fhelium.experimental.bootstrap.reduction.exponential

Exponential-seed periodic reduction with repeated squaring.

ExponentialSquaringReduction

class View source

ExponentialSquaringReduction(input_bound: int, degree: int, evaluator: Any = BalancedPowerEvaluator(), fuse_input_normalization: bool = False)

Reduce one real branch through a truncated exponential and squaring.

Let $B=$ input_bound, $x=r/B\in [-1,1]$, and $K={\log }_{2}B$. A power-basis polynomial first approximates

$$z_0(x)=\exp (i\pi x).$$

Repeated squaring computes $z_K\approx \exp (i\pi Bx)$, and

$$\frac{z_K-\overline{z_K}}{2i\pi }\approx \frac{\sin (\pi Bx)}{\pi }={\rho }_B(x).$$

reference(values) always consumes normalized $x$. evaluate(...) consumes raw $r$ and divides by $B$ when fuse_input_normalization=False; with fusion enabled its caller must provide $x$ directly. The raw range $|r|\le B$ is a caller precondition and is not inspected in ciphertexts.

The ciphertext state, axes, functional behavior, depth transition, and depth-dependent scale schedule are the same as for CosineDoubleAngleReduction. This strategy additionally requires a conjugation key. It returns a two-component coefficient-domain, standard-RNS Q ciphertext at ciphertext.depth + required_depths, with the corresponding active prime_ids and the arithmetic owner's target scale.

Attributes

Name	Type	Default/value
requires_relinearization		True
input_bound	int
degree	int
evaluator	Any	BalancedPowerEvaluator()
fuse_input_normalization	bool	False

polynomial

property

polynomial: PolynomialApproximation

Return ascending power coefficients for $\exp (i\pi x)$.

Entry $n$ is $(i\pi {)}^n/n!$, so the stored polynomial is $\sum _{n=0}^d(i\pi {)}^n{x}^n/n!$ in normalized coordinate $x$.

squaring_iterations

property

squaring_iterations: int

Return ${\log }_{2}B$, the repeated-squaring count.

fused_input_divisor

property

fused_input_divisor: float

Return $B$ for fused $x=r/B$, or $1$ for explicit division.

required_depths

property

required_depths: int

Count normalization, polynomial DAG, squarings, and sine scaling.

evaluate

method

def evaluate(arithmetic: BootstrapArithmetic, ciphertext: Ciphertext, *, relinearization_key: RelinearizationKey | None, conjugation_key: ConjugationKey | None=None) -> Ciphertext: ...

Evaluate ${\rho }_B$ homomorphically under the class coordinate convention.

Non-fused evaluation first maps raw $r$ to $x=r/B$. After evaluating the power series, ${\log }_{2}B$ ciphertext squarings restore frequency. Finally $(z-\bar{z})/(2i\pi )$ isolates and scales the sine branch. The result represents $\sin (\pi r)/\pi$; input storage is unchanged.

reference

method

def reference(values: np.ndarray) -> np.ndarray: ...

Evaluate the plaintext oracle on normalized coordinates $x$.

The input shape is preserved. This method never divides by $B$; pass raw coordinates as values / input_bound. It models polynomial truncation and repeated squaring but not CKKS error.