Neural Operations

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Integer-only neural network primitives for encrypted inference. Every activation function, every arithmetic operation – zero floating-point.

Source files: ops/neural.rs, arithmetic/mq_relu.rs, arithmetic/pade_engine.rs, arithmetic/integer_softmax.rs, arithmetic/cyclotomic_phase.rs, arithmetic/mobius_int.rs.

FHENeuralEvaluator

The unified interface for neural network operations on encrypted data.

use nine65::ops::neural::{FHENeuralEvaluator, ActivationType};

let evaluator = FHENeuralEvaluator::new(modulus, plaintext_mod);

// Or with custom softmax scale
let evaluator = FHENeuralEvaluator::with_softmax_scale(modulus, plaintext_mod, 1_000_000_000_000);

ActivationType 

pub enum ActivationType {
    None,       // Linear pass-through
    ReLU,       // max(0, x) via MQ-ReLU O(1) threshold
    LeakyReLU,  // Configurable leak coefficient
    Sigmoid,    // 1/(1+exp(-x)) via Pade [4/4]
    Tanh,       // (exp(x)-exp(-x))/(exp(x)+exp(-x)) via Pade
    Softmax,    // Exact sum guarantee
    GELU,       // x * sigmoid(1.702x) approximation
}

Dense Layer Forward Pass 

let output = evaluator.dense_forward(
    &input,      // &[MobiusInt]
    &weights,    // &[Vec<MobiusInt>]
    &bias,       // &[MobiusInt]
    ActivationType::ReLU,
);

Computes output[i] = activation(sum_j(weights[i][j] * input[j]) + bias[i]) using MobiusInt arithmetic.

MQ-ReLU: O(1) Sign Detection

Defined in arithmetic/mq_relu.rs. Replaces traditional comparison circuits with a simple threshold check: if value > q/2, it is “negative.”

Coq-verified: MQReLU.v – sign_detection_correct, mq_relu_correct, speedup_is_2000x.

Sign Enum 

pub enum Sign {
    Positive,  // value in [1, q/2)
    Negative,  // value in [q/2, q)
    Zero,      // value == 0
}

MQReLU Type 

use nine65::arithmetic::MQReLU;

let relu = MQReLU::new(modulus);

// Sign detection (constant-time via subtle crate)
let sign = relu.detect_sign(value); // -> Sign::Positive, Negative, or Zero

// ReLU: max(0, x) -- returns value if positive, 0 otherwise
let activated = relu.apply_scalar(value);

// Apply to entire polynomial
let activated_poly = relu.apply_polynomial(&coeffs);

// Leaky ReLU: positive -> value, negative -> |x| * leak_num / leak_den
let leaky = relu.leaky_relu_scalar(value, 1, 10); // leak = 0.1
let leaky_poly = relu.leaky_relu_polynomial(&coeffs, 1, 10);

// Custom threshold
let custom = MQReLU::with_threshold(modulus, custom_threshold);

Performance: ~20ns per coefficient. The sign detection uses subtle::ConstantTimeLess and subtle::ConstantTimeEq for constant-time operation on the sensitive computation path.

PadeEngine: Transcendentals via Rational Approximation

Defined in arithmetic/pade_engine.rs. Computes exp, sin, cos, ln, sigmoid, tanh using Pade [4/4] rational functions with integer coefficients.

Coq-verified: PadeEngine.v.

PADE_SCALE Constant 

pub const PADE_SCALE: i128 = 1_000_000_000; // 10^9 represents 1.0

All inputs and outputs are scaled integers. A value x in the real domain is represented as x * PADE_SCALE.

Core Functions 

use nine65::arithmetic::PadeEngine;

let pade = PadeEngine::default(); // uses PADE_SCALE

// Exponential: input x (scaled), output exp(x/SCALE) * SCALE
let exp_val = pade.exp_integer(500_000_000); // exp(0.5) * 10^9

// Sine and Cosine
let sin_val = pade.sin_integer(x);
let cos_val = pade.cos_integer(x);

// Natural logarithm
let ln_val = pade.ln_integer(x); // x must be > 0

// Sigmoid: 1 / (1 + exp(-x))
let sig = pade.sigmoid_integer(x);

// Tanh: (exp(x) - exp(-x)) / (exp(x) + exp(-x))
let th = pade.tanh_integer(x);

When the exact_rational feature is enabled, the engine delegates to the exact_transcendentals backend for higher accuracy. Otherwise, it uses the built-in Pade [4/4] approximation.

Pade [4/4] Coefficients for exp(x) 

P(x) = 1680 + 840x + 180x^2 + 20x^3 + x^4
Q(x) = 1680 - 840x + 180x^2 - 20x^3 + x^4
exp(x) ~ P(x) / Q(x)
Evaluation uses Horner’s method (integer only). Accuracy: error < 10^-8 for x < 1 in the scaled domain.

Performance: ~200ns per evaluation, zero drift, fully reproducible.

Via FHENeuralEvaluator 

let sigmoid = evaluator.sigmoid(x); // i128 -> i128
let tanh = evaluator.tanh(x);
let exp = evaluator.exp(x);
let gelu = evaluator.gelu(x);       // x * sigmoid(1.702 * x)

IntegerSoftmax: Exact Sum Guarantee

Defined in arithmetic/integer_softmax.rs. Computes softmax using Pade exp() and guarantees sum(output) == SOFTMAX_SCALE exactly.

Coq-verified: IntegerSoftmax.v.

SOFTMAX_SCALE Constant 

pub const SOFTMAX_SCALE: u128 = 1_000_000_000_000; // 10^12

The sum of all softmax outputs equals this value exactly – not approximately.

Usage 

use nine65::arithmetic::IntegerSoftmax;

let softmax = IntegerSoftmax::new();

// Input: logits as scaled integers
let logits = vec![1000i128, 2000, 3000];
let probs = softmax.compute(&logits);

// probs[0] + probs[1] + probs[2] == SOFTMAX_SCALE exactly
let sum: u128 = probs.iter().sum();
assert_eq!(sum, 1_000_000_000_000);

// Custom scale
let softmax = IntegerSoftmax::with_scale(1_000_000);

Algorithm

  1. Find max logit (integer comparison)
  2. Shift inputs for numerical stability: shifted[i] = logits[i] - max
  3. Compute exp() via Pade [4/4] for each shifted value
  4. Sum all exp values
  5. Divide each by total
  6. Adjust rounding to guarantee exact sum (the key innovation)

The adjustment step distributes any rounding remainder across the largest output values, ensuring the sum is always exactly SOFTMAX_SCALE.

Via FHENeuralEvaluator 

let probs = evaluator.softmax(&logits); // Vec<u128>

CyclotomicPhase: Ring-Native Trigonometry

Defined in arithmetic/cyclotomic_phase.rs. The ring R_q[X]/(X^N + 1) has native trigonometric properties because X^N = -1, so X^k is a phase rotation by k * (pi/N).

Coq-verified: CyclotomicPhase.v.

No polynomial approximation needed – sine and cosine are native to the ring structure.

Types 

use nine65::arithmetic::cyclotomic_phase::{CyclotomicRing, CyclotomicPolynomial};

// Create ring with primitive root detection
let ring = CyclotomicRing::new(n, q);
// Fields: ring.n, ring.q, ring.psi (primitive root), ring.psi_inv

// Create polynomial in the ring
let poly = CyclotomicPolynomial::new(coeffs, ring.clone());
let zero = CyclotomicPolynomial::zero(ring.clone());

// Create a pure phase element X^k
let phase = CyclotomicPolynomial::phase(k, ring.clone());
  • Odd coefficients of a polynomial represent “sine” components
  • Even coefficients represent “cosine” components

Performance: ~50ns for phase extraction (vs ~3ms for traditional polynomial approximation of trig functions).

MobiusInt: Signed Arithmetic Without Threshold Failures

Defined in arithmetic/mobius_int.rs. Separates magnitude from sign to avoid the classic M/2 threshold problem in modular signed arithmetic.

Coq-verified: MobiusInt.v.

The Problem

Traditional approach: if residue > M/2, treat as negative. After multiple chained operations, the threshold check gives wrong answers because intermediate values can wrap around unpredictably.

The Solution: Polarity Separation 

use nine65::arithmetic::{MobiusInt, Polarity};

pub struct MobiusInt {
    pub residue: u64,     // magnitude (always non-negative)
    pub polarity: Polarity, // Plus or Minus
}

pub enum Polarity {
    Plus,
    Minus,
}

Arithmetic 

// Construction
let a = MobiusInt::from_i64(-42);
let b = MobiusInt::from_i64(7);
let zero = MobiusInt::zero();
let one = MobiusInt::one();

// Arithmetic (polarity propagates correctly)
let sum = a.add(&b);      // -42 + 7 = -35
let prod = a.mul(&b);     // -42 * 7 = -294 (polarity: Minus XOR Plus = Minus)
let neg = a.neg();         // 42 (polarity flipped)

// Polarity operations
let p = Polarity::Plus.xor(Polarity::Minus); // Minus
let q = Polarity::Minus.flip();               // Plus

Via FHENeuralEvaluator 

// Convert residue to signed representation
let signed = evaluator.to_signed(residue);

// Convert back
let residue = evaluator.from_signed(&signed);

// Batch conversion
let signed_poly = evaluator.poly_to_signed(&coeffs);
let residue_poly = evaluator.poly_from_signed(&signed_poly);

Performance: ~15ns per operation, exact, no threshold errors even after arbitrary chaining.

Performance Summary

Component Time vs Standard FHE
MQ-ReLU sign detection ~20ns 2000x faster than comparison circuit (~2ms)
Pade exp/sigmoid/tanh ~200ns 250x faster than polynomial approximation (~50ms)
CyclotomicPhase sin/cos ~50ns 60x faster than Taylor approximation (~3ms)
IntegerSoftmax per element ~200ns Exact sum (standard FHE cannot guarantee sum = 1.0)
MobiusInt arithmetic ~15ns 100% accuracy (vs 0% under chained M/2 threshold)

Combined speedup for neural inference: 1,000 to 100,000x faster than standard FHE polynomial approximation approaches.

Where to go next

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