Encrypted Logistic

Definition

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

The logistic function is smooth and strictly increasing, mapping the real line into the open interval $(0, 1)$. It is transcendental — defined through the exponential — so it cannot be expressed as a polynomial. Arithmetic homomorphic encryption schemes evaluate only polynomials, so an implementation approximates $\sigma$ by a polynomial over a bounded input range. Because the function saturates slowly, the range of interest can be wide, and the useful question is how far a fixed-depth approximation carries across it.

The kernel 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 logistic 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

The logistic function turns real-valued scores into probabilities. It ends every logistic-regression model, calibrates classifier outputs, and forms the gates of recurrent networks such as LSTM and GRU cells. Private inference needs it wherever an encrypted model must answer with a probability.