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How a Threshold Scheme Works
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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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The Key Distribution Problem and Public-Key Cryptography
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Figure 4-28 The Shamir threshold scheme
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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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To solve the key distribution problem, you can use public-key cryptography. With the RSA algorithm, the data encrypted by the public key can be decrypted only by the private key. To securely transmit the session key, you can use a digital envelope. With Diffie-Hellman or Elliptic Curve Diffie-Hellman, you can use public-key technology to generate a shared secret. Only the correspondents can create this secret value, which can then be used as a session key. Each of the three algorithms has its advantages and disadvantages, so it s not really possible to say that one or the other is better. But any one algorithm may be better suited for a specific application. It s possible to lose cryptographic keys by forgetting a PBE password, losing the token where they re stored, and so on. In addition, a company may want to be able to recover material encrypted by an employee who, for example, has left the firm. For these reasons, many organizations implement a key recovery plan. Generally, key recovery involves the use of an RSA digital envelope, encrypting keys with a recovery agent s public key. The key recovery agent might be an individual or a group of trustees. Threshold schemes offer an attractive means of implementing key recovery with checks and balances. With a threshold algorithm (also known as secret sharing or secret splitting), a secret such as an RSA private key is split into a number of shares. To recover the secret, a minimum number of shares must be collected. This method prevents one individual from obtaining keys surreptitiously, while making it possible to reconstruct the keys even if one or more trustees is absent.
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