Detailed comparisons

Comprehensive performance comparison of PVAC-HFHE against BFV, BGV, CKKS, TFHE, and FHEW

This page presents detailed benchmark results comparing PVAC-HFHE against production-optimized FHE schemes from OpenFHE.

Schemes compared

Scheme	Type	Implementation	Security
BFV	RLWE, exact integer	OpenFHE 1.2	128-bit
BGV	RLWE, exact integer	OpenFHE 1.2	128-bit
CKKS	RLWE, approximate	OpenFHE 1.2	128-bit
TFHE	Bit-level	OpenFHE 1.2	128-bit
FHEW	Bit-level, GINX	OpenFHE 1.2	128-bit
PVAC-HFHE	LPN, exact uint64	PoC/research	128-bit (est)

All schemes are configured for 128-bit security. PVAC-HFHE security is based on the Learning Parity with Noise (LPN) problem, which is less studied than RLWE but undergoes active cryptanalytic evaluation.

Scalar multiplication (ct × ct)

Scheme	Mode	Time (ms)	vs PVAC
PVAC-HFHE	scalar	2.47	1.0x
BFV	shallow (d=1)	7.23	2.9x slower
BFV	leveled (d=5)	18.28	7.4x slower
BGV	leveled	17.61	7.1x slower
CKKS	leveled	35.23	14.3x slower

BFV plaintext modulus comparison

BFV performance varies with plaintext modulus selection:

Modulus	Bits	Ring dim	Mul time (ms)	CT size
65537	17	8192	8.81	384 KB
786433	20	8192	8.12	384 KB
2013265921	31	16384	19.75	1024 KB

BFV requires NTT-friendly primes (p-1 must be divisible by 2×ring_dim). PVAC-HFHE has no such constraint and works with arbitrary uint64 values.

Scalar addition (ct + ct)

Homomorphic addition performance:

Scheme	Time (ms)	vs PVAC
PVAC-HFHE	0.012	1.0x
BFV	0.124	10x slower
BGV	0.552	46x slower
CKKS	1.050	87x slower

PVAC-HFHE’s addition operation is extremely fast, ranging from 10x to 87x faster than RLWE schemes.

Ciphertext size

Fresh ciphertext comparison

Scheme	Mode	Size	vs PVAC
PVAC-HFHE	scalar	42 KB	1.0x
BFV	shallow	256 KB	6x larger
BFV	leveled	1024 KB	24x larger
BGV	leveled	1792 KB	43x larger
CKKS	leveled	3584 KB	85x larger

PVAC-HFHE ciphertexts are dramatically smaller, ranging from 6x to 85x smaller than RLWE schemes for fresh encryptions.

PVAC-HFHE ciphertext growth with depth

PVAC-HFHE ciphertext size grows exponentially with circuit depth in the current PoC implementation.

Depth	Time (ms)	Size	Growth factor
d0 (fresh)	-	42 KB	1.0x
d1	2.68	34 KB	0.8x
d2	10.34	136 KB	3.2x
d3	31.46	441 KB	10.5x
d4	97.11	1359 KB	32x
d5	285.83	4112 KB	98x

PVAC-HFHE ciphertext size exceeds BFV leveled at depth 4.

Circuit depth performance

Performance comparison across different multiplicative depths:

Depth	PVAC-HFHE	BFV	BGV	CKKS	Fastest
d1	2.68 ms	19.54 ms	17.40 ms	35.85 ms	PVAC 7.3x
d2	10.34 ms	14.38 ms	15.11 ms	31.22 ms	PVAC 1.4x
d3	31.46 ms	13.98 ms	14.39 ms	30.71 ms	BFV 2.3x
d4	97.11 ms	13.84 ms	11.10 ms	21.83 ms	BGV 8.7x
d5	285.83 ms	11.37 ms	9.50 ms	18.93 ms	BGV 30x

PVAC-HFHE (PoC) exhibits exponential performance degradation with depth, while RLWE schemes maintain near-constant performance through modulus switching and other optimizations.

Dot product (scalar vectors)

Vector dot product performance for various vector sizes:

Size (n)	PVAC-HFHE	BFV	BGV	CKKS	Speedup
4	9.61 ms	73.24 ms	74.55 ms	156.53 ms	7.6x
8	19.08 ms	149.68 ms	152.55 ms	308.24 ms	7.8x
16	38.49 ms	297.02 ms	294.65 ms	605.52 ms	7.7x
32	80.27 ms	598.94 ms	626.17 ms	1218.69 ms	7.5x

PVAC-HFHE maintains a consistent 7.5-7.8x speedup across all vector sizes.

Polynomial evaluation

Evaluating f(x) = 3x³ + 2x² + 5x + 7 (requires depth 3):

Scheme	Time (ms)	vs PVAC
PVAC-HFHE	62.88	1.0x
BFV	71.72	1.1x slower
BGV	92.79	1.5x slower
CKKS	182.35	2.9x slower

For degree-3 polynomials, PVAC-HFHE maintains competitive performance despite depth limitations.

Bit-level FHE comparison

NAND gate performance

Single NAND gate evaluation:

Scheme	Mode	Keygen (ms)	NAND (ms)
FHEW	GINX	374	79.30
TFHE	std128	440	81.71
binfhe_ap	AP	785	102.56

Derived 64-bit multiplication

These estimates are derived by multiplying NAND gate latency by the number of gates required for 64-bit schoolbook multiplication (24,576 gates) without optimizations. This comparison is primarily academic, as bit-level FHE and scalar FHE solve different problems.

Scheme	64-bit mul time	vs PVAC
PVAC-HFHE	2.47 ms	1.0x
FHEW	32.48 min	789,000x slower
TFHE	33.47 min	813,000x slower
binfhe_ap	42.01 min	1,020,000x slower

TFHE-rs GPU comparison

Comparison with TFHE-rs on both CPU and GPU for 64-bit integer operations:

Operation	PVAC-HFHE (PoC)	TFHE-rs CPU	TFHE-rs GPU	vs CPU	vs GPU
Addition	0.012 ms	109 ms	8.97 ms	9,083x	747x
Subtraction	0.012 ms	109 ms	8.97 ms	9,083x	747x
Multiplication	2.47 ms	402 ms	31.9 ms	163x	13x

Source: TFHE-rs official benchmarks

SIMD and batch throughput

RLWE SIMD performance

RLWE schemes support native SIMD operations:

Scheme	Slots	Mul time (ms)	Per-slot (μs)
BFV	8192	17.85	2.18
CKKS	4096	35.49	8.66

PVAC-HFHE parallel throughput

PVAC-HFHE parallel multiplication performance (8 threads):

Operations	Sequential (ms)	Parallel (ms)	Speedup	Throughput
512	1391	189	7.4x	2711 ops/s
2048	4963	795	6.2x	2575 ops/s
8192	19904	2608	7.6x	3141 ops/s

Throughput comparison

RLWE schemes achieve significantly higher throughput through native SIMD support, while PVAC-HFHE relies on multi-threading.

Mode	Ops/second	Relative
BFV SIMD (8192 slots)	~459,000	146x faster
PVAC-HFHE parallel (8 threads)	~3,141	1.0x

Key generation and encryption

Setup and encryption operation performance:

Scheme	Keygen (ms)	Encrypt (ms)	Decrypt (ms)
BFV	38.43	10.91	2.54
BGV	62.03	12.70	3.48
CKKS	143.61	23.34	10.37
PVAC-HFHE	858.95	84.11	13.38

PVAC-HFHE key generation is 22x slower and encryption is 8x slower than BFV. This is acceptable for a proof of concept and is primarily due to unoptimized initialization. However, key generation typically only needs to be performed once.

Key sizes

Public key and ciphertext size comparison:

Scheme	Public key size	CT size
PVAC-HFHE	8 MB	42 KB
BFV	-	1024 KB
BGV	-	1792 KB
CKKS	-	3584 KB

PVAC-HFHE has a larger public key (8 MB) but much smaller ciphertexts for fresh encryptions.

Performance summary

Where PVAC-HFHE excels

• Scalar multiplication (2.9-14.3x faster)
• Scalar addition (10-87x faster)
• Dot products (7.5-7.8x faster)
• Fresh ciphertext size (6-85x smaller)
• Shallow circuits (depth 1-2)

Where RLWE schemes excel

• Deep circuits (depth ≥ 3)
• SIMD batch processing (146x higher throughput)
• Ciphertext size at depth ≥ 4
• Key generation and encryption speed

Choose PVAC-HFHE for applications requiring fast scalar arithmetic at shallow depths with minimal ciphertext size. Choose RLWE schemes for deep circuits or batch processing workloads.