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The basis for this question is a 1-dimensional NTRU cryptosystem.
After some literature inspection I have found out it can be also generalised into higher algebras: quaternions (QTRU) and octonions (OTRU) while keeping the same logic of operations in a modular quotient ring. One may therefore encrypt multiple messages at once. It appears to me that such a system, based on an algebra of dimension $2^x$, should be more secure than parallel encryption of $2^x$ messages due to obvious cross-dependence inside each element. However, I cant really put my finger on it... If so - maybe it is even better to split a given message $M$ into separate $2^x$ chunks, treat those chunks as separate messages: $m_1...m_{2^x}$ and encrypt in a higher algebra?

Could someone please help me out in nailing down the security benefits of a high-dimensional NTRU variants?

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