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I've come accross the following expression in a paper:

$$A_i=(C_2 * a_0)^{1/e_i} \bmod n$$

How to calculate $1/e_i$ in modular arithmetic?

Or does it have some different meaning that I do not know?


The paper is "A Practical and Provably Secure Coalition-Resistant Group Signature Scheme" by Giuseppe Ateniese, Jan Camenisch, Marc Joye, and Gene Tsudik", on page no.9, step 4 of JOIN.

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In this context $1/e_i$ (more commonly written as $e_i^{-1}$) stands for the multiplicative inverse of $e_i$ modulo the relevant modulus, which in this case is $\lambda(n) = \text{lcm}(p-1,q-1)$. That is, $1/e_i$ is that value $f$ such that $f \cdot e_i = k \cdot \text{lcm}(p-1, q-1) + 1$, for some integer $k$.

Given the factorization of $n$, we can compute $1/e_i$ using the Extended Euclidean method.

We also believe that that $(C_2 \cdot a_0)^{1/e_i} \bmod n$ cannot be computed without knowing the factorization $n = p \cdot q$ (for general $C_2, a_0, e_i$); in fact, this is precisely the RSA problem.

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