It is shown that the commuting probability of a finite ring cannot be a fraction with square-free denominator, resolving a conjecture of Buckley and MacHale.
Whilst lattice-based cryptosystems are believed to be resistant to quantum attack, they are often forced to pay for that security with inefficiencies in implementation. This problem is overcome by ringand module-based schemes such as Ring-LWE or Module-LWE, whose keysize can be reduced by exploiting its algebraic structure, allowing for neater and faster computations. Many rings may be chosen to define such cryptoschemes, but cyclotomic rings, due to their cyclic nature allowing for easy multiplication, are the community standard. However, there is still much uncertainty as to whether this structure may be exploited to an adversary's benefit. In this paper, we show that the decomposition group of a cyclotomic ring of arbitrary conductor may be utilised in order to significantly decrease the dimension of the ideal (or module) lattice required to solve a given instance of SVP. Moreover, we show that there exist a large number of rational primes for which, if the prime ideal factors of an ideal lie over primes of this form, give rise to an "easy" instance of SVP. It is important to note that the work on ideal SVP does not break Ring-LWE, since its security reduction is from worst case ideal SVP to average case Ring-LWE, and is one way. However, by [32], we know RLWE and MLWE are (polynomial time) equivalent, and that MLWE and module SVP are equivalent, so our algorithm for module SVP may indeed have consequences for the security of RLWE and MLWE.
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