Concepts

Concepts explain the mental models that remain useful across APIs and implementations. FHElium's foundation consists of two use models, shared CKKS value and operation semantics, and a Backend execution interface. Eager evaluates one requested operation at a time. Compile represents and transforms a Program before linking it for execution. Manual Program/Compilation/Pipeline execution and callable specialization are equally supported paths through that Compile machinery.

Core model

Expanded diagram

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Eager and Compile differ in when CKKS state and scheduling decisions are made. They converge on registered implementations that consume Tensor operands and non-Tensor execution handles. A Compilation keeps one live material-binding dictionary alongside its Program, workspace, and pass reports; linking prepares host calls to selected native or generated implementations. Runtime, distributed execution, persistence, and Residency compose with these paths without becoming hidden properties of a ciphertext or Program.

Choose a starting point

Understand the architectureSee how values, Eager, Compile, Backend implementations, runtime mechanisms, and native execution form one stack.Reason about CKKS valuesStart from value identity, represented state, operation transitions, and the scale-depth lifecycle.Understand a ProgramLearn how mixed-level SSA, partial CKKS state, symbolic references, and structural validity coexist in one neutral representation.Compose a Compile pipelineUnderstand manual and callable Compile, shared transformation state, Tensor materials, implementation selection, linking, and specialization.Trace system ownershipUnderstand process-local ownership, placement, policy, and lifetime responsibilities.Use multiple GPUsUnderstand rank-local values, application-owned partitioning, and collective ordering before choosing a communication schedule.Repeat execution within bounded memoryConnect value signatures, reusable buffers, CUDA Graph execution, and Residency lifetimes.Optimize a workloadEstablish the CKKS cost model before selecting lowering, implementation, batching, communication, or scheduling choices.Use terminology and notationLook up FHElium terms, value-state coordinates, tensor layouts, arithmetic laws, and distinctions among execution layers.

Concept families

Family	Questions answered
Architecture	How do Eager, Compile, shared operation semantics, Backend implementations, runtime mechanisms, and native execution fit together?
CKKS semantics	What mathematical and representational state does a value carry, and how does each operation change it?
Program representation	What can a partially specified, mixed-level Program represent before it is executable?
Compile pipelines	How do source capture, caller-composed passes, Backend linking, and runtime specialization remain separable?
Distributed execution	How do rank-local values and collective order define a multi-process computation?
Execution and lifecycle	How are repeated execution, buffers, CUDA Graphs, persistence, and live-value retention modeled?
Performance	Which arithmetic, memory, launch, and communication costs determine a useful optimization?
Advanced CKKS mechanisms	How are bootstrapping components and other research mechanisms composed without redefining the core execution model?
Terminology and mathematical model	What do specialized terms, symbols, layouts, and cross-layer identities mean?

Suggested reading sequences

For immediate evaluator work:

Architecture
→ Value model and identity
→ Evaluator operation transitions
→ Scale and depth lifecycle
→ Key lifecycle

For Program transformation and execution:

Architecture
→ Neutral IR programs
→ Open compiler stack
→ State transitions and orthogonality
→ System ownership

For runtime and multi-device work:

Architecture
→ Rank-local SPMD
→ Communication semantics
→ Signatures and buffers
→ Residency lifetimes