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The threshold scheme has many advantages over the key recovery programs described earlier, and it eliminates some of the disadvantages. First, no one person can recover keys; it takes a group acting together. Anyone attempting to be dishonest must find some co-conspirators. Second, if one of the trustees is unavailable, it s still possible to perform the operation. Third, if one of the trustees leaves the company, the secret is still safe, and there s no need to restart the key recovery process from the beginning. A disadvantage is that if one trustee leaves the company, his or her share is still valid. By itself, this share can t do anything, but if a threshold number of people leave the company, this group of unauthorized people would have the power to recover the company s secrets. For example, suppose that Warren, Maria, and Julia leave the company, either all at once or over a period of time. They might form their own company, start working for another firm, or work for different companies. If the three of them decide to steal their former employer s secrets, they could re-create the key recovery private key. Of course, that private key won t do them any good without the digital envelopes protecting the session keys of all the employees. So if they want to steal secrets, they still have to find the floppy disks or tokens storing the encrypted session keys. But a company that wants to eliminate such an attack would generate a new key pair and restart the key recovery program from scratch. Fortunately, with a threshold scheme, this step is not necessary every time a trustee leaves but only when several of them leave.



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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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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.

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