Privacy-preserving computation for matrix workloads

Linear algebra,
without exposing the data.

Encrypted linear algebra enables matrix-vector multiplication, matrix factorization, private inference, secure analytics, and collaborative scientific computing while keeping inputs, models, or intermediate states protected.

Explore the workflow View supported operations

HE Compute directly on ciphertexts

MPC Split secrets across parties

SIMD Pack many slots into one encrypted vector

secure_matrix_pipeline.ela

⟦a₁₁⟧

⟦a₁₂⟧

⟦a₁₃⟧

⟦a₂₁⟧

⟦a₂₂⟧

⟦a₂₃⟧

⟦a₃₁⟧

⟦a₃₂⟧

⟦a₃₃⟧

⟦x₁⟧

⟦x₂⟧

⟦x₃⟧

⟦·⟧

1client uploads encrypted vector enc_x
2server encodes matrix diagonals as plaintext masks
3acc = Σ rotate(enc_x, i) · diag_i(A)
4optional: rescale, relinearize, refresh, or open shares
5return encrypted result ⟦A x⟧
6// raw values remain hidden throughout the pipeline

Motivation

Why encrypted linear algebra matters

Modern data systems are built on vectors, matrices, and tensors. Once these objects contain medical records, mobility traces, financial indicators, genomic features, or proprietary model parameters, ordinary linear algebra becomes a privacy problem.

Data utility without raw disclosure

Organizations can outsource computation or collaborate on analytics while keeping sensitive vectors, labels, or model states protected from the computing party.

Foundation for private AI

Neural inference, recommendation, clustering, graph analytics, and federated optimization all rely on repeated linear algebra kernels.

Composable security design

Encrypted kernels can be combined with access control, audit trails, differential privacy, secure hardware, and verifiable computation.

Cryptographic primitives

Two major routes: homomorphic encryption and secure multi-party computation

Both approaches support privacy-preserving linear algebra, but they expose different engineering trade-offs in latency, communication, trust assumptions, and deployment complexity.

Approach	Core idea	Strengths	Typical challenges
Homomorphic Encryption	Compute on ciphertexts so that decryption reveals the same result as computing on plaintext data.	Non-interactive server-side compute Natural outsourcing model SIMD ciphertext packing	Expensive rotations and multiplications, limited multiplicative depth, approximation error in CKKS, key management.
Secure MPC	Split values into shares held by multiple parties; parties jointly compute without revealing their private inputs.	High throughput for additions Flexible protocols Strong collaborative setting	Communication rounds, online availability, network latency, protocol-specific assumptions, secure preprocessing.
Hybrid HE + MPC	Use HE for compact encrypted transmission or outsourcing, and MPC for interactive nonlinear or high-depth subroutines.	Balanced performance Protocol specialization Better end-to-end design	Conversion between ciphertexts and shares, security proof complexity, implementation overhead.

Reference architecture

A practical encrypted computation workflow

A robust system usually separates cryptographic setup, data encoding, encrypted kernel execution, result release, and auditability.

Encode and protect inputs

Vectors and matrices are packed into ciphertext slots or secret shares according to the target kernel.

Execute encrypted kernels

The compute party evaluates additions, scalar products, rotations, multiplications, and reductions under protection.

Manage depth and precision

HE systems schedule rescaling, relinearization, modulus switching, and optional bootstrapping; MPC systems manage rounds and preprocessing.

Release only authorized outputs

The final encrypted result is decrypted by the data owner or reconstructed by authorized parties.

Client

Private inputs

Vectors, matrices, features, labels, or model parameters.

Protection

Encryption / sharing

HE ciphertexts, additive shares, garbled values, or hybrid encodings.

Compute

Encrypted kernels

MatVec, MatMul, convolution, least squares, PCA, factor models.

Output

Protected result

The server learns no raw values; only authorized users recover outputs.

Governance

Audit and policy

Access control, key rotation, logs, and compliance boundaries.

Core operations

Encrypted linear algebra is built from a small set of kernels

Once these kernels are efficient, larger workloads such as private neural inference, secure recommendation, encrypted graph analytics, and confidential scientific simulation become much easier to assemble.

Vector addition

Elementwise addition of encrypted vectors; typically cheap in both HE and MPC.

Low cost

Scalar product

Multiply slots elementwise and reduce the sum using rotations, tree reductions, or MPC opening protocols.

Reduction-heavy

Matrix-vector multiplication

A central primitive for linear models, neural layers, iterative solvers, and graph propagation.

Packing-sensitive

Matrix multiplication

Can be implemented through blocked packing, diagonal methods, outer-product decompositions, or share-based protocols.

Bandwidth-aware

Polynomial evaluation

Approximates nonlinear steps around linear algebra pipelines, including activations, normalizations, and inverse-square-root routines.

Depth-critical

Iterative solvers

Gradient descent, conjugate gradient, EM updates, and power iteration require careful scheduling of depth, communication, and precision.

System-level

Interactive mini demo

Plaintext simulation of a protected dot product

This toy widget does not perform real encryption. It illustrates the workflow: inputs are hidden, computation is performed over protected representations, and only the final authorized output is revealed.

Private vector a

Private vector b

Encoded inputs Waiting for computation...

Protected elementwise products Waiting for computation...

Authorized result Waiting for computation...

Use cases

Where encrypted linear algebra fits naturally

The strongest applications are those where the computation is valuable, the data is sensitive, and the linear algebra structure is regular enough to optimize.

Private model inference

Linear layers, convolutions, attention projections, and recurrent updates can be evaluated over encrypted features or protected model weights.

Biomedical analytics

Hospitals and laboratories can run risk scoring, similarity search, regression, and cohort analysis without pooling raw patient records.

FIN

Financial collaboration

Institutions can compute exposure metrics, fraud signals, or shared risk models while preserving proprietary client-level data.

IoT

Edge and vehicle data

Encrypted sensor features can be processed by cloud services while reducing the leakage of location traces, diagnostics, or behavior profiles.

SCI

Scientific computing

Private matrix computations support cross-institutional research on genomic, chemical, environmental, or industrial datasets.

REC

Secure recommendation

Matrix factorization and similarity-based ranking can be redesigned so that user preferences and item embeddings are not directly exposed.

Design principles

Build for the algebra, not just the cryptography

The best systems co-design packing, numerical approximation, kernel scheduling, network behavior, and security boundaries. A mathematically elegant protocol can still fail in practice if it ignores rotations, memory pressure, precision loss, or round complexity.

Packing first Minimize rotations Control multiplicative depth Exploit sparsity Benchmark end-to-end

Profile the target workload

Identify dominant matrix kernels, tensor shapes, precision requirements, and allowed interaction patterns.

Choose the protection model

Select HE, MPC, or a hybrid architecture based on trust assumptions, latency, and communication constraints.

Co-design encoding and algorithms

Map algebraic structure to ciphertext slots or secret shares, then redesign kernels around the representation.

Validate security and utility

Measure leakage assumptions, numerical error, throughput, memory, bandwidth, and application-level accuracy.

Linear algebra,without exposing the data.