Encrypted Sign

Definition

The sign function applied element by element to a secret vector. For a real vector $x = (x_0, \ldots, x_{N-1})$, the result $s$ has

Each output element depends only on the corresponding input element. There is no interaction between positions and no reduction across the vector.

Mathematical Domain

Sign is a step function: discontinuous at the origin and constant on either side of it. Arithmetic homomorphic encryption schemes evaluate polynomials — additions and multiplications — and nothing else, so the step cannot be represented exactly. It is instead approximated by a polynomial over a bounded input range, and the quality of that approximation and the multiplicative depth it consumes are what a specification measures.

Sign is the base comparison primitive under encryption: a comparison is $\bigl(\operatorname{sign}(a-b)+1\bigr)/2$, and maximum, ReLU, and max-pooling are all built on top of it. The kernel itself fixes neither the arithmetic scheme, the input range, the vector length, nor the packing; these are defined by individual specifications.

Parameters Defined by Specifications

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

• the element domain and encryption scheme (for example real values under CKKS)
• the input range on which the approximation must hold
• the vector length $N$
• the ciphertext packing and layout
• the correctness or accuracy rule and how it is scored
• which stages of the computation are timed

Applications

Sign underpins every decision-shaped operation in private computation: comparison and thresholding, argmax and maximum selection, ReLU and max-pooling in private neural-network inference, decision-tree evaluation, and comparison-based sorting over encrypted data.