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Suppose I have a secret value $a$ which maps to a public point on an elliptic curve $A = a \cdot G$, where $G$ is a generator of the elliptic curve of prime order $q$.

Can I prove to someone that the least significant bit of $a$ is $0$ using a non-interactive proof that does not expose the value of $a$? If I can, what would be the most efficient (in terms of proof size) way to do it?


Edit: this can probably be done using bit commitments (something similar to Confidential Transactions), but in this case, proof sizes are likely to be close to 10KB. Is there a way to do this more efficiently? Maybe somehow relying on the fact that $a$ is even?


Edit2: I've been reading a bit about Division Polynomials and I'm wondering if it is possible to use them somehow to prove that $a$ is even.

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  • $\begingroup$ I assume $A$ is public? You can certainly do this using standard generic zero-knowledge proofs, since this is an NP statement. There might be a much more efficient way for this specific example, though. $\endgroup$ – Noah Stephens-Davidowitz Jan 11 at 7:42
  • $\begingroup$ Yes - $A$ is public. I updated the question with this and one other clarification. $\endgroup$ – irakliy Jan 11 at 8:12
  • $\begingroup$ How are considering to do this with CT? $\endgroup$ – Youssef El Housni Jan 15 at 10:09
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Take a look at SNARGs, e.g. Bulletproofs (which are range proofs). The proof may be slightly too large though.

Assuming that $a \in \{0, \ldots, q-1\}$ is your representative, you want to prove is that you know a bit decomposition of $a$ with $a = a_1 d_1 + a_2 d_2 + ... + a_k d_k$ where all $d_i$ are even and $\sum d_i = p-1$. E.g. $d_i = 2^i$ but $d_k < 2^k$ otherwise you can get wraparound. This trick is found in some range-proof paper(s). Without this, it may be possible to find, for odd $a$, a bit decomposition $a_i$ with $a' = \sum a_i d_i$, and $a'$ even (due to wraparound, $a = a' \mod q$). Hence you don't get a proof for "(representative of) a is even".

One question though: Why is the last bit zero? If you just want to split the exponents into two groups, using squares and non-squares is more efficient w.r.t. ZK. It's simple and efficient to prove that $a = b^2$ (with standard sigma protocols).

It seems hard to somehow express "is divisible by 2" efficiently, because all calculations happen over finite fields, hence every (non-zero) element is invertible. So there is no such thing as "even and odd". (Also: $1$ is odd, $1 + p$ is even, but $1 + q \equiv 1 (q)$.) But: Polynomials make sense, e.g. squares $f(x, y) = x^2 - y$ or bits $f(b) = b (b-1)$. The vanishing/zero sets of polynomials are "algebraic", which helps in the group setting.

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  • $\begingroup$ Using bit decomposition will work - but I was trying to see if there is a more efficient way to do it somehow (it appears there might not be). The reason I wanted to do this is: if $a$ is divisible by 2, then $a/2 < q / 2$. This makes commitment to $a$ additively homomorphic. Meaning, I can check that $A + B = C$ and not worry about overflows and underflows as long as each of these points is backed by values that is less than $q/2$. $\endgroup$ – irakliy Jan 21 at 7:58
  • $\begingroup$ What do you think of this approach: crypto.stackexchange.com/questions/66656/… ? $\endgroup$ – irakliy Jan 21 at 18:21
  • $\begingroup$ As I said in the last paragraph, unless you can make "divisible by 2" an algebraic property (over finite fields you're working on) somehow, I do not expect such approaches to work. $\endgroup$ – mk. Jan 23 at 17:53

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