bsy is now at UCSD, and this page is no longer being maintained here.

Abstractly, a zero knowledge proof is an interactive proof with a prover and a verifier, where the prover convinces the verifier of a statement (with high probability) without revealing any information about how to go about proving that statement. Hopefully the following example will make it all clear.

First, our assumptions. We're going to arithmetic modulo
`n`, where `n = pq`, and `p` and
`q` are primes. Factoring `n` is assumed to be
intractible.

Rabin showed in [RabinFunc] that finding square roots modulo
`n` is equivalent to factoring `n`. That is, if
you have an algorithm that can find a square root of a number modulo
`n`, then you can use that algorithm to factor
`n`. Our zero knowledge proof will consist of rounds of
interaction which shows that the prover knows a square root of a
published number, where we do not reveal any new information about the
square root. It is known that there exists a square root to this
number (public knowledge), i.e., it is a quadratic residue. The
factors of the modulus `n` may be entirely secret. (U.
Feige et al show a refinement in [FFS] which allows the published
number to be a non-quadratic residue of a particular form as well,
thus revealing less information; in either case, runs of the protocol
itself reveals no new information.)

The prover, *P*, publishes the quadratic residue `v`
for which *P* claims to know a root `s`.

When *P* wishes to prove its knowledge of `s` to the
verifier, *V*, *P* runs several rounds of interaction. In
each round, *P* choses a new random number `r` and
sends `x = r^2 mod n` to *V*. Now, *V* choses a
random bit `b`, and sends it to *P*. *P* replies
with `y = r s^b`. To verify *P*'s claim, *V*
computes `y^2` and compares it with `x v^b`.

Now, let's do the analysis. The first claim is that only *P* can
successfully complete the protocol for both possible values of
`b`. This is clear since knowing `y_1 = r s`
when `b = 1` and `y_2 = r` when `b =
0` means you also know `s`, since `y_1/y_2 =
s`. The second claim is that an imposter *P'* who does not
actually know `s` can succeed with a probability of exactly
`1/2` each round: to see this, notice that if *P'*
guesses correctly that `b = 0`, then it can just follow the
protocol and succeed; on the other hand, if *P'* guesses that
`b = 1`, *P'* can generate `x` by chosing a
random number `t` and setting `x = t^2 / v`.
The response is y = t. The third claim is that no new
information is released. To see this, consider what an eavesdropper
*E* hears. In the case of the random bit `b = 0`,
*E* sees a random numer `r` and its square
`x`; in the case of `b = 1`, *E* sees the
numbers `rs` and `x = (rs)^2/v`. These are
numbers that the eavesdropper could have generated in a closet. More
precisely, a simulator *S* can run both sides of the protocol,
and by using advanced information as to the value of the random bit
(model is a poly-time TM with an auxiliary input tape of random bits),
*S* can simulate the protocol without knowledge of
`s`.

Each round of the proof shows that there is a `1/2` chance
that a prover *P''* might not actually know `s`.
Iterating `20` times gives a probability of less than
`2^-20` or `.0000009536` that *P''* does
not actually know `s`.

Such zero knowledge proofs can be used for authentication -- the value
of `v` can be generated from a randomly chose
`s`, and `v` is widely published. A successful
zero knowledge proof showing knowledge of `s` authenticates
identity. In [StrongboxIn25th], Doug Tygar and I show how to obtain
superexponential scaling in security modulo the factorization
assumption, run the protocol in constant rounds while retaining the
zero knowledge property, and simultaneously perform key exchange.

-bsy

-------------------- bibtex format bibliographic entries -------------------- @TechReport(RabinFunc, Author="Michael Rabin", Institution="Laboratory for Computer Science, Massachusetts Institute of Technology", Title="Digitalized Signatures and Public-Key Functions as Intractable as Factorization", Key="Rabin", Year=1979, Month="January", Number="MIT/LCS/TR-212") @InProceedings(FeigeFiatShamir, Key="Feige", Author="Uriel Feige and Amos Fiat and Adi Shamir", Title="Zero Knowledge Proofs of Identity", Year=1987, Pages="210-217", Booktitle="Proceedings of the 19th ACM Symp. on Theory of Computing", Month="May") @Inproceedings(StrongboxIn25th, Key="Tygar and Yee", Author="J. D. Tygar and Bennet S. Yee", Title="Strongbox: A System for Self Securing Programs", Organization = "ACM", Booktitle="CMU Computer Science: 25th Anniversary Commemorative", Year = 1991)

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bsy@cs.cmu.edu / bsy@cse.ucsd.edu, last updated 8 October 1996.

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