(a >> 1); // 10 trace2(a << 1); // 1000 trace16(a << 8); //0x400 var b:uint = 0xb; trace2(b); // 1011 b >>= 2; trace2(b); // 10

110 111 7 5 8 7 4 1 6 4 7 8 5 2 0 6 8 8 1 5 7 7 1 6 5 1 2 3

concyclic) and the coordinates of their linking point are recorded as the properties of the line between the two nodes in the line data file If not, the distance between the two curvature centers is calculated When the distance is equal to the sum of their curvature radii, they are externally tangential to each other If the distance is equal to the difference of their curvature radii, they are internally tangential to each other Otherwise, they intersect each other The type of their relationship and the coordinates of their tangential point or intersecting point are thus recorded as the properties of the line between the two nodes in the line data file If one represents a straight line and another an arc, calculate the distance between the straight line and the curvature center of the arc to determine the type of their relationship.

// 1000 } public function trace2(n:uint):void { trace(n.toString(2)); } public function trace16(n:uint):void { trace("0x" + n.toString(16)); } } }

b <<= 2; trace2(b);

C; I 80 85 90 95 T(C,) 50 52 51 56

Notice that there s no decimal place to shift past when you re dealing with integers in binary. So values shifted to the right, like b here, lose whatever digits are truncated forever. Bit shifting is useful to create one long binary string out of several bytes. You can shift the digits to the left or right, and then add them, to get them to t together. This is how, if you know the red, green, and blue components of a color, you can reassemble them into one RGB pixel value. Just adding them up isn t right; the blue value is supposed to occupy the last byte, so this would just ll up the blue bucket with all three values. No, you have to align them to bytes, as shown in Example 13-2.

EXAMPLE 13-2

100 60

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