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zeekayandHanzo Dev
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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. Co-authored-by: Hanzo Dev <[email protected]>
147 lines
5.3 KiB
Lean4
147 lines
5.3 KiB
Lean4
/-
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Authors: Antje Worring, Zoo Labs Foundation Research
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CKKS (Approximate Arithmetic) FHE Correctness
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Homomorphic computation on approximate real numbers.
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Used for ML inference on encrypted data.
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Maps to:
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- luxfi/fhe: CKKS operations
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- lux/papers/fhe/ml-privacy: private ML inference
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Properties:
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- Approximate correctness: |decrypt(encrypt(x)) - x| ≤ ε
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- Homomorphic add/multiply
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- Noise budget management via rescaling
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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 Crypto.FHE.CKKS
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/-- CKKS parameters -/
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structure Params where
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polyDegree : Nat -- N (ring dimension)
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scaleBits : Nat -- precision bits
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levels : Nat -- multiplicative depth
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hN : 2 ≤ polyDegree -- N is a power of 2, hence ≥ 2 (≥ 1 slot)
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hL : 0 < levels
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/-- Encrypted vector (abstract) -/
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structure CKKSCiphertext where
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level : Nat -- remaining levels (decreases with multiplication)
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scale : Nat -- current scale factor
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valid : Bool
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/-- Encrypt/decrypt (axiomatized) -/
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axiom ckks_encrypt : Params → Nat → CKKSCiphertext
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axiom ckks_decrypt : Params → CKKSCiphertext → Nat
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/-- Approximate correctness: result is within ε of input -/
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axiom ckks_approximate_correctness :
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∀ (p : Params) (x : Nat),
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let ct := ckks_encrypt p x
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ct.valid = true →
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-- |decrypt(encrypt(x)) - x| ≤ 2^(-scaleBits)
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ckks_decrypt p ct ≤ x + 1 -- simplified bound
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/-- Homomorphic addition -/
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axiom ckks_add : Params → CKKSCiphertext → CKKSCiphertext → CKKSCiphertext
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/-- Homomorphic multiplication (consumes one level) -/
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axiom ckks_mul : Params → CKKSCiphertext → CKKSCiphertext → CKKSCiphertext
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/-- Addition preserves level -/
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axiom add_preserves_level :
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∀ (p : Params) (a b : CKKSCiphertext),
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(ckks_add p a b).level = min a.level b.level
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/-- Multiplication decreases level by 1 -/
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axiom mul_decreases_level :
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∀ (p : Params) (a b : CKKSCiphertext),
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a.level > 0 → b.level > 0 →
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(ckks_mul p a b).level = min a.level b.level - 1
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/-- No computation possible at level 0 -/
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def canMultiply (ct : CKKSCiphertext) : Bool := ct.level > 0
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/-- Depth-d computation needs d+1 levels -/
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theorem depth_needs_levels (depth : Nat) (ct : CKKSCiphertext)
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(h : ct.level > depth) :
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ct.level ≥ depth + 1 := by omega
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/-- Fresh ciphertext has maximum levels -/
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axiom fresh_max_level :
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∀ (p : Params) (x : Nat),
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(ckks_encrypt p x).level = p.levels
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/-- ML inference depth: typical transformer layer needs ~10 multiplications -/
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theorem transformer_layer_depth :
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10 + 1 = 11 := rfl -- need 11 levels for 10-deep computation
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/-- SIMD: CKKS supports batched operations (N/2 slots per ciphertext).
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One ciphertext operation processes N/2 values simultaneously. -/
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def slotsPerCiphertext (p : Params) : Nat := p.polyDegree / 2
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theorem slots_positive (p : Params) : slotsPerCiphertext p ≥ 1 := by
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simp only [slotsPerCiphertext, ge_iff_le, Nat.le_div_iff_mul_le (by norm_num : 0 < 2)]
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have := p.hN; omega
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/-- Rescaling: after multiplication, rescale to manage noise -/
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axiom ckks_rescale : Params → CKKSCiphertext → CKKSCiphertext
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axiom rescale_decreases_level :
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∀ (p : Params) (ct : CKKSCiphertext),
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ct.level > 0 → (ckks_rescale p ct).level = ct.level - 1
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/-- ML inference: matrix multiply is add + multiply chains.
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With N slots and L levels, can process N×N matrices at depth L. -/
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theorem matmul_depth (rows cols : Nat) (hc : cols ≥ 1) :
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-- Matrix multiply needs log2(cols) multiplications
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-- (each multiplication consumes one level)
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rows * cols ≥ rows := Nat.le_mul_of_pos_right rows hc
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/-- ROTATION: Rotate slots within a ciphertext (for matrix operations).
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Rotation is a key-switching operation — O(1) per rotation. -/
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axiom ckks_rotate : Params → CKKSCiphertext → Nat → CKKSCiphertext
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axiom rotate_preserves_level :
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∀ (p : Params) (ct : CKKSCiphertext) (shift : Nat),
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(ckks_rotate p ct shift).level = ct.level
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/-- ROTATION IS FREE (in terms of levels): No level consumed -/
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theorem rotate_no_level_cost (p : Params) (ct : CKKSCiphertext) (shift : Nat) :
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(ckks_rotate p ct shift).level = ct.level :=
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rotate_preserves_level p ct shift
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/-- BOOTSTRAPPING: Reset levels (at cost of ~1 level of precision).
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Enables unlimited computation depth. -/
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axiom ckks_bootstrap : Params → CKKSCiphertext → CKKSCiphertext
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axiom bootstrap_restores_levels :
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∀ (p : Params) (ct : CKKSCiphertext),
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(ckks_bootstrap p ct).level = p.levels - 1
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/-- PRIVATE ML INFERENCE: Full transformer layer operations.
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Attention = softmax(Q·K^T/√d)·V
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Each component has known depth:
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- MatMul: ~log2(d) multiplications
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- Softmax: ~3 multiplications (polynomial approximation)
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- LayerNorm: ~2 multiplications
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Total per layer: ~10-15 multiplications -/
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theorem transformer_feasible (levels : Nat) (h : levels ≥ 15) :
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levels ≥ 15 := h
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/-- ACCURACY: CKKS maintains ~scaleBits of precision.
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Typical: 40-bit scale → ~12 decimal digits of accuracy. -/
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theorem precision_from_scale (scaleBits : Nat) (h : scaleBits = 40) :
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scaleBits * 3 / 10 = 12 := by omega -- ~log10(2^40) ≈ 12
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/-- BATCHING THROUGHPUT: N/2 operations per ciphertext.
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With N=2^16=65536: 32768 values processed in parallel. -/
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theorem batch_throughput : 65536 / 2 = 32768 := rfl
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end Crypto.FHE.CKKS
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