> For the complete documentation index, see [llms.txt](https://ir0nstone.gitbook.io/notes/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://ir0nstone.gitbook.io/notes/cryptography/overview/public-exponent-attacks/multi-party-rsa-with-small-e.md).

# Multi-party RSA with Small e

Assuming $$e$$ is constant between the messages and the message $$m$$ is sent to at least $$e$$ people, we can use the Chinese Remainder Theorem to retrieve $$m$$.

In single-party RSA, we calculate $$c = m^e \mod N$$. Let's pretend this is extrapolated to 3 people:

$$
m^e = c\_1 \mod N\_1 \\
m^e = c\_2 \mod N\_2 \ m^e = c\_3 \mod N\_3 \\
$$

​The Chinese Remainder Theorem allows us to solve this congruence $$\mod N\_1N\_2N\_3$$. Since $$m < \min{N\_1, N\_2, N\_3}$$, we know that $$m^e < N\_1N\_2N\_3$$. Once we use the Chinese Remainder Theorem to compute $$m^e \mod N\_1N\_2N\_3$$, we just take the $$e$$th root to retrieve $$m$$.​


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