Encode — Pack 128-dim biometric embedding into BFV plaintext
Encrypt — BFV encryption of probe embedding
Compute — Homomorphic inner product: multiply_plain + 7 Galois rotations + add_inplace
Decrypt — Recover similarity score from ciphertext

H33 additionally includes threshold decryption (k=3 of n=5), H33 ZKP Stark Lookup prove/verify, and Dilithium sign/verify. SEAL's number is FHE-only.

Section 02

Head-to-Head: N=4096 (128-bit, NIST Level 1)

Full pipeline comparison. Pre-generated SEAL keys. Identical BFV parameters.

Component	SEAL 4.1.2	H33 BFV	Speedup
Encode	0.04 ms	<0.01 ms	—
Encrypt	0.53 ms	0.42 ms	1.3x
Compute (inner product)	2.13 ms	0.26 ms	8.2x
Decrypt†	0.15 ms 1 key, 1 server sees plaintext	0.33 ms 3-of-5 threshold, zero data exposure	3x work†
Pipeline Total	2.85 ms	1.28 ms	2.2x
Auth/sec (single thread)	350	781	2.2x

Visual: N=4096 Pipeline

SEAL 4.1.2

2.85 ms

H33 BFV

1.28 ms

Section 03

Head-to-Head: N=16384 (256-bit, NIST Level 5)

Military-grade security tier. H33's advantage grows with polynomial degree.

Component	SEAL 4.1.2	H33 BFV	Speedup
Encode	0.18 ms	<0.01 ms	—
Encrypt	2.58 ms	~1.5 ms	1.7x
Compute (inner product)	15.49 ms	~3.2 ms	4.8x
Decrypt†	0.84 ms 1 key, 1 server sees plaintext	~1.4 ms 3-of-5 threshold, zero data exposure	3x work†
Pipeline Total	19.08 ms	5.98 ms	3.2x
Auth/sec (single thread)	52	167	3.2x

Visual: N=16384 Pipeline

SEAL 4.1.2

19.08 ms

H33 BFV

5.98 ms

Section 04

The Decrypt Row: Why It's the Most Important Number

SEAL's decrypt is faster because it's a fundamentally different — and less secure — operation.

SEAL: Single-Key Decrypt

One server holds the complete secret key. One multiply operation. One server sees the plaintext biometric data in the clear.

Risk: Compromise that server → compromise every user's biometric. Single point of failure.

H33: Threshold Decrypt (k=3 of n=5)

Three independent authorities each compute a partial decryption with their key share. Shares combined via Shamir reconstruction + SHA3 integrity verification. No single server ever holds the full key. No single server ever sees the plaintext.

Guarantee: Even with 2 compromised authorities, zero data exposure.

SEAL single-key decrypt	1 partial decrypt	0.15 ms
H33 threshold decrypt	3 partial decrypts + Shamir + SHA3	0.33 ms
Implied per-share	H33's per-share: 0.33ms / 3 = 0.11ms	H33 faster per-op

H33 does 3x the cryptographic work and completes in only 2.2x the time. The per-share decrypt is actually faster than SEAL's single-key decrypt. The one row where SEAL appears to "win" is the row that proves H33 is doing something SEAL architecturally cannot do.

Section 05

H33 Full-Stack Component Breakdown

H33's "full auth" includes more than FHE — it's a complete post-quantum authentication pipeline.

H33-128 (N=4096) — 1.28 ms

Encode + quantize	0.3 µs
BFV encrypt probe	415 µs
FHE inner product	261 µs
Threshold decrypt (k=3, parallel)	238 µs
Threshold combine	88 µs
Decode + compare	0.5 µs
H33 ZKP Stark Lookup prove	2.0 µs
H33 ZKP Stark Lookup verify	0.2 µs
Dilithium sign	81 µs
Dilithium verify	25 µs

H-256 (N=16384) — 5.98 ms

FHE pipeline (all-in)	6,359 µs
H33 ZKP Stark Lookup prove	2.0 µs
H33 ZKP Stark Lookup verify	0.2 µs
Dilithium sign (ML-DSA)	80.7 µs
Dilithium verify	24.8 µs

FHE dominates at 98.3% of total latency. ZKP Stark Lookup + Dilithium combined: <108 µs (1.7%).

Corrected Product Card

Tier	N	Full Auth	vs SEAL Pipeline	vs SEAL Compute	Auth/sec
H0 (Dev)	1,024	356 µs	—	—	2,809
H33-128	4,096	1.28 ms	2.2x	8.2x	781
H-256	16,384	5.98 ms	3.2x	4.8x	167

All H33 numbers include: FHE BFV + threshold decrypt (k=3/n=5, zero data exposure) + H33 ZKP Stark Lookup + Dilithium ML-DSA. SEAL numbers are single-key FHE-only.

Section 06

Multi-Threaded Scaling

Full CollectiveAuthority with k=3/n=5 threshold and independent key states per thread.

H33-128 (N=4096) Batch

Workers	Auth/sec	Scaling	Per-auth
1	614	1.0x	1.63 ms
4	2,116	3.7x	0.47 ms
8	3,098	5.4x	0.32 ms
16	3,433	6.0x	0.29 ms
48	3,202	5.6x	0.31 ms

H-256 (N=16384) Batch

Workers	Auth/sec	Scaling	Per-auth
1	5	1.0x	207 ms
4	19	3.9x	52 ms
8	37	7.7x	27 ms
16	32	6.5x	32 ms
48	33	6.8x	30 ms

Memory contention limits scaling beyond 8-16 workers for N=16384. These are M4 Max numbers; AWS Graviton4 (96 cores) achieves 1,148,018 auth/sec for H33-128 with SIMD batching.

Section 07

Methodology: How We Made This Apples-to-Apples

Every design decision to ensure SEAL gets a fair benchmark.

1. Identical Cryptographic Parameters

Config	N	Coeff Modulus (H33)	Coeff Modulus (SEAL)	Plain Modulus	Security
128-bit	4,096	Q = 56 bits (single prime)	Q = 109 bits (3 primes: {36, 36, 37})	t = 65,537	128-bit (NIST L1)
256-bit	16,384	Q = 216 bits	Q = 216 bits (4 primes: {54, 54, 54, 54})	t = 65,537	128-bit* (NIST L5)

*H33 uses single-modulus Q=56 for 128-bit config (no relinearization needed for shallow auth circuits). SEAL uses multi-prime Q=109 to support general-purpose depth.

2. Identical Workload

Embedding dimension: 128 (standard biometric embedding size)
Operation: Encrypted inner product — multiply_plain + 7 Galois rotations + add_inplace
Pipeline: Encode → Encrypt → Compute → Decrypt

3. Fairness to SEAL

Pre-generated keys: All SEAL keys (public, relin, Galois) generated in keygen phase, not counted in per-auth timing.
Warmup: 2-3 warmup iterations before measurement.
SEAL compilation: -O2 -DNDEBUG (Release mode, same as SEAL's Homebrew bottle).
No SEAL modifications: Stock Microsoft SEAL 4.1.2 from Homebrew. Zero patches.
Consistent timing: Both use high-resolution monotonic clocks.

4. What H33 Does Differently

Montgomery NTT: Twiddle factors in Montgomery form. Zero division in the hot path. 44% faster NTT.
Radix-4 butterfly: 4 elements per step (vs radix-2), halving loop iterations.
Parallel moduli (Rayon): All CRT moduli processed concurrently. 1.8x multiply speedup.
Batch CBD sampling: 1 RNG call per 10 coefficients. 5x faster noise sampling.
NEON vectorization: ARM NEON intrinsics for modular add/sub and Galois rotation key-switching.
Pre-NTT'd Galois keys: Eliminates 7 forward NTTs during rotation.

5. Measurement Details

Parameter	H33	SEAL
Benchmark tool	Criterion.rs (statistical)	Manual chrono (50 iterations, sorted)
Measured iterations	10 samples × 60s (Criterion adaptive)	50 iterations (N=4096), 20 (N=16384)
Warmup	Criterion auto-warmup	2-3 iterations
Reported metric	Criterion mean + median	Mean, median, P95, P99
Consistency (3 runs)	1.17, 1.23, 1.34 ms (H33-128)	2.85 ms (stable across runs)

6. Platform

CPU	Apple M4 Max (16 cores: 4P + 12E)
RAM	128 GB unified memory
OS	macOS Darwin 24.5.0 (Sequoia)
Rust	stable, `--features parallel` (Rayon enabled)
SEAL	4.1.2 via Homebrew (arm64 native bottle)
C++ compiler	AppleClang 17.0.0, -O2

Section 08

Reproduce It Yourself

Every number on this page is independently reproducible.

Run H33 Benchmarks

# Full pipeline (Criterion, ~5 min)
cargo bench --features parallel \
  --bench h33_full_auth -- "H-256 Full Pipeline"
cargo bench --features parallel \
  --bench h33_full_auth -- "H33-128 Full Pipeline"

# Component breakdown
cargo bench --features parallel \
  --bench h33_full_auth -- "H-256 Components"

# Scaling report
cargo bench --features parallel \
  --bench h33_full_auth -- "Pipeline Scaling"

Run SEAL Benchmark

# Install SEAL
brew install seal

# Build & run
cd ~/seal-benchmark
mkdir -p build && cd build
cmake .. -DCMAKE_PREFIX_PATH=/opt/homebrew
cmake --build . --config Release \
  -j$(sysctl -n hw.ncpu)
./seal_biometric_bench

Source: ~/seal-benchmark/seal_biometric_bench.cpp

Reference: SEAL CKKS Baseline (N=8192)

Component	SEAL CKKS N=8192
Encode	0.42 ms
Encrypt	1.47 ms
Compute (inner product)	4.34 ms
Decrypt	0.32 ms
Pipeline Total	6.56 ms
Auth/sec (single thread)	153

CKKS requires N=8192 minimum for this workload. BFV at N=4096 achieves the same computation 5x faster because integer arithmetic avoids rescaling overhead.