Field256 arithmetic
Dedicated 256-bit finite-field operations provide the base for the selected curve computations.
Pythagorean Sieve Enterprise RSA-Oriented & ECC-Oriented Engineering Technologies
Pythagorean Sieve Enterprise / ECC
Explicit elliptic-curve arithmetic for ECC-oriented engineering.
A compact 256-bit finite-field implementation for two fixed computationally qualified research domains arising from the canonical specialization d = 1.
Technology profile
Finite-field operations with a controlled qualification boundary.
The ECC line implements field arithmetic, point representation in Jacobian coordinates, elliptic-curve addition and doubling, scalar multiplication and an endomorphism-assisted path. The two paths were checked for functional equivalence on the adopted test set. Qualification applies only to the two fixed research domains considered in the computational work.
Controlled ECC integration
A stable customer-facing boundary around internal curve arithmetic.
The ECC technology is exposed through a stable C API and ABI rather than through direct access to internal Field256 and curve-arithmetic components. Separately packaged ECC v1 and ECC v2 generations support controlled evaluation under a common integration boundary.
The available profiles include Managed Ephemeral and Customer-Provisioned private-key ownership, validated public-key processing, transcript-bound ECDH+HKDF application-key derivation, role-bound key confirmation and explicit clearing. Private keys are not exported through the API, and raw ECDH output is not exposed to the calling application.
Jacobian group operations
Projective representation supports curve addition, doubling and scalar-processing paths.
Domain qualification
Checks cover the field prime, nonsingularity, curve cardinality, subgroup order, cofactor and generator order.
Point validation
Curve membership, subgroup membership and public-point validation form part of the tested integration boundary.
Endomorphism-assisted path
The implementation includes scalar decomposition, the curve endomorphism and simultaneous multiplication.
Embedded-oriented review
Bounded builds examine functional behavior together with Flash, stack, heap and integration constraints.
Operational sequence
Key ownership, validated exchange and derived application keys.
The stable C API keeps the internal Field256 and curve implementation behind an explicit customer-facing boundary.
Select Managed Ephemeral generation or Customer-Provisioned private-key ownership.
Decode and validate curve and subgroup membership before use.
Perform the internal elliptic-curve operation without exporting private material.
Derive application keys while binding the agreed exchange context.
Apply role-bound confirmation to the derived-key workflow.
Clear sensitive state through the defined API lifecycle.
Domain and point qualification
A fixed checklist for each adopted research domain.
Field-prime verificationCurve nonsingularityCurve-cardinality determinationPrime subgroup order and cofactorGenerator-order verificationCurve and subgroup membershipPublic-point validationScalar and GLV path equivalence
Two packaged generations
ECC v1 and ECC v2 under one stable integration boundary.
ECC v1 and ECC v2 are supplied as separately packaged generations under the stable ECC C API v1 and ABI. This permits controlled comparison and migration without exposing internal Field256 or curve-arithmetic components.
The exact kit, platform profile and permitted evaluation scope are identified in the written materials supplied for each engagement.
Stable ECC C API v1
Explicit ownership and fixed-capacity outputs.
The same 17-function API and ABI is retained across the separately packaged ECC generations. Package identity fixes the generation; public-key blobs are not interchangeable across generations.
ECC v1 and ECC v2 retain the same stable 17-function ECC C API v1 boundary. They support Managed Ephemeral keys and validated import of exactly 32 customer-provisioned bytes, export a versioned 66-byte public-key blob, and return only a 32-byte application key and a 32-byte confirmation tag.
Boundary itemPublic contractProtected internal state
Private-key ownershipManaged Ephemeral or exactly 32 customer-provisioned bytesNo private-key export
Public keyVersioned 66-byte blob with strict validationNo runtime domain selector
Derived material32-byte application key and 32-byte confirmation tagNo raw ECDH value
Arithmetic controlsOpaque handles and explicit lifecycleNo Field256 type, curve parameter or GLV control
The API exposes no private-key export, raw ECDH value, runtime domain selector, Field256 type, curve parameter or GLV control. Generation is fixed by package identity, and public-key blobs are not interchangeable across generations.
Hosted performance and operational footprint
Separate evidence for hosted and constrained environments.
The promoted GLV path reduced median scalar-multiplication time by approximately 49–50% relative to the retained recoded scalar path in the recorded hosted configurations, without a material increase in process RSS.
Separate ARM and RISC-V evaluations provide Flash, stack, caller-state, heap-allocation and binary-hygiene evidence. This distinction is important because hosted Linux RSS is not treated as a substitute for embedded RAM or static-footprint measurements.
Potential ECC deployment value
Compact arithmetic for constrained integration studies.
The evaluated ECC profiles combine low hosted RSS with small static Flash and stack estimates and zero measured heap allocation in the recorded embedded-oriented boundary. These characteristics can be economically relevant when multiplied across devices, instances or long-lived deployments.
A compact Flash and memory boundary may widen the set of microcontrollers or embedded platforms suitable for customer-side assessment.
Fixed-capacity execution and the recorded zero-heap profile can simplify memory budgeting in constrained environments.
A stable opaque C API limits the customer-facing surface while keeping Field256, domain and GLV internals outside application code.
Even a small verified saving per unit can become material across a high-volume device fleet, but it must be measured on the intended platform.
ECC hosted benchmark evidence
Hosted scalar performance, resident memory and timing diagnostics.
Hosted GCC/Clang results are kept separate from ARM/RISC-V Flash, stack, workspace and package-footprint evidence.
Hosted measurement and timing-diagnostic interpretation are summarized in the Technical FAQ.
Technical brochure