Seth Schoen's Hard to Verify Signatures

Hal Finney hal at
Tue Sep 7 20:52:38 EDT 2004

Seth Schoen of the EFF proposed an interesting cryptographic primitive
called a "hard to verify signature" in his blog at .
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, 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

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

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