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How to compute $\frac{1 + a_1 \times \dots \times a_n}{b_1 \times \dots \times b_n}$ where $a_i$ and $b_i$ are secret?

One straightforward way of doing this using the Paillier encryption scheme and $n$ honest-but-curious participants is as follows: a broker (such as participant $n$) generates a pair of secret primes $...
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