Paper 2026/450

A unified toolkit for advanced arithmetic in FHE

Abstract

We suggest an extension of discrete--CKKS that enables advanced computations over real, Boolean and large integers (e.g., 64/128-bit), including standard arithmetic operations, comparison, shift, bit length, quotient, square root, and other non linear functions based on binary representation. The flexibility of our approach lies in the synergy between the integer and the real arithmetic layers. At the core of our approach lies indeed a novel non-iterative binary decomposition technique extracting the $i$-th bit of $x$ by evaluating an approximation of $\lfloor x/2^i \rfloor \bmod 2$, very efficient when $x$ has at most 8 bits. We show how such procedure can decompose not only a value $x$, but also a more general $g(x)$ at no additional cost by generalizing the evaluation to $\lfloor g(x)/2^i \rfloor$, enabling fast LUT evaluation over integers. Interestingly, the output of our LUTs can be arbitrarily large at the cost of requiring redundant slots: this unlocks the computation of large seeds for Newton--Raphson algorithms, enabling for the first time the practical evaluation of (exact) non linear functions over large integers such as square root and quotient within the discrete--CKKS family of schemes. By supporting computations over different domains, our framework is usable in various applications: smart contracts naturally fit our framework, as it efficiently supports both arbitrary logical gates and operations over large integers. It also allows integer computations on the output class of a real-valued neural network, such as metadata lookup. Compared to the state-of-the-art by Gao and Zheng (CRYPTO '26), our solution does not require ad-hoc encoding or expensive integer to Boolean conversions, making it more suitable when some non arithmetic structure is involved. %has lower latency on all standard integer operations (additions, multiplications, comparisons and logical shifts), and we additionally support nonlinear integer operations. Our current limitation lies in the throughput, which is lower. We additionally provide an open source proof of concept for GPU, showing that our work has better latency and throughput than TFHE-rs on most arithmetic and non-arithmetic operations.

Note: This is an updated version of the ePrint 2026/420 originally named ‟A flexible and polynomial framework for integer arithmetic in CKKS"