Last year, Stefan Brands announced that he had come up with improved versions of Chaumian cash and credentialling protocols which were smaller, faster, and had provable correctness. He still hasn't gone public with them, but I thought I'd write up an introduction to his earlier work so people can see what direction things are going. IMO, if he plays his cards right his technology could be the foundation for electronic commerce. OTOH if he is too greedy he'll be bypassed. It appears he is seeking patents on everything, a necessary step for commercial interest, but we'll see how he markets it.

This is based on Brands' "An Efficient Off-line Electronic Cash System Based on the Representation Problem", which was available on the net for a while before he took it off. I'm not sure what its status is now. Perhaps he removed it pending release of his improved version.

Brands' work is based on discrete logs rather than RSA. The discrete logarithm problem is the "other" widely-used foundation for crypto primitives, underlying Diffie-Hellman key exchange, ElGamal, Schnorr, and DSS signatures, and many others. I'll do a brief intro to using discrete logs and then get to Brands' cash.

Discrete-log based cryptosystems generally work with a modulus n which is prime, along with a "generator" g < n such that the series g^0, g^1, g^2, ... , includes all values from 1 to n-1. It is pretty straightforward to find such n's and g's. It is easy to compute g^x for any x, but intractable to calculate x given just g^x. (Notation: ^ represents exponentiation, and all math is implicitly mod n). x is called the discrete log (to the base g) of g^x and the difficulty of solving this is the foundation of these protocols. Note that unlike RSA, where taking eth roots is hard for everyone except the owner of the secret key, taking discrete logs is hard for everyone, without exception. There is no trap door here.

Diffie-Hellman key exchange

As an introduction, consider Diffie-Hellman key exchange. In this protocol, two people, Alice and Bob, want to publicly exchange data and end up with a secret value which only they know.

  1. Alice chooses a random x and sends GX = g^x to Bob. Bob chooses a random y and sends GY = g^y to Alice.
  2. Alice calculates GY^x, which is g^(y*x). Bob calculates GX^y, which is g^(x*y).
  3. These are equal, so they use them as their shared secret value.

    An observer sees only GX and GY, and without knowledge of x and y is unable to calculate g^(x*y).

    DH-based identification protocol

    An identification protocol allows someone to prove that he is really who he claims. In this context, the prover Paul will convince the verifier Vicki that he knows the secret key corresponding to Paul's established public key. In this and the following systems, Paul has a secret key x<n, and a public key GX = g^x. Again, note that it is impossible to go from GX to x assuming discrete logs are hard.

    1. Vicki chooses a random y and sends GY = g^y to Paul.
    2. Paul calculates GYX = GY^x = g^(y*x) and sends that back to Vicki.
    3. Vicki confirms that GYX = GX^y; both should be g^(x*y).

      This is like DH except that Paul exposes the secret information he calculated, and only he could have done this. One problem with this protocol is that perhaps Paul calculating xth powers for Vicki might reveal something about x. The next protocol solves that:

      Schnorr identification protocol

      This comes from Schnorr, Journal of Cryptology, v4 n3, 1991.

      1. Paul chooses a random w and sends GW = g^w to Vicki.
      2. Vicki chooses a random c and sends it to Paul.
      3. Paul calculates r = cx+w and sends that to Vicki.
      4. Vicki confirms that g^r = (GX^c)*GW. Both should be g^(cx+w).

        The extra step of Paul sending g^w for a random w makes this protocol reveal less information about x. For any one run of the protocol, there is some value of w which would produce a given r for any x, so knowing r and c doesn't tell you anything about x.

        Chaum discrete-log interactive signature protocol

        This is the basic signature used by Brands, but I believe it comes from Chaum&Pederson, Crypto 92. It is an extension of the previous protocol to allow signatures. A digital signature on a value m is a certificate which could only have been produced by the owner of a particular public key.

        In this protocol, a message m (<n) is being signed. The basic signature value is MX = m^x, which Paul sends. By itself, though, this signature is not obviously correct. Without knowing x, Paul's secret key, there is no way to confirm it. So, Vicki must engage in an interactive protocol with Paul in which he will prove that MX is equal to m^x. It is very similar to the previous one:

        1. Paul chooses a random w and sends GW = g^w and MW = m^w to Vicki.
        2. Vicki chooses a random c and sends it to Paul.
        3. Paul calculates r = cx+w and sends that to Vicki.
        4. Vicki confirms that g^r = (GX^c)*GW. Both should be g^(cx+w). She also confirms that m^r = (MX^c)*MW. Both should be m^(cx+w).

          This is the previous protocol plus one extra number, MW. The fact that the same r is used for both m and g shows that m was raised to the same power as g in creating MX vs GX.

          Chaum Discrete-Log Signature Protocol

          Interactive signature protocols may have advantages in some circumstances, but in most cases we would prefer a signature which can be checked without help from Paul. There is a simple trick which can turn most interactive signature protocols into regular signatures. The idea is that instead of c being chosen at random by Vicki, it is calculated by using a cryptographically strong hash function (such as MD4, MD5, or DHS) on the values which are publicly known by that point in the protocol: m, MX, GW and MW. Since both Paul and Vicki could calculate the hash, there is no need for Vicki to send c to Paul. Instead, he can do everything in one step. This leads to:

          1. Paul calculates MX = m^x. He then chooses a random w and calculates GW = g^w and MW = m^w. He then calculates c = hash(m,MX,GW,MW). He calculates r = cx+w, and sends MX, GW, MW and r to Vicki. The tuple (MX,GW,MW,r) is the signature on m.
          2. Vicki calculates c = hash(m,MX,GW,MW). She then verifies, as before, that g^r = (GX^c)*GW. Both should be g^(cx+w). She also confirms that m^r = (MX^c)*MW. Both should be m^(cx+w).

            This protocol is not interactive. Once Paul has completed step 1, the signature can be published and anyone can play Vicki's part in checking it. Such a signature is functionally similar to the PGP signatures we see on messages on the net, but as you can see the mathematics behind it is completely different.

            OK, that's all for now. Next comes the good part: blind signatures. Unlike Chaum's original blind signatures, which are the foundation of his cash system, the discrete-log blind signatures have "restrictive blinding", where there are limitations on what kinds of changes can be made to the number being signed during the blinding process. This allows Brands to dispense with the clumsy cut-and-choose techniques Chaum was forced to use in his advanced cash and credential systems. I'll write more about this later today or tomorrow.

            Hal <>