IR operations and implementations

This index maps operation families to their Compile transformations and Backend execution owners. Paths below are relative to fhelium/. An IR abstraction level describes the amount of computational detail represented by an operation; CKKS modulus-chain depth is a separate value property.

The operation definitions specify operands, attributes, equations, and state transitions: CKKS, RNS, NTT, and core data references. Numerical parameters, tables, and keys enter execution as Tensor operands. Non-Tensor execution handles use resource references.

Program structure, binding, and interoperability

Operation	Responsibility	Direct owner
fhelium.constant	Materialize a represented literal in generated host code	compile/passes/codegen/_host.py
fhelium.material.ref	Read the Tensor bound to a Program symbol	compile/passes/backend/_operations.py, _prepare_host.py
fhelium.resource.ref	Bind a supplied execution handle	compile/passes/backend/_operations.py, _link_program.py
torch.call	Execute supported public Tensor calls	backend/torch.py
torch.tensor_call	Execute supported public Tensor calls	backend/torch.py
fhelium_fusion.yield	Return the region's selected result values	compile/passes/codegen/_host.py, backend/triton/_expressions.py
fhelium_fusion.execute	Execute a selected generated region	backend/triton/_pointwise.py, _tensor.py

Compile selects fusion regions in compile/passes/fusion.py. The Triton backend owns expression lowering, generated kernels, compilation, and launch. A region may contain several kernels, including native NTT middle stages.

Frontend intent and operand roles

Operation	Responsibility	Direct owner
fhelium_semantic.add	Classify addition operands by encrypted and public roles	compile/passes/frontend/_lower_semantic_to_logical.py
fhelium_semantic.subtract	Classify subtraction operands by encrypted and public roles	compile/passes/frontend/_lower_semantic_to_logical.py
fhelium_semantic.multiply	Classify multiplication operands by encrypted and public roles	compile/passes/frontend/_lower_semantic_to_logical.py
fhelium_semantic.negate	Classify unary negation by its input role	compile/passes/frontend/_lower_semantic_to_logical.py
fhelium_semantic.roll	Classify slot rotation by its input role	compile/passes/frontend/_lower_semantic_to_logical.py
fhelium_logical.add.encrypted_encrypted	Construct CKKS add for encrypted/encrypted operands	compile/passes/ckks/_lower_logical_to_ckks.py
fhelium_logical.add.encrypted_public	Construct CKKS add for encrypted/public operands	compile/passes/ckks/_lower_logical_to_ckks.py
fhelium_logical.add.public_encrypted	Construct CKKS add for public/encrypted operands	compile/passes/ckks/_lower_logical_to_ckks.py
fhelium_logical.subtract.encrypted_encrypted	Construct CKKS subtract for encrypted/encrypted operands	compile/passes/ckks/_lower_logical_to_ckks.py
fhelium_logical.subtract.encrypted_public	Construct CKKS subtract for encrypted/public operands	compile/passes/ckks/_lower_logical_to_ckks.py
fhelium_logical.subtract.public_encrypted	Construct CKKS subtract for public/encrypted operands	compile/passes/ckks/_lower_logical_to_ckks.py
fhelium_logical.multiply.encrypted_encrypted	Construct CKKS multiply for encrypted/encrypted operands	compile/passes/ckks/_lower_logical_to_ckks.py
fhelium_logical.multiply.encrypted_public	Construct CKKS multiply for encrypted/public operands	compile/passes/ckks/_lower_logical_to_ckks.py
fhelium_logical.multiply.public_encrypted	Construct CKKS multiply for public/encrypted operands	compile/passes/ckks/_lower_logical_to_ckks.py
fhelium_logical.negate.encrypted	Construct CKKS negate for encrypted operands	compile/passes/ckks/_lower_logical_to_ckks.py
fhelium_logical.roll.encrypted	Resolve rotation keys and the coefficient rotation route	compile/passes/ckks/_resolve_rotation_keys.py

The shared construction helpers preserve value identity through transparent type references. Arithmetic lowering retains unknown facts until they can be derived or assigned; a type cast does not perform an NTT or change residue data.

Boundary conversion and cryptography

Operation	Responsibility	Direct owner
fhelium_ckks.encode	Embed message slots and quantize scaled coefficients	backend/ckks/codec/_implementation.py, _codec.py, _embedding.py
fhelium_ckks.decode	Reconstruct message slots from scaled coefficients	backend/ckks/codec/_implementation.py, _codec.py, _embedding.py
fhelium_ckks.integer_coefficients_to_rns	Reduce integer coefficients into supplied prime rows	backend/ckks/codec/_implementation.py, backend/rns/context.py
fhelium_ckks.prepare_compressed_plaintext	Prepare periodic compact plaintexts	backend/ckks/codec/_periodic.py
fhelium_ckks.encrypt	Sample and assemble ciphertext components	backend/ckks/crypto/_encryption.py
fhelium_ckks.decrypt	Evaluate the ciphertext phase and reconstruct coefficients	backend/ckks/crypto/_decryption.py

Key generation is a data-provision API in backend/ckks/crypto/_key_generation.py. Decryption's numerical reconstruction tables live in crypto/_tables.py; crypto/_resources.py defines the encryption sampler's execution-resource names.

Ciphertext arithmetic and component composition

Operation	Responsibility	Direct owner
fhelium_ckks.add	Lower ciphertext addition to row-wise RNS addition	compile/passes/lowering/_arithmetic.py
fhelium_ckks.subtract	Lower ciphertext subtraction to row-wise RNS subtraction	compile/passes/lowering/_arithmetic.py
fhelium_ckks.negate	Lower ciphertext negation to row-wise RNS negation	compile/passes/lowering/_arithmetic.py
fhelium_ckks.multiply	Whole CT2 convolution or RNS component composition	backend/ckks/arithmetic.py, compile/passes/lowering/_arithmetic.py
fhelium_ckks.add_scalar	Encode and add a scalar at the ciphertext scale	backend/ckks/scalar.py
fhelium_ckks.multiply_scalar	Encode and multiply by a scalar with its represented scale	backend/ckks/scalar.py
fhelium_ckks.multiply_integer_scalar	Multiply by an unscaled integer while preserving actual scale	backend/ckks/scalar.py
fhelium_rns.add_standard	Add corresponding residues modulo each supplied prime	backend/rns/operations.py
fhelium_rns.subtract_standard	Subtract corresponding residues modulo each supplied prime	backend/rns/operations.py
fhelium_rns.negate_standard	Negate residues modulo each supplied prime	backend/rns/operations.py
fhelium_rns.montgomery_multiply	Multiply corresponding Montgomery residues	backend/rns/operations.py
fhelium_rns.add_montgomery_lazy	Accumulate Montgomery residues in the declared lazy range	backend/rns/operations.py
fhelium_rns.extract_component	Select one polynomial component from a Tensor bundle	backend/rns/operations.py
fhelium_rns.pack_two_components	Assemble a two-component Tensor bundle	backend/rns/operations.py
fhelium_rns.pack_three_components	Assemble a three-component Tensor bundle	backend/rns/operations.py

Public and plaintext preparation and mixed arithmetic

Operation	Responsibility	Direct owner
fhelium_ckks.prepare.add.message	Prepare message data for CKKS add arithmetic	compile/passes/ckks/_lower_message_preparation.py
fhelium_ckks.prepare.add.plaintext	Prepare plaintext data for CKKS add arithmetic	compile/passes/ckks/_lower_message_preparation.py
fhelium_ckks.prepare.add.static	Prepare static data for CKKS add arithmetic	compile/passes/ckks/_lower_message_preparation.py
fhelium_ckks.prepare.multiply.message	Prepare message data for CKKS multiply arithmetic	compile/passes/ckks/_lower_message_preparation.py
fhelium_ckks.prepare.multiply.plaintext	Prepare plaintext data for CKKS multiply arithmetic	compile/passes/ckks/_lower_message_preparation.py
fhelium_ckks.prepare.multiply.static	Prepare static data for CKKS multiply arithmetic	compile/passes/ckks/_lower_message_preparation.py
fhelium_ckks.add_plaintext	Lower addition of a supplied plaintext to component zero	compile/passes/lowering/_arithmetic.py
fhelium_ckks.multiply_plaintext	Lower multiplication of ciphertext components by a supplied plaintext	compile/passes/lowering/_arithmetic.py
fhelium_ckks.add_compressed_plaintext	Add compact plaintext values to component zero through encoded-axis indexing	backend/ckks/arithmetic.py, backend/triton/_expressions.py
fhelium_ckks.multiply_compressed_plaintext	Multiply ciphertext components by compact plaintext values through encoded-axis indexing	backend/ckks/arithmetic.py, backend/triton/_expressions.py
fhelium_rns.add_plaintext	Add the supplied plaintext polynomial to component zero	backend/rns/operations.py
fhelium_rns.multiply_plaintext	Multiply each ciphertext component by the supplied plaintext polynomial	backend/rns/operations.py
fhelium_rns.montgomery_weighted_sum	Accumulate one weighted sum of polynomial Tensor products	backend/rns/operations.py
fhelium_rns.montgomery_weighted_sums	Apply several plaintext weight rows to shared polynomial Tensor inputs	backend/rns/operations.py

Polynomial and residue representation transforms

Operation	Responsibility	Direct owner
fhelium_ckks.to_ntt	Lower a forward polynomial-domain transition to NTT operations	compile/passes/lowering/_representation.py
fhelium_ckks.from_ntt	Lower an inverse polynomial-domain transition to NTT operations	compile/passes/lowering/_representation.py
fhelium_ckks.to_montgomery_residues	Lower standard-to-Montgomery residue conversion	compile/passes/lowering/_representation.py
fhelium_ckks.to_standard_residues	Lower Montgomery-to-standard residue conversion	compile/passes/lowering/_representation.py
fhelium_rns.standard_to_montgomery	Convert standard residues to Montgomery form	backend/rns/operations.py
fhelium_rns.montgomery_to_standard	Convert Montgomery residues to standard form	backend/rns/operations.py
fhelium_ntt.coefficient_standard_to_ntt_montgomery	Forward NTT with standard input	backend/ntt/operations.py, backend/ntt/executors/
fhelium_ntt.coefficient_montgomery_to_ntt_montgomery	Forward NTT with Montgomery input	backend/ntt/operations.py, backend/ntt/executors/
fhelium_ntt.ntt_montgomery_to_coefficient_standard	Normalized inverse NTT with standard output	backend/ntt/operations.py, backend/ntt/executors/
fhelium_ntt.ntt_montgomery_to_coefficient_montgomery	Normalized inverse NTT retaining Montgomery output	backend/ntt/operations.py, backend/ntt/executors/

backend/ntt/plans/ constructs twiddle and index tables; backend/ntt/tables.py materializes their device-specific views. Native transitions execute through backend/ntt/operations.py. Generated NTT regions use backend/triton/_ntt.py and _ntt_codegen.py.

Modulus-chain and scale transitions

Operation	Responsibility	Direct owner
fhelium_ckks.rescale	Lower a rounded quotient over the next complete Q depth group	compile/passes/lowering/_representation.py
fhelium_ckks.mod_switch	Lower restriction to the selected active Q rows	compile/passes/lowering/_representation.py
fhelium_ckks.reinterpret_scale	Lower a represented-scale change that preserves residue data	compile/passes/lowering/_representation.py
fhelium_rns.rescale_drop_leading_primes	Divide by a dropped Q group with the selected rounding	backend/rns/rescale.py
fhelium_rns.restrict_depth	Select the active rows at the requested depth	backend/rns/operations.py
fhelium_rns.reinterpret_scale	Update represented scale while preserving residue data	backend/rns/operations.py

Modulus switching to fewer Q rows and ModDown by the auxiliary P product have different equations. They remain separate operations. Numerical inverse tables for rescaling and ModDown are constructed in backend/rns/tables.py.

Key switching, automorphisms, and rotation

Operation	Responsibility	Direct owner
fhelium_ckks.switch_key	Assemble ciphertext components under the supplied destination key relation	backend/ckks/key_switch.py
fhelium_ckks.conjugate	Compose conjugation with the supplied key-switch relation	backend/ckks/key_switch.py
fhelium_ckks.relinearize	Reduce a three-component ciphertext through a supplied relinearization key	backend/ckks/key_switch.py
fhelium_ckks.rotate	Compose a slot-selected automorphism and key switch	backend/ckks/rotation/operations.py
fhelium_ckks.hoisted_rotate_many	Reuse input-derived digit preparation across keys	backend/ckks/rotation/operations.py, _hoisted.py
fhelium_ckks.grouped_rotation_weighted_sum	Compose a selected rotation group with plaintext products	backend/ckks/rotation/operations.py
fhelium_rns.hybrid_modup_digit	Decompose and extend one digit into QP	backend/rns/modup.py
fhelium_rns.key_switch_digit_product	Multiply one prepared digit by its supplied key rows	backend/rns/key_product.py
fhelium_rns.moddown_qp_to_q	Remove the auxiliary P basis in coefficient representation	backend/rns/moddown.py
fhelium_rns.moddown_ntt_qp_to_q	Remove the auxiliary P basis while retaining Q NTT evaluations	backend/rns/moddown.py
fhelium_rns.coefficient_automorphism	Apply a supplied polynomial Galois element	backend/rns/automorphism.py

The CKKS slot-step convention is defined in backend/ckks/rotation/_galois.py. Coefficient and NTT permutation builders accept the polynomial Galois element in backend/rns/automorphism.py and backend/ntt/automorphism.py respectively. Neither requires a key Tensor or a slot-rotation convention.

compile/passes/lowering/_keyswitch.py exposes key switching as RNS/NTT dataflow. The whole-operation and lowered routes use the same numerical owners. backend/rns/_preparation.py describes hybrid arithmetic tables and ModDown requirements; backend/ckks/_key_switch_preparation.py declares whole CKKS implementation requirements. backend/ckks/tables.py assembles the table views used by those whole implementations. Material lookup does not generate keys.

Placement and memory movement

Operation	Responsibility	Direct owner
fhelium_memory.transfer	Move a Tensor to the supplied destination handle	backend/memory/operations.py

backend/memory/resources.py owns the device resource kind. Tensor placement and storage policy are caller choices.

Rank-local SPMD and collectives

Operation	Responsibility	Direct owner
fhelium_dist.rank	Read the process-group-relative rank	backend/distributed/operations.py
fhelium_dist.group_size	Read the participating process-group size	backend/distributed/operations.py
fhelium_dist.broadcast	Broadcast a rank-local Tensor from the selected group root	backend/distributed/operations.py
fhelium_dist.all_reduce	Reduce rank-local Tensors through a supplied combine region	backend/distributed/operations.py
fhelium_dist.all_reduce_add_ciphertext	Execute ciphertext-add reduction or expose a generic combine region	backend/distributed/operations.py, compile/passes/distributed/_lower_specialized_collectives.py
fhelium_dist.yield	Return a combine region's Tensor result	compile/passes/codegen/_host.py

backend/distributed/resources.py owns process-group handles. The distributed schedule must preserve cross-rank operation order and the combine region's algebraic requirements.

Upstream structured execution substrate

Operation	Responsibility	Direct owner
builtin.unrealized_conversion_cast	Forward a one-input, one-result type reference	compile/passes/codegen/_host.py
func.return	Return function results	compile/passes/codegen/_host.py
arith.constant	Materialize a scalar integer or floating-point constant	compile/passes/codegen/_host.py
arith.addi	Add scalar indices	compile/passes/codegen/_host.py
arith.subi	Subtract scalar indices	compile/passes/codegen/_host.py
arith.muli	Multiply scalar indices	compile/passes/codegen/_host.py
arith.divui	Evaluate unsigned scalar quotient control flow	compile/passes/codegen/_host.py
arith.divsi	Evaluate signed scalar quotient control flow	compile/passes/codegen/_host.py
arith.remui	Evaluate unsigned scalar remainder control flow	compile/passes/codegen/_host.py
arith.remsi	Evaluate signed scalar remainder control flow	compile/passes/codegen/_host.py
arith.minui	Select the minimum scalar index	compile/passes/codegen/_host.py
arith.maxui	Select the maximum scalar index	compile/passes/codegen/_host.py
arith.index_cast	Adapt a scalar index	compile/passes/codegen/_host.py
arith.select	Select one scalar value from a condition	compile/passes/codegen/_host.py
arith.cmpi	Compare scalar indices using the represented predicate	compile/passes/codegen/_host.py
scf.if	Execute the selected conditional region	compile/passes/codegen/_host.py
scf.for	Execute a loop with carried values	compile/passes/codegen/_host.py
scf.yield	Return region values to structured control flow	compile/passes/codegen/_host.py

compile/passes/backend/_prepare_host.py binds the emitted host function. ProgramExecutable invokes it with the linked Tensor materials and execution resources.

Continue

• Operation registration and implementation selection
• IR, capture, effects, and open state
• Values, operation semantics, and Eager dispatch
• Distributed internals
• RNS and NTT internals