Seth Schoen of the EFF proposed an interesting cryptographic primitive called a "hard to verify signature" in his blog at http://vitanuova.loyalty.org/weblog/nb.cgi/view/vitanuova/2004/09/02 . The idea is to have a signature which is fast to make but slow to verify, with the verification speed under the signer's control. He proposes that this could be useful with trusted computing to discourage certain objectionable applications. The method Seth describes is to include a random value in the signature but not to include it in the message. He shows a sample signature with 3 decimal digits hidden. The only way to verify it is to try all possibilities for the random values. By controlling how much data is hidden in this way, the signer can control how long it will take to verify the signature. This idea is nice and simple, but one disadvantage is that it is probabilistic. It works on average, but occasionally someone might choose an n digit value which happens to be low (assuming the values are chosen at random). Then they don't get as much protection. They could fix this by eliminating too-low values, but then verifiers might exploit that by doing low values last in their testing. Another problem is that this method is inherently parallelizable, so that someone with N computers could solve it N times faster, by having each computer test a subset of the values. An alternative is based on the paper, "Time-lock puzzles and timed release Crypto", by Rivest, Shamir and Wagner, from 1996, http://theory.lcs.mit.edu/~rivest/RivestShamirWagner-timelock.pdf or .ps. They are looking more at the problem of encrypting data such that it can be decrypted only after a chosen amount of computing time, but many of their techniques are applicable to signatures. The first solution they consider is essentially the same as Seth's, doing an encryption where n bits of the encryption key are unknown, and letting people search for the decryption key. They identify the problems I noted above (which I stole from their paper). They also point out BTW that this concept was used by Ralph Merkle in his paper which basically foreshadowed the invention of public key cryptography. Merkle had to fight for years to get his paper published, otherwise he would be known as the father of the field rather than just a pioneer or co-inventor. The next method they describe can be put into signature terms as follows. Choose the number of modular squarings, t, that you want the verifier to have to perform. Suppose you choose t = 1 billion. Now you will sign your value using an RSA key whose exponent e = 2^t + 1. (You also need to make sure that this value is relatively prime to p-1 and q-1, but that is easily arranged, for example by letting p and q be strong primes.) The way you sign, even using such a large e, is to compute phi = (p-1)*(q-1) and to compute e' = e mod phi, which can be done using about 30 squarings of 2 mod phi. You then compute the secret exponent d as the multiplicative inverse mod phi of e', in the standard way that is done for RSA keys. Using this method you can sign about as quickly as for regular RSA keys. However, the verifier has a much harder problem. He does not know phi, hence cannot reduce e. To verify, he has to raise the signature to the e power as in any RSA signature, which for the exponent I described will require t = 1 billion modular squaring operations. This will be a slow process, and the signer can control it by changing the size of t, without changing his own work factor materially. The authors also point out that modular squaring is an intrinsically sequential process which cannot benefit from applying multiple computers. So the only speed variations relevant will be those between individual computers. Another idea I had for a use of hard to verify signatures would be if you published something anonymously but did not want to be known as the author of it until far in the future, perhaps after your death. Then you could create a HTV signature on it, perhaps not identifying the key, just the signature value. Only in the future when computing power is cheap would it be possible to try verifying the signature under different keys and see which one worked. Hal Finney