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The dual-tone multifrequency (DTMF) signal was originally developed just over 25 years ago. This was before the U.S. government forced Bell Telephone to break up, allowing the company to expand into other markets. DTMF is commonly known as touch-tone dialing. The standard DTMF signal is composed of two audio tones generated from a group of eight possible tone frequencies. The eight frequencies are divided into two equal groups, a low-frequency

One of the first threshold algorithms was developed in 1979 by Adi Shamir (the S in RSA). It s probably the easiest to understand. Consider the case of a key recovery scheme that uses three shares with a threshold of two that is, three shares are created, any two of which can recover the secret. You can think of the secret as a point on an (x, y) graph. Any point on the graph can be represented by two numbers: the x-coordinate and the y-coordinate. In Figure 4-28, the secret is the point (0, S). For the Shamir algorithm, the secret is always a point on the y-axis. So let s consider the secret a number, call it S, and then use the point (0, S). Now you generate a random or pseudo-random line that runs through that point. Next, you find three random or pseudo-random points on that line. In Figure 4-28, these points the shares are labeled S1, S2, and S3.

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The while-loop on Line 27 ending on Line 43 surrounds the remainder of the routine causing that code to be repeatedly executed (once for every object that is drawn) until a Return statement is executed 5313 Reading Mouse Data The next section of code (Lines 29 to 31) is a repeat-until loop that executes the ReadMouse command until the user clicks the left or right mouse button In this example, ReadMouse places the current coordinates of the mouse into the variables x and y and assigns a number to the variable m that speci es if and which buttons were pressed on the mouse A value of 1 indicates the left button was clicked A value of 2 means the right button was clicked Notice the use of the logical OR in the until-statement.

To recover the secret, you take two of the points and find the line that runs through them. You might recall from high school algebra that any two points uniquely define a line. With the line just created, you next determine where it crosses the y-axis. That s the secret. It doesn t matter which points are used: S1 and S2, or S1 and S3, or S2 and S3. Each pair of points generates the same line. If your scheme uses more than three shares, you simply find additional random or pseudo-random points on the line. To create a line, however, you need at least two points. One point is not enough because an infinite number of lines can run through any single point. Which one is the correct line It s impossible to tell, and that s why one share alone won t recover the secret. If you use a threshold of three, instead of a line, the algorithm generates a parabola (a curve of degree 2) that intersects the y-axis at the secret. Any three points on a parabola uniquely define it, so any three shares (points on the parabola) can re-create the curve. With the curve, if you find the point where it intersects the y-axis, you find the secret. For any threshold count, then, you simply generate a random curve of the appropriate degree (the degree of the curve will be 1 less than the threshold count) that intersects the y-axis at the secret. Each share will be a random point on that curve. Of course, a program executing the Shamir algorithm will not do this graphically; instead, it will do all the work using math equations.

group and a high-frequency group (see Table 5.2). The DTMF signal is an algebraic sum of two tone frequencies, one tone from each frequency group (see Figs. 5.13 through 5.15). If we do the math, we see that there are 4 4 16 possible combinations.

It causes the loop to wait until either of these events occurs and then execution continues with the if-statement on Line 32..

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