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You must choose q so that the noise in the ciphertext doesn't overflow. For example, if $p = 2$ and $n = 256$ you can use $q = 7681$ (taken from Kyber). There are many possible instantiations, and the important point is that the norm $||c_0 - c_1 s||_\infty = ||p (e r + e_2 - e_1 s) + m||_\infty$ is less than $q/2$.


Fermat theorem Lies behind this second factorization scheme, known as pollard p-1 method. suppose odd composite integer n to be factored has prime divisor n, with the property that p-1 is a product of relatively small primes. Let q be then any integer such that (p-1)|q. For instance q could be either k! or the least common multiple of first k positive ...

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