Multiplicative inverse protocol for MPC that outputs 0 when input is 0

anticommutative

8/30/24, 12:00 AM

Setting: Shamir secret-sharing over the field $GF(p)$, $p$ a prime.

For $a\in GF(p)$, I would like a protocol that takes a sharing $[a]$ as input, and outputs: $[a^{-1}]$ if $a\neq 0$, and $[0]$ if $a=0$.

I cannot think of an easy way to modify the well-known protocol that computes $a^{-1}$ using a multiplicative mask, and cannot think of another approach.

Any ideas are welcome!

0 + 5

multiparty-computation

secret-sharing

fgrieu

8/30/24, 7:09 AM

What about turning the exponent to positive in $a^{-1}$ using Fermat's little theorem?

Geoffroy Couteau

8/30/24, 7:49 AM

I think fgrieu's suggestion is the best choice here, unless $p$ is extremely large. Given a secure multiplication protocol that computes $[xy]$ from $[x]$ and $[y]$, compute $[a^{p-2}]$ via the square-and-multiply protocol, in about $1.5\log p$ invocations of the secure multiplication. This will be $a^{-1}$ if $a$ is nonzero, and zero otherwise.

anticommutative

8/30/24, 3:21 PM

Thanks for the responses! I like the idea, but $p$ may be quite large ($\approx 2000$ bits).

Geoffroy Couteau

8/30/24, 4:38 PM

Ok then, how many parties do you have? What is the corruption threshold used for the Shamir secret sharing? I assume semi-honest security is enough? In the two party setting I can think of much more efficient solutions, but in the multiparty setting, it requires a bit of thinking, though there might be solutions.

anticommutative

8/30/24, 5:37 PM

Currently, $3$ parties with threshold $2$, and semi-honest security. I'd be curious to hear any solutions requiring 2PC.

Elon Musk

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EN: Multiplicative inverse protocol for MPC that outputs 0 when input is 0

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