Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 2 / 12K / b / ii / Solution

= RSA common-modulus attack
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If the same invertible message $m$ is sent under one RSA modulus with coprime public exponents $e_1,e_2$, choose integers $a,b$ with $ae_1+be_2=1$. From $c_i=m^{e_i}\pmod N$ one recovers
$$
m\equiv c_1^ac_2^b\pmod N,
$$
using modular inverses for negative exponents.


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 Incoming links: Common-modulus RSA attack