genotype_phenotype_covariance — Blind Machine protocol (v1 encrypted×encrypted anchor)

tenseal-BFV, multiplication-supporting params, depth 1. The protocol that
justifies shipping the multiplication tier at all: the server derives a genuine
ciphertext × ciphertext product under encryption (enc(g) · enc(y)), which
the additive tier structurally cannot do. Same trust loop as the flagship
(freeze cohort → encrypted moments → min-N release → certificate), one
relinearized multiplicative level deeper. See docs/protocol_catalog.md §6.

What it computes

Each contributor holds a genotype dosage vector g ∈ {0,1,2}^L over a fixed,
published coordinate definition (ordered variants (chrom,pos,ref,alt); missing
calls → 0) and an integer-coded phenotype scalar y (binary
case/control in {0,1} by default, or a quantized trait in {0..Q}). Both are
encrypted and co-packed into one blob per contributor (a BMCT1 container
carrying that owner’s (cipher_g, cipher_y) pair) so the server forms the product
itself. One blob per contributor is the platform’s canonical contribution shape —
it keeps the pair inseparable through the hosted worker’s digest-sorting Stager
(see Contribution shape). The cohort
aggregate released is four moments:

sum_g[j]  = Σ_i g_ij                 (additive)
sum_gy[j] = Σ_i g_ij · y_i           (ciphertext × ciphertext, depth 1)
sum_y     = Σ_i y_i                  (additive, read from a broadcast slot)
sum_y2    = Σ_i y_i²                 (ciphertext × ciphertext, depth 1)

and, post-decrypt, the per-variant genotype/phenotype covariance:

cov[j] = sum_gy[j]/N − (sum_g[j]/N)·(sum_y/N)

(plus the cohort phenotype mean sum_y/N and variance sum_y2/N − (sum_y/N)²).

Phenotype broadcast. At encode time the scalar y is broadcast to all L
slots, so the element-wise cipher_g · cipher_y yields g_j · y per coordinate
with no cross-slot rotation — which is why this protocol needs relin keys
but no Galois keys. sum_y / sum_y² are therefore constant across the leading
slots (any slot is the scalar); decode reads one and cross-checks uniformity.

Exactness: BFV is exact in Z_t. With a binary phenotype the largest moment
is ~2N (sum_gy, g≤2·y≤1), so plain_modulus t = 786433 (a 20-bit batching
prime valid at poly=16384) stays exact for N up to ~196k. tolerance: 0 — the
encrypted integer moments equal the cleartext moments bit-for-bit. A quantized
trait y ∈ {0..Q} raises the envelope to ~N·Q² and needs a larger t (a per-
deployment build decision; see SECURITY.md).

Append-1 sentinel: encryption appends a trailing 1 to BOTH the genotype and
the broadcast-phenotype vectors, so all four moments’ last slot decrypts to
exactly N (sum_g/sum_y: Σ1=N; sum_gy/sum_y2: Σ 1·1 = N). decode
cross-checks that all four sentinels agree — a stronger integrity check than the
single-sentinel additive flagship — but it is an integrity check, not a MAC
(see SECURITY.md).

Contribution shape — one packed (g,y) blob. Each contributor uploads a
single ciphertext blob that co-packs its (cipher_g, cipher_y) pair into a
deterministic BMCT1 container (magic BMCT1\n, names {g, y} — the same
container format stage 30 uses for the four moment ciphertexts). This is
load-bearing, not cosmetic: the hosted worker’s Stager digest-sorts every
staged ciphertext (worker/lib/blind_worker/stager.rb), so two separate g and
y ciphertexts would be reordered into an arbitrary permutation and the server’s
positional pairing would break — silently, because every moment’s append-1
sentinel still reconciles to N. Co-packing the pair at encrypt time makes that
mis-pairing structurally impossible: the digest-sort can only permute whole
contributors, never split a pair, and the moment folds in stage 30 are order-
independent across contributors (pinned by
test_result_is_order_independent_under_digest_sort).

Why encrypted × encrypted (honest note)

For a single contributor who holds both g and y, the product g·y is
client-precomputable — so the same covariance could be served by additive BFV
with a client-supplied g·y. v1 ships the multiplicative version deliberately, and
the paper states the actual benefit plainly (docs/protocol_catalog.md §6):
server-derived-quantity integrity (the server, not a possibly-malformed client,
forms the product), minimal contributor payload (one packed blob carrying two
ciphertexts, no precomputed cross-terms), and it is the bridge to future
cross-party products
where g and y are held by different parties and no single client can
precompute g·y. It is not mathematical necessity — it is the least-powerful
configuration that exercises the encrypted-multiply path v2 will need.

Stage lifecycle & I/O contract

The author’s logic lives in three pure-function files, grouped by role: server.py
(compute, the only server-side function), local_project_owner.py
(keygen/decrypt/decode), and local_data_owner.py (encode/encrypt) —
these are what sibling tests/ import. The six numbered files are materialized into signed/ at run time and are
kit-owned shims (thin argparse wrappers; do not edit) that map each stage’s CLI
(python NN_*.py --help) onto those functions, keeping the lifecycle visible
without opening a subdirectory.

Inter-stage formats: contexts and ciphertexts are TenSEAL’s raw serialized bytes
(binary); raw is {"genotype":[…], "phenotype":y}; encoded is {"g":[…], "y":[…]}; plain is a labelled dict of four int vectors; the released result is
JSON with sum_g, sum_gy, sum_y, sum_y2, mean_g, mean_y, var_y,
covariance, n_contributors.

Server I/O contract preserved. Stage 30 keeps the flagship’s exact
--context/--inputs/--out CLI (that is what the server worker invokes), writing
ONE opaque result.bin FILE that the hosted worker content-addresses. The four
moment ciphertexts are packed into that single --out artifact as a
deterministic, self-describing binary container — magic BMCT1\n (Blind Machine
multi-CipherText container v1), a uint8 count then, in fixed
MOMENT_ORDER = (sum_g, sum_gy, sum_y, sum_y2), each moment as a length-prefixed
name + length-prefixed raw ciphertext (pack_results/unpack_results). This is
the SAME container format allele_frequency_with_variance uses (each bundle
carries its own verbatim copy — bundles are self-contained). The moments cannot
be folded into one ciphertext without cross-slot masking (rotation/Galois), which
this protocol deliberately avoids, so one artifact carries four labelled
ciphertexts.

server.py’s compute is written once against an abstract evaluator E
(add/mul), so docs/simulation_mode.md’s cleartext correctness oracle swaps a
PlaintextEvaluator for the same compute and cannot drift from this encrypted
path. Determinism (BFV add and relinearized multiply are deterministic; the
container order is fixed) gives verify-by-re-execution: the same ordered
ciphertexts in → a bit-identical result digest out.

Run the full loop by hand

cd protocols/genotype_phenotype_covariance
D=/tmp/gpc && mkdir -p "$D"
R() { (cd signed && uv --project env run python "$@"); }

R 00_keygen.py --out-dir "$D"
inputs=()
for i in 00 01 02 03; do
  R 10_encode.py  --raw ../tests/vectors/contributor_$i.json --length 16 --out "$D/enc_$i.json"
  R 20_encrypt.py --context "$D/public_context.tenseal" --encoded "$D/enc_$i.json" \
    --out "$D/ct_$i.bin"          # ONE packed (g,y) blob per contributor
  inputs+=("$D/ct_$i.bin")
done
R 30_compute_encrypted.py --context "$D/public_context.tenseal" \
  --inputs "${inputs[@]}" --out "$D/result.bin"   # input order does not matter
R 40_decrypt.py --context "$D/secret_context.tenseal" --result "$D/result.bin" --out "$D/plain.json"
R 50_decode.py  --plain "$D/plain.json" --length 16 --out "$D/result.json"
cat "$D/result.json"

Test (local-loop equivalence)

uv --project signed/env run --group dev python -m pytest tests/

Proves keygen → encode → encrypt (≥3 synthetic contributors, one packed (g,y) blob
each) → compute (a real ct×ct product) → decrypt → decode equals the cleartext
moment oracle exactly, that the SAME compute() run over a PlaintextEvaluator
agrees with a direct cleartext oracle (the abstract-evaluator seam), that the
sentinel decrypts to exactly N in all four moments (including that dropping one
upload yields N−1 and removes exactly that contributor’s moments), and that
digest-sorting the contributor blobs — the exact reordering the hosted Stager
performs — does not change the decoded result. Skips with a clear reason only if
TenSEAL cannot be imported.

Crypto parameters

--security {128,192,256} (default 128)

00_keygen.py accepts --security to select the coefficient-modulus chain. N
(16384) and plain_modulus (786433) are fixed — only the chain moves the achieved
level, which flows unchanged through every later stage (they all
ts.context_from(...)).

The intentional inversion: at fixed N, security level == the q-band, so a
smaller coefficient modulus is more secure. The depth-1 noise floor for the
binary payload sits in the 256 band, so certifying 128/192 means spending surplus
modulus (bigger, slower ciphertexts). The 128-bit config is therefore the largest
and slowest, not the cheapest — correct RLWE behaviour, not a bug (see
SECURITY.md). All three decrypt bit-exact vs the cleartext oracle
(test_local_loop_bit_exact_at_every_security_level, TenSEAL 0.3.16), with the
append-1 sentinel recovering exactly N in all four moments at each level.

python 00_keygen.py --out-dir "$D" --security 192   # 192-bit context

Escape hatch (§3, quantized trait / oversized cohort). A quantized phenotype
y ∈ {0..Q} needs t > ~N·Q² (a ~30-bit batching prime), whose fatter depth-1
noise budget breaches the 256 cap at n=16384 and forces N=32768. That per-
deployment build is reachable by overriding the fixed knobs directly:
--poly-modulus-degree 32768 --plain-modulus 537133057 --coeff-mod-bit-sizes 60 40 40 60
(the explicit --coeff-mod-bit-sizes overrides --security). Not part of the
default table.

Coordinate definition & synthetic data

For the synthetic v1 demo the L=1000 coordinate list and the phenotype coding
scheme are generated/declared deterministically from manifest.yml (input
block) rather than enumerated inline. The invariant that matters is that every
contributor encodes against the same published definition (variants +
phenotype coding) and that definition is folded into the bundle SHA-256. All data
here is synthetic integer vectors; no real genomic or phenotype data is used
anywhere.