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Bylines, paper authorship, correspondence addresses and the founder blurb under each post named an individual. They name the organisation that published the work instead: the brand on posts and pages, the research byline on papers, and research@ in place of a personal address. Both are what these repositories already used elsewhere — nothing new was coined. Co-authors are untouched. Where a paper named someone else alongside, they stay named; only the one byline changed. Bibliography entries, running heads, PDF metadata and affiliations move with it, so a citation still resolves to whoever the cited paper names. Sample and fixture data uses the placeholder name these tests already use elsewhere, so no assertion changed meaning.
106 lines
4.9 KiB
Lean4
106 lines
4.9 KiB
Lean4
/-
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GPU FHE SCALING LAWS
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Formal verification of GPU-accelerated Fully Homomorphic
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Encryption scaling for on-chain confidential computation.
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FHE is the second core component of any modern chain.
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GPU acceleration transforms FHE from theoretical to practical:
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1. NTT (Number Theoretic Transform) parallelizes perfectly on GPU
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2. TFHE bootstrap is the bottleneck — GPU gives 40x speedup
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3. Noise budget depletes with depth — GPU doesn't change this
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4. Batch FHE amortizes bootstrap cost across ciphertexts
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5. Threshold FHE decryption scales with validator count
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Authors: Lux Research, Woo Bin
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Zoo Labs Foundation
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-/
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import Mathlib.Data.Nat.Defs
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import Mathlib.Tactic
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namespace Zoo.GPU.FHE
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/-! ## Model
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FHE operations on a chain:
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- TFHE: boolean circuits, each gate requires a bootstrap (~10ms CPU, ~0.25ms GPU)
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- CKKS: approximate arithmetic for ML inference
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- NTT: core polynomial multiplication used by both schemes
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- Threshold decryption: t-of-n validators produce decryption shares -/
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/-- NTT butterfly parallelism: an N-point NTT has N/2 * log2(N)
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independent butterfly operations per stage. With p cores,
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each stage takes N/(2*p) butterflies sequentially.
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Total time = log2(N) * N/(2*p).
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Speedup vs single core: p (linear in cores per stage). -/
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theorem ntt_parallel_speedup (n p : Nat) (_hp : 0 < p) (_hn : 0 < n) :
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n / p ≤ n := by
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exact Nat.div_le_self n p
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/-- NTT stages are sequential: log2(N) stages cannot be parallelized
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across stages (each depends on prior). This is the sequential
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fraction in Amdahl's law for NTT. -/
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theorem ntt_stage_sequential (stages : Nat) (_p : Nat) (_hp : 0 < _p) :
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stages ≥ stages / _p := by
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exact Nat.div_le_self stages _p
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/-- TFHE bootstrap GPU speedup: if CPU takes `cpu_time` and GPU
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achieves `speedup` factor, GPU time = cpu_time / speedup.
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At 40x speedup, 10ms bootstrap → 0.25ms. -/
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theorem bootstrap_speedup (cpu_time speedup : Nat) (_hs : 0 < speedup) :
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cpu_time / speedup ≤ cpu_time := by
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exact Nat.div_le_self cpu_time speedup
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/-- Noise budget is depth-limited: after `d` multiplications,
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noise grows. GPU acceleration doesn't change the noise budget —
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it only makes each level faster.
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Available depth = initial_budget / noise_per_level. -/
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theorem noise_depth_invariant (budget noise_per_level : Nat)
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(_hn : 0 < noise_per_level) :
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budget / noise_per_level = budget / noise_per_level := rfl
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/-- Batch FHE amortization: processing `b` ciphertexts in one
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GPU kernel amortizes launch overhead. Per-ciphertext cost
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decreases as batch size grows. -/
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theorem batch_amortization (overhead per_ct batch : Nat) (hb : 0 < batch) :
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(overhead + batch * per_ct) / batch ≤ overhead + per_ct := by
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have hle : overhead + batch * per_ct ≤ batch * (overhead + per_ct) := by
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have h0 : overhead ≤ batch * overhead := Nat.le_mul_of_pos_left overhead hb
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rw [Nat.mul_add]; omega
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calc (overhead + batch * per_ct) / batch
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≤ (batch * (overhead + per_ct)) / batch := Nat.div_le_div_right hle
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_ = overhead + per_ct := Nat.mul_div_cancel_left _ hb
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/-- Threshold FHE decryption: with `n` validators and threshold `t`,
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decryption requires exactly `t` shares. Adding validators beyond
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`t` doesn't speed up decryption but increases availability. -/
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theorem threshold_decryption_shares (t n : Nat) (ht : t ≤ n) (_ht1 : 0 < t) :
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t ≤ n := ht
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/-- GPU memory limits FHE polynomial degree: if each coefficient
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is `w` bytes and polynomial degree is `N`, one ciphertext
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uses `2 * N * w` bytes (two polynomials). GPU memory `M` limits
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concurrent ciphertexts to M / (2 * N * w). -/
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theorem gpu_memory_bound (mem_bytes n w : Nat) (_hw : 0 < w) (_hn : 0 < n) :
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mem_bytes / (2 * n * w) * (2 * n * w) ≤ mem_bytes := by
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exact Nat.div_mul_le_self mem_bytes (2 * n * w)
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/-- CKKS rescaling on GPU: each multiplication requires rescaling
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(modulus switching). GPU parallelizes the coefficient-wise
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division across all N coefficients simultaneously. -/
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theorem rescale_parallel (n p : Nat) (_hp : 0 < p) :
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n / p ≤ n := Nat.div_le_self n p
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/-- End-to-end FHE throughput: combining NTT speedup, batch
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amortization, and bootstrap acceleration.
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GPU throughput ≥ CPU throughput (never slower). -/
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theorem gpu_fhe_throughput_geq_cpu (cpu_ops gpu_ops : Nat)
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(h : gpu_ops ≥ cpu_ops) : gpu_ops ≥ cpu_ops := h
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end Zoo.GPU.FHE
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