Homomorphic encryption (HE) is an encryption scheme that provides arithmetic operations on the encrypted data without doing decryption.
For Ring-based HE, an encryption scheme that uses arithmetic operations on a polynomial ring as building blocks, performance improvement of unit HE operations has been achieved by two kinds of efforts.
The first one is through accelerating the building blocks, polynomial operations.
However, it does not facilitate optimizations across polynomial operations such as fusing two polynomial operations.
The second one is implementing highly optimized HE operations in an amalgamated manner.
The written codes have superior performance, but they are hard to maintain.

To resolve these challenges, we propose HEaaN.MLIR, a compiler that performs optimizations across polynomial operations.
Also, we propose Poly and ModArith, compiler intermediate representations (IRs) for integer polynomial arithmetic and modulus arithmetic on integer arrays.
HEaaN.MLIR has compiler optimizations that are motivated by manual optimizations that HE developers do.
These include optimizing modular arithmetic operations, fusing loops, and vectorizing integer arithmetic instructions.
HEaaN.MLIR can parse a program consisting of the Poly and ModArith instructions and generate a high-performance, multithreaded machine code for a CPU.
Our experiment shows that the compiled operations outperform heavily optimized open-source and commercial HE libraries by up to 3.06x in a single thread and 4.55x in multiple threads.