fhelium.ir.dialects.ntt

Logical number-theoretic-transform operations used below CKKS lowering.

The number-theoretic transform (NTT) evaluates a polynomial in ${\mathbb{Z}}_{q_i}[X]/(X^N+1)$ at roots of unity modulo each active prime $q_i$, turning negacyclic multiplication into pointwise multiplication. Residue number system (RNS) rows hold those per-prime polynomials. Montgomery representation stores a residue $x_i$ as $x_i{R}_{i}mod{q}_i$ for radix $R_i$.

The transform operations distinguish the complete source and destination residue representations needed by ciphertext and plaintext transitions. The parameter operand carries the RNS arithmetic parameters. Additional Tensor operands carry transform tables after their layout is selected. An operation can leave its algorithm and execution implementation unassigned; selection does not change its residue transitions or frequency ordering.

FHEliumNtt

data View source

FHEliumNtt = Dialect('fhelium_ntt', [*_NTT_OPERATION_TYPES, AutomorphismOp], [])

AutomorphismOp

class View source

AutomorphismOp(value, indices, result_type)

Bases: IRDLOperation

Gather NTT evaluations through supplied Galois permutation indices.

For bit-reversed evaluation coordinates, indices[j] selects the evaluation of the input polynomial at the root obtained by raising the destination root to the Galois exponent. The gather preserves components, prime rows, batch axes, scale, and Montgomery residues. The index Tensor is a live operand; the result owns storage independent of the input.

Attributes

Name	Type	Default/value
name		'fhelium_ntt.automorphism'
value		operand_def(RnsBundleType)
indices		operand_def()
result		result_def(RnsBundleType)
traits		traits_def(Pure())

dependencies

method

def dependencies() -> OperationDependencies: ...

CoefficientMontgomeryToNttMontgomeryOp

class View source

CoefficientMontgomeryToNttMontgomeryOp()

Bases: _NttOp

Compute the forward negacyclic NTT of Montgomery coefficients.

Each input coefficient row represents $a_j{R}_{i}mod{q}_i$. The transform is applied independently for every active prime and returns ${\mathrm{NTT}}_{q_i}(a)R_{i}mod{q}_i$. The operation preserves prime rows, depth, component axes, and CKKS scale.

Attributes

Name	Type	Default/value
name		'fhelium_ntt.coefficient_montgomery_to_ntt_montgomery'

dependencies

method

def dependencies() -> OperationDependencies: ...

Describe result reads in this operation's value coordinates.

CoefficientStandardToNttMontgomeryOp

class View source

CoefficientStandardToNttMontgomeryOp()

Bases: _NttOp

Compute the forward negacyclic NTT from standard residues.

For every active prime $q_i$, the input row contains coefficient residues $a_{j}mod{q}_i$ for a polynomial in $R_{q_i}={\mathbb{Z}}_{q_i}[X]/(X^N+1)$. The result contains its number-theoretic transform ${\mathrm{NTT}}_{q_i}(a)$, multiplied by the Montgomery radix $R_i$ modulo $q_i$. Prime rows, CKKS depth, component axes, and scale do not change.

Attributes

Name	Type	Default/value
name		'fhelium_ntt.coefficient_standard_to_ntt_montgomery'

dependencies

method

def dependencies() -> OperationDependencies: ...

Describe result reads in this operation's value coordinates.

NttMontgomeryToCoefficientMontgomeryOp

class View source

NttMontgomeryToCoefficientMontgomeryOp()

Bases: _NttOp

Compute the inverse negacyclic NTT and retain Montgomery residues.

For every active prime $q_i$, the input ${\mathrm{NTT}}_{q_i}(a)R_i$ becomes coefficient data $a_j{R}_{i}mod{q}_i$. The prime-row set, depth, component axes, and CKKS scale remain unchanged.

Attributes

Name	Type	Default/value
name		'fhelium_ntt.ntt_montgomery_to_coefficient_montgomery'

dependencies

method

def dependencies() -> OperationDependencies: ...

Describe result reads in this operation's value coordinates.

NttMontgomeryToCoefficientStandardOp

class View source

NttMontgomeryToCoefficientStandardOp()

Bases: _NttOp

Compute the inverse negacyclic NTT and leave standard residues.

For each active prime $q_i$, an NTT/Montgomery row ${\mathrm{NTT}}_{q_i}(a)R_i$ is mapped to coefficient residues $a_{j}mod{q}_i$. The inverse transform includes the Montgomery and transform normalization factors. Prime rows, depth, components, and scale are preserved.

Attributes

Name	Type	Default/value
name		'fhelium_ntt.ntt_montgomery_to_coefficient_standard'

dependencies

method

def dependencies() -> OperationDependencies: ...

Describe result reads in this operation's value coordinates.

OPERATION_SPECS

constant View source

OPERATION_SPECS: tuple[OperationSpec, ...] = tuple((registered_operation_spec(operation_type, 'ntt', validator=flat_with_attributes('ntt_backend', 'ntt_algorithm', 'ntt_group_width', 'ntt_radix', 'ntt_table_layout', 'ckks_config')) for operation_type in _NTT_OPERATION_TYPES)) + (registered_operation_spec(AutomorphismOp, 'ntt'),)