Encrypted Matrix Inverse

Definition

The inverse of a square matrix $A$ of order $N$ is the unique matrix $A^{-1}$ satisfying

where $I_N$ is the $N\times N$ identity. It exists precisely when $A$ is nonsingular, that is $\det A \ne 0$. This kernel takes an encrypted square matrix and returns its encrypted inverse.

Mathematical Domain

Matrix inversion is defined over any field of entries. Under a homomorphic scheme the entries are held encrypted and the arithmetic is the scheme's arithmetic. Over the reals under CKKS this arithmetic is approximate, so the returned matrix is a numerical approximation of the true inverse.

Inversion is harder to realize homomorphically than a product because division and pivoting are not native operations: a homomorphic scheme offers additions and multiplications, but not exact reciprocals, comparisons, or branching. Inversion over encrypted data is therefore typically approached iteratively — by a scheme of repeated multiply-and-add steps that converge to the inverse using only ring operations — rather than by direct elimination. The kernel prescribes no particular method; the number of steps and the accuracy reached depend on the technique and on how well-conditioned $A$ is.

Parameters Defined by Specifications

A specification narrows the general operation into a concrete problem by fixing:

• the entry domain and homomorphic scheme (for example f64 under CKKS)
• the matrix order $N$ and the requirement that $A$ be invertible
• any assumptions on the range or conditioning of $A$ that make the accuracy well-defined
• the accuracy rule and how closeness is scored
• which evaluation keys an answer may declare and use

Applications

A matrix inverse underlies the solution of linear systems and least-squares problems (the normal equations $x = (A^\top A)^{-1}A^\top b$), covariance whitening and Mahalanobis distances, Gaussian-process and Kalman-filter updates, and portfolio and regression models. Performing it over encrypted data lets these analytics run on confidential inputs — medical, financial, or otherwise sensitive matrices — without ever exposing the matrix or its inverse.