I have doubts about whether the protocol below for LSB extraction (denoted Mod2) from the paper Improved Primitives for Secure Multiparty Integer Computation is correct. Clarifications will be appreciated.

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The protocol above calls PRandM(k, 1) in line 1 to generate random shared values $r''$ and $r'_0$ (it ignores $r'$) and then uses them in line 2 to blind $a$ and then reveals it to extract the LSB of the blinded value after which it computes the random secretly shared value $a_0$.

The protocol $looks$ correct because the LSB of $c$ is essentially $r_0\oplus a_0$ and then in line 3 we XOR it again with $r'_0$.

That is... in case $2^{k-1}+[a]+2\cdot [r'']+[r'_0]$ doesn't overflow the prime field $p$. When it overflows the prime field, then the LSB flips and then the math is no longer true.

We also don't know whether $c$ overflows or not, because $[a]$ can be $p-1$ and in that case $c$ will be bigger than $2^{k-1}$ .

Any idea what I am missing here?


1 Answer 1


The protocol is only designed to work when $a$ is in the range $\{-2^{k-1},\dots,2^{k-1}-1\}$. This ensures that $c$ doesn't overflow mod $p$, as long as $p$ is sufficiently large.

In the paper they ensure this by choosing $p > 2^{k+\kappa+\log n}$, where $\kappa$ is a statistical security parameter and $n$ is the number of parties. See the discussions on p3-5.


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