================
ECC in Libraries
================

.. contents:: Table of Contents
   :backlinks: none
   :depth: 1
   :local:

This page collects information about ECC implementations in open-source software. It was extracted
by source code analysis (i.e. we looked at the code and tried really hard to name what we see), so it
may or may not match the reality (whether due to a mistake or due to different naming choices).

We restricted ourselves to ECDH/ECDSA (on prime field curves) and x25519/Ed25519. In case libraries contain multiple
implementations we tried to document them clearly, along with the rules on how they pick one
(e.g. curve-based/architecture-based). Naming of scalar multipliers is tricky. Naming of coordinate systems
and formulas is taken from the `EFD <https://www.hyperelliptic.org/EFD/index.html>`__.

BouncyCastle
============

| Version: ``1.76`` (tag r1rv76)
| Repository: https://github.com/bcgit/bc-java/
| Docs: https://bouncycastle.org/docs/docs1.8on/index.html

Primitives
----------

Supports short-Weierstrass curves for the usual (ECDSA, ECDH).
Supports X25519, Ed25519.
Also more exotic stuff like ECMQV, GOST key exchange and signatures
and lots of others.

Lots of `scalarmults <https://github.com/bcgit/bc-java/tree/r1rv76/core/src/main/java/org/bouncycastle/math/ec>`__ available:
 - `Comb (w=6 for > 250 bits else w=5) <https://github.com/bcgit/bc-java/blob/r1rv76/core/src/main/java/org/bouncycastle/math/ec/FixedPointCombMultiplier.java>`__
 - `GLV <https://github.com/bcgit/bc-java/blob/r1rv76/core/src/main/java/org/bouncycastle/math/ec/GLVMultiplier.java>`__
 - `Window NAF L2R <https://github.com/bcgit/bc-java/blob/r1rv76/core/src/main/java/org/bouncycastle/math/ec/WNafL2RMultiplier.java>`__
 - `Window "tau" NAF <https://github.com/bcgit/bc-java/blob/r1rv76/core/src/main/java/org/bouncycastle/math/ec/WTauNafMultiplier.java>`__

Several `coordinate systems <https://github.com/bcgit/bc-java/blob/r1rv76/core/src/main/java/org/bouncycastle/math/ec/ECCurve.java#L27>`__ supported:
 - Affine
 - Projective (Homogenous)
 - Jaobian
 - Jacobian-Chudnovsky
 - Jacobian-Modified
 - Lambda-Affine? (binary-field curves only)
 - Lambda-Projective? (binary-field curves only)
 - Skewed? (binary-field curves only)

Some curve-custom code in:
https://github.com/bcgit/bc-java/tree/r1rv76/core/src/main/java/org/bouncycastle/math/ec/custom/sec
Specifically, fast-prime modular reduction for SECG curves, and (weirdly) a short-Weierstrass implementation of Curve25519.

Ed25519 based on `Mike Hamburg's work <https://eprint.iacr.org/2012/309.pdf>`__.

ECDH
^^^^

KeyGen:
 - Short-Weierstrass
 - `Comb <https://github.com/bcgit/bc-java/blob/r1rv76/core/src/main/java/org/bouncycastle/crypto/generators/ECKeyPairGenerator.java#L94>`__ via ``ECKeyPairGenerator.generateKeyPair -> ECKeyPairGenerator.createBasePointMultiplier``.
 - `Jacobian-Modified <https://github.com/bcgit/bc-java/blob/r1rv76/core/src/main/java/org/bouncycastle/math/ec/ECCurve.java#L676>`__ via ``ECCurve.FP_DEFAULT_COORDS``.
   SECP curves use Jacobian, SECT curves use Lambda-Projective.
 - Formulas unknown: `add-bc-r1rv76-jac <https://github.com/J08nY/pyecsca/blob/master/test/data/formulas/add-bc-r1rv76-jac.op3>`__,
   `dbl-bc-r1rv76-jac <https://github.com/J08nY/pyecsca/blob/master/test/data/formulas/dbl-bc-r1rv76-jac.op3>`__,
   `add-bc-r1rv76-mod <https://github.com/J08nY/pyecsca/blob/master/test/data/formulas/add-bc-r1rv76-mod.op3>`__,
   `dbl-bc-r1rv76-mod <https://github.com/J08nY/pyecsca/blob/master/test/data/formulas/dbl-bc-r1rv76-mod.op3>`__

Derive:
 - Short-Weierstrass
 - `GLV if possible, else Window NAF <https://github.com/bcgit/bc-java/blob/r1rv76/core/src/main/java/org/bouncycastle/math/ec/ECCurve.java#L154>`__ via ``ECDHBasicAgreement.calculateAgreement -> ECPoint.multiply -> ECCurve.getMultiplier -> ECCurve.createDefaultMultiplier``.
 - `Jacobian-Modified <https://github.com/bcgit/bc-java/blob/r1rv76/core/src/main/java/org/bouncycastle/math/ec/ECCurve.java#L676>`__ via ``ECCurve.FP_DEFAULT_COORDS``.
   SECP curves use Jacobian, SECT curves use Lambda-Projective.
 - Formulas same as KeyGen.

ECDSA
^^^^^

KeyGen:
 - Short-Weierstrass
 - `Comb <https://github.com/bcgit/bc-java/blob/r1rv76/core/src/main/java/org/bouncycastle/crypto/generators/ECKeyPairGenerator.java#L94>`__ via ``ECKeyPairGenerator.generateKeyPair -> ECKeyPairGenerator.createBasePointMultiplier``.
 - `Jacobian-Modified <https://github.com/bcgit/bc-java/blob/r1rv76/core/src/main/java/org/bouncycastle/math/ec/ECCurve.java#L676>`__ via ``ECCurve.FP_DEFAULT_COORDS``.
   SECP curves use Jacobian, SECT curves use Lambda-Projective.
 - Formulas same as KeyGen.

Sign:
 - Short-Weierstrass
 - `Comb <https://github.com/bcgit/bc-java/blob/r1rv76/core/src/main/java/org/bouncycastle/crypto/signers/ECDSASigner.java#L237>`__ via
   ``ECDSASigner.generateSignature -> ECDSASigner.createBasePointMultiplier``.
 - `Jacobian-Modified <https://github.com/bcgit/bc-java/blob/r1rv76/core/src/main/java/org/bouncycastle/math/ec/ECCurve.java#L676>`__ via ``ECCurve.FP_DEFAULT_COORDS``.
   SECP curves use Jacobian, SECT curves use Lambda-Projective.
 - Formulas same as KeyGen.

Verify:
 - Short-Weierstrass
 - `Multi-scalar GLV if possible, else multi-scalar Window NAF with Shamir's trick <https://github.com/bcgit/bc-java/blob/r1rv76/core/src/main/java/org/bouncycastle/math/ec/ECAlgorithms.java#L72>`__ via ``ECDSASigner.verifySignature -> ECAlgorithms.sumOfTwoMultiples``.
 - `Jacobian-Modified <https://github.com/bcgit/bc-java/blob/r1rv76/core/src/main/java/org/bouncycastle/math/ec/ECCurve.java#L676>`__ via ``ECCurve.FP_DEFAULT_COORDS``.
   SECP curves use Jacobian, SECT curves use Lambda-Projective.
 - Formulas same as KeyGen.

X25519
^^^^^^

KeyGen:
 - Twisted-Edwards
 - `Comb <https://github.com/bcgit/bc-java/blob/r1rv76/core/src/main/java/org/bouncycastle/math/ec/rfc8032/Ed25519.java#L92>`__ via
   ``X25519.generatePublicKey -> X25519.scalarMultBase -> Ed25519.scalarMultBaseYZ -> Ed25519.scalarMultBase``.
 - Many coordinate systems: Extended, half-Niels, affine.
 - Some HWCD formulas are used.

Derive:
 - Montgomery
 - `Ladder <https://github.com/bcgit/bc-java/blob/r1rv76/core/src/main/java/org/bouncycastle/math/ec/rfc7748/X25519.java#L93>`__ via
   ``X25519.calculateAgreement -> X25519.scalarMult``.
 - `xz <https://github.com/bcgit/bc-java/blob/r1rv76/core/src/main/java/org/bouncycastle/math/ec/rfc7748/X25519.java#L68>`__.
 - Unknown formulas: `ladd-bc-r1rv76-x25519 <https://github.com/J08nY/pyecsca/blob/master/test/data/formulas/ladd-bc-r1rv76-x25519.op3>`__,
   `dbl-bc-r1rv76-x25519 <https://github.com/J08nY/pyecsca/blob/master/test/data/formulas/dbl-bc-r1rv76-x25519.op3>`__.
   Code: `dbl <https://github.com/bcgit/bc-java/blob/r1rv76/core/src/main/java/org/bouncycastle/math/ec/rfc7748/X25519.java#L73>`__ and `ladd <https://github.com/bcgit/bc-java/blob/r1rv76/core/src/main/java/org/bouncycastle/math/ec/rfc7748/X25519.java#L111>`__

Ed25519
^^^^^^^

KeyGen:
 - Twisted-Edwards
 - `Comb <https://github.com/bcgit/bc-java/blob/r1rv76/core/src/main/java/org/bouncycastle/math/ec/rfc8032/Ed25519.java#L92>`__  via
   ``Ed25519.generatePublicKey -> Ed25519.scalarMultBaseEncoded -> Ed25519.scalarMultBase``.
 - Many coordinate systems: Extended, half-Niels, affine.
 - Some HWCD formulas are used.

Sign:
 - Twisted-Edwards
 - `Comb <https://github.com/bcgit/bc-java/blob/r1rv76/core/src/main/java/org/bouncycastle/math/ec/rfc8032/Ed25519.java#L92>`__ via
   ``Ed25519.sign -> Ed25519.implSign -> Ed25519.scalarMultBaseEncoded -> Ed25519.scalarMultBase``.
 - Many coordinate systems: Extended, half-Niels, affine.
 - Some HWCD formulas are used.

Verify:
 - Twisted-Edwards
 - `Multi-scalar Window-NAF with Straus's trick <https://github.com/bcgit/bc-java/blob/r1rv76/core/src/main/java/org/bouncycastle/math/ec/rfc8032/Ed25519.java#L1329>`__ via
   ``Ed25519.verify -> Ed25519.implVerify -> Ed25519.scalarMultStraus128Var``.
 - Many coordinate systems: Extended, half-Niels, affine.
 - Some HWCD formulas are used.


BoringSSL
=========

| Version: ``bfa8369`` (commit bfa8369)
| Repository: https://github.com/google/boringssl/
| Docs: https://commondatastorage.googleapis.com/chromium-boringssl-docs/headers.html

Primitives
----------

Supports P-224, P-256, P-384 and P-521.
Also Curve25519.
Uses fiat-crypto for the SECP curve field arithmetic and x25519.

P-224
^^^^^
 - Short-Weierstrass
 - `Comb <https://github.com/google/boringssl/blob/bfa8369795b7533a222a72b7a1bc928941cd66bf/crypto/fipsmodule/ec/p224-64.c#L995>`__ via ``mul_base -> ec_GFp_nistp224_point_mul_base``.
   `Fixed Window (signed, Booth) (width=5) <https://github.com/google/boringssl/blob/bfa8369795b7533a222a72b7a1bc928941cd66bf/crypto/fipsmodule/ec/p224-64.c#L947C13-L947C38>`__ via ``mul -> ec_GFp_nistp224_point_mul``.
 - `Jacobian <https://github.com/google/boringssl/blob/bfa8369795b7533a222a72b7a1bc928941cd66bf/crypto/fipsmodule/ec/p224-64.c#L580>`__,
 - Formulas unknown: `add-boringssl-p224 <https://github.com/J08nY/pyecsca/blob/master/test/data/formulas/add-boringssl-p224.op3>`__,
   `dbl-boringssl-p224 <https://github.com/J08nY/pyecsca/blob/master/test/data/formulas/dbl-boringssl-p224.op3>`__.

P-256
^^^^^
 - Short-Weierstrass
 - `Comb <https://github.com/google/boringssl/blob/bfa8369795b7533a222a72b7a1bc928941cd66bf/crypto/fipsmodule/ec/p256.c#L543>`__ via ``mul_base -> ec_GFp_nistp256_point_mul_base``.
   `Fixed Window (signed, Booth) (width=5) <https://github.com/google/boringssl/blob/bfa8369795b7533a222a72b7a1bc928941cd66bf/crypto/fipsmodule/ec/p256.c#L476>`__ via ``mul -> ec_GFp_nistp256_point_mul``.
 - `Jacobian-3 <https://github.com/google/boringssl/blob/bfa8369795b7533a222a72b7a1bc928941cd66bf/crypto/fipsmodule/ec/p256.c#L238>`__,
 - `add-2007-bl <https://github.com/google/boringssl/blob/bfa8369795b7533a222a72b7a1bc928941cd66bf/crypto/fipsmodule/ec/p256.c#L238>`__,
   `dbl-2001-b <https://github.com/google/boringssl/blob/bfa8369795b7533a222a72b7a1bc928941cd66bf/crypto/fipsmodule/ec/p256.c#L184>`__

P-384
^^^^^
 - Uses defaults (described below).

P-521
^^^^^
 - Uses defaults (described below).

ECDH
^^^^

KeyGen:
 - Short-Weierstrass
 - ``EC_KEY_generate_key -> ec_point_mul_scalar_base -> meth.mul_base``.
   Default: `Fixed Window <https://github.com/google/boringssl/blob/bfa8369795b7533a222a72b7a1bc928941cd66bf/crypto/fipsmodule/ec/simple_mul.c#L24>`__, via ``ec_GFp_mont_mul_base -> ec_GFp_mont_mul``.
 - `Jacobian <https://github.com/google/boringssl/blob/bfa8369795b7533a222a72b7a1bc928941cd66bf/crypto/fipsmodule/ec/ec_montgomery.c#L218>`__
 - `add-2007-bl <https://github.com/google/boringssl/blob/bfa8369795b7533a222a72b7a1bc928941cd66bf/crypto/fipsmodule/ec/ec_montgomery.c#L218>`__, `dbl-2001-b <https://github.com/google/boringssl/blob/bfa8369795b7533a222a72b7a1bc928941cd66bf/crypto/fipsmodule/ec/ec_montgomery.c#L329>`__

Derive:
 - Short-Weierstrass
 - ``ECDH_compute_key -> ec_point_mul_scalar -> meth.mul``.
   Default: `Fixed Window <https://github.com/google/boringssl/blob/bfa8369795b7533a222a72b7a1bc928941cd66bf/crypto/fipsmodule/ec/simple_mul.c#L24>`__, via ``ec_GFp_mont_mul``.
 - `Jacobian <https://github.com/google/boringssl/blob/bfa8369795b7533a222a72b7a1bc928941cd66bf/crypto/fipsmodule/ec/ec_montgomery.c#L218>`__
 - `add-2007-bl <https://github.com/google/boringssl/blob/bfa8369795b7533a222a72b7a1bc928941cd66bf/crypto/fipsmodule/ec/ec_montgomery.c#L218>`__, `dbl-2001-b <https://github.com/google/boringssl/blob/bfa8369795b7533a222a72b7a1bc928941cd66bf/crypto/fipsmodule/ec/ec_montgomery.c#L329>`__

ECDSA
^^^^^

KeyGen:
 - Short-Weierstrass
 - ``EC_KEY_generate_key -> ec_point_mul_scalar_base -> meth.mul_base``.
   Default: `Fixed Window <https://github.com/google/boringssl/blob/bfa8369795b7533a222a72b7a1bc928941cd66bf/crypto/fipsmodule/ec/simple_mul.c#L24>`__, via ``ec_GFp_mont_mul``.
 - `Jacobian <https://github.com/google/boringssl/blob/bfa8369795b7533a222a72b7a1bc928941cd66bf/crypto/fipsmodule/ec/ec_montgomery.c#L218>`__
 - `add-2007-bl <https://github.com/google/boringssl/blob/bfa8369795b7533a222a72b7a1bc928941cd66bf/crypto/fipsmodule/ec/ec_montgomery.c#L218>`__, `dbl-2001-b <https://github.com/google/boringssl/blob/bfa8369795b7533a222a72b7a1bc928941cd66bf/crypto/fipsmodule/ec/ec_montgomery.c#L329>`__

Sign:
 - Short-Weierstrass
 - ``ECDSA_sign -> ECDSA_do_sign -> ecdsa_sign_impl -> ec_point_mul_scalar_base -> meth.mul_base``.
   Default: `Fixed Window <https://github.com/google/boringssl/blob/bfa8369795b7533a222a72b7a1bc928941cd66bf/crypto/fipsmodule/ec/simple_mul.c#L24>`__, via ``ec_GFp_mont_mul``.
 - `Jacobian <https://github.com/google/boringssl/blob/bfa8369795b7533a222a72b7a1bc928941cd66bf/crypto/fipsmodule/ec/ec_montgomery.c#L218>`__
 - `add-2007-bl <https://github.com/google/boringssl/blob/bfa8369795b7533a222a72b7a1bc928941cd66bf/crypto/fipsmodule/ec/ec_montgomery.c#L218>`__, `dbl-2001-b <https://github.com/google/boringssl/blob/bfa8369795b7533a222a72b7a1bc928941cd66bf/crypto/fipsmodule/ec/ec_montgomery.c#L329>`__

Verify:
 - Short-Weierstrass
 - ``ECDSA_verify -> ECDSA_do_verify -> ecdsa_do_verify_no_self_test -> ec_point_mul_scalar_public -> meth.mul_public or meth.mul_public_batch``.
   Default: `Window NAF (w=4) based interleaving multi-exponentiation method <https://github.com/google/boringssl/blob/bfa8369/crypto/fipsmodule/ec/wnaf.c#L83>`__, via ``ec_GFp_mont_mul_public_batch``.
 - `Jacobian <https://github.com/google/boringssl/blob/bfa8369795b7533a222a72b7a1bc928941cd66bf/crypto/fipsmodule/ec/ec_montgomery.c#L218>`__
 - `add-2007-bl <https://github.com/google/boringssl/blob/bfa8369795b7533a222a72b7a1bc928941cd66bf/crypto/fipsmodule/ec/ec_montgomery.c#L218>`__, `dbl-2001-b <https://github.com/google/boringssl/blob/bfa8369795b7533a222a72b7a1bc928941cd66bf/crypto/fipsmodule/ec/ec_montgomery.c#L329>`__

X25519
^^^^^^

KeyGen:
 - Twisted-Edwards
 - ?? via ``X25519_keypair -> X25519_public_from_private -> x25519_ge_scalarmult_base``.
 - Has `multiple coordinate systems <https://github.com/google/boringssl/blob/bfa8369795b7533a222a72b7a1bc928941cd66bf/crypto/curve25519/internal.h#L79>`__: projective, extended, completed, Duif
 - Unknown formulas. `dbl <https://github.com/google/boringssl/blob/bfa8369795b7533a222a72b7a1bc928941cd66bf/crypto/curve25519/curve25519.c#L617>`__, `add <https://github.com/google/boringssl/blob/bfa8369795b7533a222a72b7a1bc928941cd66bf/crypto/curve25519/curve25519.c#L624>`__

Derive:
 - Montgomery
 - Ladder via ``X25519 -> x25519_scalar_mult -> x25519_NEON/x25519_scalar_mult_adx/x25519_scalar_mult_generic``
 - xz.
 - Unknown formula: `ladd-boringssl-x25519 <https://github.com/J08nY/pyecsca/blob/master/test/data/formulas/ladd-boringssl-x25519.op3>`__ from fiat-crypto.

Ed25519
^^^^^^^
Based on ref10 of Ed25519.

KeyGen:
 - Twisted-Edwards
 - ?? via ``ED25519_keypair -> ED25519_keypair_from_seed -> x25519_ge_scalarmult_base``.
 - Has `multiple coordinate systems <https://github.com/google/boringssl/blob/bfa8369795b7533a222a72b7a1bc928941cd66bf/crypto/curve25519/internal.h#L79>`__: projective, extended, completed, Duif
 - Unknown formulas. `dbl <https://github.com/google/boringssl/blob/bfa8369795b7533a222a72b7a1bc928941cd66bf/crypto/curve25519/curve25519.c#L617>`__, `add <https://github.com/google/boringssl/blob/bfa8369795b7533a222a72b7a1bc928941cd66bf/crypto/curve25519/curve25519.c#L624>`__

Sign:
 - Twisted-Edwards
 - ?? via ``ED25519_sign -> ED25519_keypair_from_seed -> x25519_ge_scalarmult_base``.
 - Has `multiple coordinate systems <https://github.com/google/boringssl/blob/bfa8369795b7533a222a72b7a1bc928941cd66bf/crypto/curve25519/internal.h#L79>`__: projective, extended, completed, Duif
 - Unknown formulas. `dbl <https://github.com/google/boringssl/blob/bfa8369795b7533a222a72b7a1bc928941cd66bf/crypto/curve25519/curve25519.c#L617>`__, `add <https://github.com/google/boringssl/blob/bfa8369795b7533a222a72b7a1bc928941cd66bf/crypto/curve25519/curve25519.c#L624>`__

Verify:
 - Twisted-Edwards
 - Sliding window (signed) with interleaving? via ``ED25519_verify -> ge_double_scalarmult_vartime``.
 - Has `multiple coordinate systems <https://github.com/google/boringssl/blob/bfa8369795b7533a222a72b7a1bc928941cd66bf/crypto/curve25519/internal.h#L79>`__: projective, extended, completed, Duif
 - Unknown formulas. `dbl <https://github.com/google/boringssl/blob/bfa8369795b7533a222a72b7a1bc928941cd66bf/crypto/curve25519/curve25519.c#L617>`__, `add <https://github.com/google/boringssl/blob/bfa8369795b7533a222a72b7a1bc928941cd66bf/crypto/curve25519/curve25519.c#L624>`__


Botan
=====

| Version: ``3.2.0`` (tag 3.2.0)
| Repository: https://github.com/randombit/botan/
| Docs: https://botan.randombit.net/handbook/

Primitives
----------

Has coordinate and scalar blinding,

ECDH
^^^^

KeyGen:
 - Short-Weierstrass
 - `Fixed Window with FullPrecomputation (no doublings) (w=3) <https://github.com/randombit/botan/blob/3.2.0/src/lib/pubkey/ec_group/point_mul.cpp#L78>`__, via ``blinded_base_point_multiply -> EC_Point_Base_Point_Precompute::mul``.
 - `Jacobian <https://github.com/randombit/botan/blob/3.2.0/src/lib/pubkey/ec_group/ec_point.cpp#L181>`__
 - `add-1998-cmo-2 <https://github.com/randombit/botan/blob/3.2.0/src/lib/pubkey/ec_group/ec_point.cpp#L181>`__

Derive:
 - Short-Weierstrass
 - Fixed Window (w=4) via ``blinded_var_point_multiply -> EC_Point_Var_Point_Precompute::mul``.
 - `Jacobian <https://github.com/randombit/botan/blob/3.2.0/src/lib/pubkey/ec_group/ec_point.cpp#L181>`__
 - `add-1998-cmo-2 <https://github.com/randombit/botan/blob/3.2.0/src/lib/pubkey/ec_group/ec_point.cpp#L181>`__,
   `dbl-1986-cc <https://github.com/randombit/botan/blob/3.2.0/src/lib/pubkey/ec_group/ec_point.cpp#L278>`__

ECDSA
^^^^^

KeyGen:
 - Short-Weierstrass
 - `Fixed Window with FullPrecomputation (no doublings) (w=3) <https://github.com/randombit/botan/blob/3.2.0/src/lib/pubkey/ec_group/point_mul.cpp#L78>`__, via ``blinded_base_point_multiply -> EC_Point_Base_Point_Precompute::mul``.
 - `Jacobian <https://github.com/randombit/botan/blob/3.2.0/src/lib/pubkey/ec_group/ec_point.cpp#L181>`__
 - `add-1998-cmo-2 <https://github.com/randombit/botan/blob/3.2.0/src/lib/pubkey/ec_group/ec_point.cpp#L181>`__

Sign:
 - Short-Weierstrass
 - `Fixed Window with FullPrecomputation (no doublings) (w=3) <https://github.com/randombit/botan/blob/3.2.0/src/lib/pubkey/ec_group/point_mul.cpp#L78>`__, via ``blinded_base_point_multiply -> EC_Point_Base_Point_Precompute::mul``.
 - `Jacobian <https://github.com/randombit/botan/blob/3.2.0/src/lib/pubkey/ec_group/ec_point.cpp#L181>`__
 - `add-1998-cmo-2 <https://github.com/randombit/botan/blob/3.2.0/src/lib/pubkey/ec_group/ec_point.cpp#L181>`__

Verify:
 - Short-Weierstrass
 - Multi-scalar (interleaved) fixed-window via ``ECDSA::verify -> EC_Point_Multi_Point_Precompute::multi_exp``.
 - `Jacobian <https://github.com/randombit/botan/blob/3.2.0/src/lib/pubkey/ec_group/ec_point.cpp#L181>`__
 - `add-1998-cmo-2 <https://github.com/randombit/botan/blob/3.2.0/src/lib/pubkey/ec_group/ec_point.cpp#L181>`__,
   `dbl-1986-cc <https://github.com/randombit/botan/blob/3.2.0/src/lib/pubkey/ec_group/ec_point.cpp#L278>`__

X25519
^^^^^^
Based on curve25519_donna.

 - Montgomery
 - Montgomery ladder (unrolled several iterations)
 - xz
 - Unknown formula: `ladd-botan-x25519 <https://github.com/J08nY/pyecsca/blob/master/test/data/formulas/ladd-botan-x25519.op3>`__

Ed25519
^^^^^^^
Based on ref10 of Ed25519.
See `BoringSSL`_.


SymCrypt
========

| Version: ``103.1.0`` (tag v103.1.0)
| Repository: https://github.com/microsoft/SymCrypt
| Docs:

Primitives
----------

Supports ECDH and ECDSA with `NIST <https://github.com/microsoft/SymCrypt/blob/v103.1.0/lib/ec_internal_curves.c#L16C19-L16C25>`__ curves (192, 224, 256, 384, 521) and Twisted Edwards `NUMS <https://github.com/microsoft/SymCrypt/blob/v103.1.0/lib/ec_internal_curves.c#L303>`__ curves (NumsP256t1, NumsP384t1, NumsP512t1).
Also custom curves.
Supports X25519.


ECDH
^^^^

KeyGen:
 - `(signed) Fixed-window <https://github.com/microsoft/SymCrypt/blob/v103.1.0/lib/ec_mul.c#L90>`__ via ``SymCryptEcpointGenericSetRandom -> SymCryptEcpointScalarMul -> SymCryptEcpointScalarMulFixedWindow``. Algorithm 1 in `Selecting Elliptic Curves for Cryptography: An Efficiency and Security Analysis <https://eprint.iacr.org/2014/130.pdf>`__.
 - NIST (Short-Weierstrass) use `Jacobian <https://github.com/microsoft/SymCrypt/blob/v103.1.0/lib/ecurve.c#L101>`__.
    - `dbl-2007-bl <https://github.com/microsoft/SymCrypt/blob/v103.1.0/lib/ec_short_weierstrass.c#L381>`__ for generic double via ``SymCryptEcpointDouble`` or a `tweak of  dbl-2007-bl/dbl-2001-b <https://github.com/microsoft/SymCrypt/blob/v103.1.0/lib/ec_short_weierstrass.c#L499>`__ formulae via ``SymCryptShortWeierstrassDoubleSpecializedAm3`` for ``a=-3``.
    - `add-2007-bl <https://github.com/microsoft/SymCrypt/blob/v103.1.0/lib/ec_short_weierstrass.c#L490>`__ via ``SymCryptEcpointAddDiffNonZero``. It also has side-channel unsafe version ``SymCryptShortWeierstrassAddSideChannelUnsafe`` and a generic wrapper for both via ``SymCryptEcpointAdd``.
 - NUMS (Twisted-Edwards) curves use `Extended projective <https://github.com/microsoft/SymCrypt/blob/v103.1.0/lib/ecurve.c#L104>`__.
    - `dbl-2008-hwcd <https://github.com/microsoft/SymCrypt/blob/v103.1.0/lib/ec_twisted_edwards.c#L195>`__ via ``SymCryptTwistedEdwardsDouble``.
    - `add-2008-hwcd <https://github.com/microsoft/SymCrypt/blob/v103.1.0/lib/ec_twisted_edwards.c#L313>`__ via ``SymCryptTwistedEdwardsAdd`` or ``SymCryptTwistedEdwardsAddDiffNonZero``.

Derive:
 - `(signed) Fixed-window <https://github.com/microsoft/SymCrypt/blob/v103.1.0/lib/ec_mul.c#L90>`__ via ``SymCryptEcDhSecretAgreement -> SymCryptEcpointScalarMul -> SymCryptEcpointScalarMulFixedWindow``. Algorithm 1 in `Selecting Elliptic Curves for Cryptography: An Efficiency and Security Analysis <https://eprint.iacr.org/2014/130.pdf>`__.
 - Same coordinates and formulas as KeyGen.


ECDSA
^^^^^

KeyGen:
 - Short-Weierstrass
 - `(signed) Fixed-window <https://github.com/microsoft/SymCrypt/blob/v103.1.0/lib/ec_mul.c#L90>`__ via ``SymCryptEcpointGenericSetRandom -> SymCryptEcpointScalarMul -> SymCryptEcpointScalarMulFixedWindow``. Algorithm 1 in `Selecting Elliptic Curves for Cryptography: An Efficiency and Security Analysis <https://eprint.iacr.org/2014/130.pdf>`__.
 - NIST (Short-Weierstrass) use `Jacobian <https://github.com/microsoft/SymCrypt/blob/v103.1.0/lib/ecurve.c#L101>`__.
    - `dbl-2007-bl <https://github.com/microsoft/SymCrypt/blob/v103.1.0/lib/ec_short_weierstrass.c#L381>`__ for generic double via ``SymCryptEcpointDouble`` or a `tweak of  dbl-2007-bl/dbl-2001-b <https://github.com/microsoft/SymCrypt/blob/v103.1.0/lib/ec_short_weierstrass.c#L499>`__ formulae via ``SymCryptShortWeierstrassDoubleSpecializedAm3`` for ``a=-3``.
    - `add-2007-bl <https://github.com/microsoft/SymCrypt/blob/v103.1.0/lib/ec_short_weierstrass.c#L490>`__ via ``SymCryptEcpointAddDiffNonZero``. It also has side-channel unsafe version ``SymCryptShortWeierstrassAddSideChannelUnsafe`` and a generic wrapper for both via ``SymCryptEcpointAdd``.
 - NUMS (Twisted-Edwards) curves use `Extended projective <https://github.com/microsoft/SymCrypt/blob/v103.1.0/lib/ecurve.c#L104>`__.
    - `dbl-2008-hwcd <https://github.com/microsoft/SymCrypt/blob/v103.1.0/lib/ec_twisted_edwards.c#L195>`__ via ``SymCryptTwistedEdwardsDouble``.
    - `add-2008-hwcd <https://github.com/microsoft/SymCrypt/blob/v103.1.0/lib/ec_twisted_edwards.c#L313>`__ via ``SymCryptTwistedEdwardsAdd`` or ``SymCryptTwistedEdwardsAddDiffNonZero``.


Sign:
 - Short-Weierstrass
 - `(signed) Fixed-window <https://github.com/microsoft/SymCrypt/blob/v103.1.0/lib/ec_mul.c#L90>`__ via ``SymCryptEcDsaSignEx -> SymCryptEcpointScalarMul -> SymCryptEcpointScalarMulFixedWindow``. Algorithm 1 in `Selecting Elliptic Curves for Cryptography: An Efficiency and Security Analysis <https://eprint.iacr.org/2014/130.pdf>`__.
 - Same coordinates and formulas as KeyGen.

Verify:
 - Short-Weierstrass
 - `Double-scalar multiplication using the width-w NAF with interleaving <https://github.com/microsoft/SymCrypt/blob/v103.1.0/lib/ec_mul.c#L90>`__ via ``SymCryptEcDsaVerify > SymCryptEcpointMultiScalarMul -> SymCryptEcpointMultiScalarMulWnafWithInterleaving``. Algorithm 9 in `Selecting Elliptic Curves for Cryptography: An Efficiency and Security Analysis <https://eprint.iacr.org/2014/130.pdf>`__.
 - Same coordinates and formulas as KeyGen.

X25519
^^^^^^

KeyGen:
 - Montgomery
 - `Ladder <https://github.com/microsoft/SymCrypt/blob/v103.1.0/lib/ec_montgomery.c#L297>`__ via
   ``SymCryptMontgomeryPointScalarMul``.
 - `xz <https://github.com/microsoft/SymCrypt/blob/v103.1.0/lib/ec_montgomery.c#L173>`__.
 - `ladd-1987-m-3 <https://github.com/microsoft/SymCrypt/blob/v103.1.0/lib/ec_montgomery.c#L151>`__  via ``SymCryptMontgomeryDoubleAndAdd``.


Derive:
 - Same as Keygen.


fastecdsa
=========

| Version: ``v2.3.1``
| Repository: https://github.com/AntonKueltz/fastecdsa/
| Docs: https://fastecdsa.readthedocs.io/en/latest/index.html

Primitives
----------

Offers only ECDSA.
Supported `curves <https://github.com/AntonKueltz/fastecdsa/blob/v2.3.1/fastecdsa/curve.py>`__: all SECP curves (8) for 192-256 bits, all (7) Brainpool curves as well as custom curves.


ECDSA
^^^^^

KeyGen:
 - Short-Weierstrass
 - `Ladder <https://github.com/AntonKueltz/fastecdsa/blob/v2.3.1/src/curveMath.c#L124>`__ via ``get_public_key -> pointZZ_pMul``.
 -  Affine and schoolbook `add <https://github.com/AntonKueltz/fastecdsa/blob/v2.3.1/src/curveMath.c#L68>`__ and `double <https://github.com/AntonKueltz/fastecdsa/blob/v2.3.1/src/curveMath.c#L2>`__.

Sign:
 - Short-Weierstrass
 - Same ladder as Keygen via ``sign``.

Verify:
 - Short-Weierstrass
 - `Shamir's trick <https://github.com/AntonKueltz/fastecdsa/blob/v2.3.1/src/curveMath.c#L163>`__ via ``verify -> pointZZ_pShamirsTrick``.


micro-ecc
=========

| Version: ``v1.1``
| Repository: https://github.com/kmackay/micro-ecc/
| Docs:

Primitives
----------

Offers ECDH and ECDSA on secp160r1, secp192r1, secp224r1, secp256r1, and secp256k1.

ECDH
^^^^

KeyGen:
 - Short-Weierstrass
 - `Ladder (coZ, with subtraction) <https://github.com/kmackay/micro-ecc/blob/v1.1/uECC.c#L862>`__ via ``uECC_make_key -> EccPoint_compute_public_key -> EccPoint_mult`` (also has coordinate randomization).
 - `Jacobian coZ coordinates (Z1 == Z2) <https://github.com/kmackay/micro-ecc/blob/v1.1/uECC.c#L748>`__ from https://eprint.iacr.org/2011/338.pdf.
 - `coZ formulas <https://github.com/kmackay/micro-ecc/blob/v1.1/uECC.c#L793>`__ from https://eprint.iacr.org/2011/338.pdf.

Derive:
 - Short-Weierstrass
 - `Ladder (coZ, with subtraction) <https://github.com/kmackay/micro-ecc/blob/v1.1/uECC.c#L862>`__ via ``uECC_shared_secret -> EccPoint_compute_public_key -> EccPoint_mult`` (also has coordinate randomization).
 - Same coords and formulas as KeyGen.

ECDSA
^^^^^

Keygen:
 - Same as ECDH.

Sign:
 - Short-Weierstrass
 - `Ladder (coZ, with subtraction) <https://github.com/kmackay/micro-ecc/blob/v1.1/uECC.c#L862>`__ via ``uECC_sign -> uECC_sign_with_k_internal -> EccPoint_mult`` (also has coordinate randomization).
 - Same coords and formulas as KeyGen.

Verify:
 - Short-Weierstrass
 - `Shamir's trick <https://github.com/kmackay/micro-ecc/blob/v1.1/uECC.c#L1558>`__ via ``uECC_verify``.
 - Same coords and formulas as KeyGen.


Intel IPP cryptography
======================

| Version: ``2021.9.0``
| Repository: https://github.com/intel/ipp-crypto/
| Docs: https://www.intel.com/content/www/us/en/docs/ipp-crypto/developer-reference/2021-8/overview.html

Primitives
----------

Supports "ECC (NIST curves), ECDSA, ECDH, EC-SM2".
Also ECNR.

ECDH
^^^^

KeyGen:
 - Short-Weierstrass
 - `(signed, Booth) Fixed Window with full precomputation? (width = 5) <https://github.com/intel/ipp-crypto/blob/ippcp_2021.9.0/sources/ippcp/pcpgfpec_mulbase.c#L34>`__ via ``ippsGFpECPublicKey -> gfec_MulBasePoint -> gfec_base_point_mul or gfec_point_mul``.
    - Has special functions for NIST curves, but those implement the same scalarmult.
 - `Jacobian coords <https://github.com/intel/ipp-crypto/blob/ippcp_2021.9.0/sources/ippcp/pcpgfpecstuff.h#L76>`__
 - `add-1998-cmo-2 <https://github.com/intel/ipp-crypto/blob/ippcp_2021.9.0/sources/ippcp/pcpgfpec_add.c#L35>`__
   `dbl-1998-cmo-2 <https://github.com/intel/ipp-crypto/blob/ippcp_2021.9.0/sources/ippcp/pcpgfpec_dblpoint.c#L36>`__
 - Weirdly mentions "Enhanced Montgomery Multiplication" DOI:10.1155/2008/583926 in each of the formulas.
   Does actually use Montgomery arithmetic.

Derive:
 - Short-Weierstrass
 - `(signed, Booth) Fixed Window (width = 5) <https://github.com/intel/ipp-crypto/blob/ippcp_2021.9.0/sources/ippcp/pcpgfpec_mul.c#L36>`__ via ``ippsGFpECSharedSecretDH -> gfec_MulPoint -> gfec_point_mul``.
 - Has special functions for NIST curves, but those implement the same scalarmult.
 - Same coordinates and formulas as KeyGen.

ECDSA
^^^^^

KeyGen:
 - Same as ECDH.

Sign:
 - Short-Weierstrass
 - `(signed, Booth) Fixed Window with full precomputation? (width = 5) <https://github.com/intel/ipp-crypto/blob/ippcp_2021.9.0/sources/ippcp/pcpgfpec_mulbase.c#L34>`__ via ``ippsGFpECSignDSA -> gfec_MulBasePoint -> gfec_base_point_mul or gfec_point_mul``.
 - Same coordinates and formulas as KeyGen (and ECDH).

Verify:
 - Short-Weierstrass
 - `(signed, Booth) Fixed window (width = 5) interleaved multi-scalar <https://github.com/intel/ipp-crypto/blob/ippcp_2021.9.0/sources/ippcp/pcpgfpec_prod.c#L36>`__ via ``ippsGFpECVerifyDSA -> gfec_BasePointProduct -> either (gfec_base_point_mul + gfec_point_mul + gfec_point_add) or (gfec_point_prod)``.
 - Same coordinates and formulas as KeyGen (and ECDH).


x25519
^^^^^^

KeyGen:
 - Montgomery
 - `Some Full precomputation <https://github.com/intel/ipp-crypto/blob/ippcp_2021.9.0/sources/ippcp/crypto_mb/src/x25519/ifma_x25519.c#L1596>`__ via ``mbx_x25519_public_key``
 - xz
 - Unknown formulas: `add-ipp-x25519 <https://github.com/J08nY/pyecsca/blob/master/test/data/formulas/add-ipp-x25519.op3>`__, `dbl-ipp-x25519 <https://github.com/J08nY/pyecsca/blob/master/test/data/formulas/dbl-ipp-x25519.op3>`__

Derive:
 - Montgomery
 - `? <https://github.com/intel/ipp-crypto/blob/ippcp_2021.9.0/sources/ippcp/crypto_mb/src/x25519/ifma_x25519.c#L1140>`__ via ``mbx_x25519 -> x25519_scalar_mul_dual``
 - xz
 - Unknown formulas.

Ed25519
^^^^^^^

KeyGen:
 - Twisted-Edwards
 - `Fixed window with full precomputation? (width = 4) <https://github.com/intel/ipp-crypto/blob/ippcp_2021.9.0/sources/ippcp/crypto_mb/src/ed25519/ifma_arith_ed25519.c#L287>`__ via ``mbx_ed25519_public_key -> ifma_ed25519_mul_basepoint``
 - Mixes coordinate models::

    homogeneous: (X:Y:Z) satisfying x=X/Z, y=Y/Z
    extended homogeneous: (X:Y:Z:T) satisfying x=X/Z, y=Y/Z, XY=ZT
    completed: (X:Y:Z:T) satisfying x=X/Z, y=Y/T
    scalar precomputed group element: (y-x:y+x:2*t*d), t=x*y
    mb precomputed group element: (y-x:y+x:2*t*d), t=x*y
    projective flavor of the mb precomputed: (Y-X:Y+X:2*T*d:Z), T=X*Y

Add::

    fe52_add(r->X, p->Y, p->X);      // X3 = Y1+X1
    fe52_sub(r->Y, p->Y, p->X);      // Y3 = Y1-X1
    fe52_mul(r->Z, r->X, q->yaddx);  // Z3 = X3*yplusx2
    fe52_mul(r->Y, r->Y, q->ysubx);  // Y3 = Y3*yminisx2
    fe52_mul(r->T, q->t2d, p->T);    // T3 = T1*xy2d2
    fe52_add(t0, p->Z, p->Z);        // t0 = Z1+Z1
    fe52_sub(r->X, r->Z, r->Y);      // X3 = Z3-Y3 = X3*yplusx2 - Y3*yminisx2 = (Y1+X1)*yplusx2 - (Y1-X1)*yminisx2
    fe52_add(r->Y, r->Z, r->Y);      // Y3 = Z3+Y3 = X3*yplusx2 + Y3*yminisx2 = (Y1+X1)*yplusx2 + (Y1-X1)*yminisx2
    fe52_add(r->Z, t0, r->T);        // Z3 = 2*Z1 + T1*xy2d2
    fe52_sub(r->T, t0, r->T);        // T3 = 2*Z1 - T1*xy2d2

Dbl::

    fe52_sqr(r->X, p->X);
    fe52_sqr(r->Z, p->Y);
    fe52_sqr(r->T, p->Z);
    fe52_add(r->T, r->T, r->T);
    fe52_add(r->Y, p->X, p->Y);
    fe52_sqr(t0, r->Y);
    fe52_add(r->Y, r->Z, r->X);
    fe52_sub(r->Z, r->Z, r->X);
    fe52_sub(r->X, t0, r->Y);
    fe52_sub(r->T, r->T, r->Z);

Sign:
 - Twisted-Edwards
 - `Fixed window with full precomputation? (width = 4) <https://github.com/intel/ipp-crypto/blob/ippcp_2021.9.0/sources/ippcp/crypto_mb/src/ed25519/ifma_arith_ed25519.c#L287>`__ via ``mbx_ed25519_sign -> ifma_ed25519_mul_basepoint``
 - Same as KeyGen.

Verify:
 - Twisted-Edwards
 - `Fixed window with full precomputation? (width = 4) <https://github.com/intel/ipp-crypto/blob/ippcp_2021.9.0/sources/ippcp/crypto_mb/src/ed25519/ifma_arith_ed25519.c#L287>`__ for base point mult, then just Fixed window (width = 4) for the other mult, all via ``mbx_ed25519_verify -> ifma_ed25519_prod_point -> ifma_ed25519_mul_point + ifma_ed25519_mul_basepoint``
 - Same as KeyGen.

LibreSSL
========

| Version: ``v3.8.2``
| Repository: https://github.com/libressl/portable
| Docs:

Primitives
----------

Supports ECDH, ECDSA as well as x25519 and Ed25519.

ECDH
^^^^

KeyGen:
 - Short-Weierstrass
 - `Simple Ladder <https://github.com/libressl/openbsd/blob/libressl-v3.8.2/src/lib/libcrypto/ec/ecp_smpl.c#L1305>`__ via ``kmethod.keygen -> ec_key_gen -> EC_POINT_mul -> method.mul_generator_ct -> ec_GFp_simple_mul_generator_ct -> ec_GFp_simple_mul_ct``.
   Also does coordinate blinding and fixes scalar bit-length.
 - Jacobian coordinates.
 - Unknown formulas: `add-libressl-v382 <https://github.com/J08nY/pyecsca/blob/master/test/data/formulas/add-libressl-v382.op3>`__,
   `dbl-libressl-v382 <https://github.com/J08nY/pyecsca/blob/master/test/data/formulas/dbl-libressl-v382.op3>`__

Derive:
 - Short-Weierstrass
 - `Simple Ladder <https://github.com/libressl/openbsd/blob/libressl-v3.8.2/src/lib/libcrypto/ec/ecp_smpl.c#L1305>`__ via ``kmethod.compute_key -> ecdh_compute_key -> EC_POINT_mul -> method.mul_single_ct -> ec_GFp_simple_mul_single_ct -> ec_GFp_simple_mul_ct``.
   Also does coordinate blinding and fixes scalar bit-length.
 - Same as KeyGen.


ECDSA
^^^^^

KeyGen:
 - Same as ECDH.

Sign:
 - Short-Weierstrass
 - `Simple Ladder <https://github.com/libressl/openbsd/blob/libressl-v3.8.2/src/lib/libcrypto/ec/ecp_smpl.c#L1305>`__ via ``ECDSA_sign -> kmethod.sign -> ecdsa_sign -> ECDSA_do_sign -> kmethod.sign_sig -> ecdsa_sign_sig -> ECDSA_sign_setup -> kmethod.sign_setup -> ecdsa_sign_setup -> EC_POINT_mul -> method.mul_generator_ct -> ec_GFp_simple_mul_generator_ct -> ec_GFp_simple_mul_ct``.
 - Same as ECDH.

Verify:
 - Short-Weierstrass
 - Window NAF interleaving multi-exponentiation method ``ECDSA_verify -> kmethod.verify -> ecdsa_verify -> ECDSA_do_verify -> kmethod.verify_sig -> ecdsa_verify_sig -> EC_POINT_mul -> method.mul_double_nonct -> ec_GFp_simple_mul_double_nonct -> ec_wNAF_mul``.
   Refers to http://www.informatik.tu-darmstadt.de/TI/Mitarbeiter/moeller.html#multiexp and https://www.informatik.tu-darmstadt.de/TI/Mitarbeiter/moeller.html#fastexp
 - Same coordinates and formulas as ECDH.


X25519
^^^^^^
Based on ref10 of Ed25519.
See `BoringSSL`_. Not exactly the same.

Ed25519
^^^^^^^
Based on ref10 of Ed25519.
See `BoringSSL`_. Not exactly the same.


libtomcrypt
===========

| Version: ``v1.18.2``
| Repository: https://github.com/libtom/libtomcrypt/
| Docs:

Primitives
----------

Offers ECDH and ECDSA on the `curves <https://github.com/libtom/libtomcrypt/blob/v1.18.2/src/pk/ecc/ecc.c>`__: SECP112r1, SECP128r1, SECP160r1, P-192, P-224, P-256, P-384, P-521.

ECDH
^^^^

KeyGen:
 - Short-Weierstrass
 - `Simple ladder <https://github.com/libtom/libtomcrypt/blob/v1.18.2/src/pk/ecc/ltc_ecc_mulmod_timing.c#L35>`__ via ``ecc_make_key -> ecc_make_key_ex -> ecc_ptmul -> ltc_ecc_mulmod_timing``.
 - jacobian, `dbl-1998-hnm <https://github.com/libtom/libtomcrypt/blob/v1.18.2/src/pk/ecc/ltc_ecc_projective_dbl_point.c#L32>`__ via ltc_ecc_projective_dbl_point
 - jacobian, `add-1998-hnm <https://github.com/libtom/libtomcrypt/blob/v1.18.2/src/pk/ecc/ltc_ecc_projective_add_point.c#L33>`__ via ltc_ecc_projective_add_point

Derive:
 - Same as Keygen via ``ecc_shared_secret -> ecc_ptmul -> ltc_ecc_mulmod_timing``.

ECDSA
^^^^^

Keygen:
 - Same as ECDH.

Sign:
 - Same as Keygen via ``ecc_sign_hash -> _ecc_sign_hash -> ecc_make_key_ex``.

Verify:
 - `Shamir's trick <https://github.com/libtom/libtomcrypt/blob/v1.18.2/src/pk/ecc/ltc_ecc_mul2add.c#L35>`__ via ``ecc_verify_hash -> _ecc_verify_hash -> ecc_mul2add`` or two separate sliding windows.
 - Same coords and formulas as KeyGen.

wolfSSL
=======

OpenSSL
=======

| Version: ``3.1.4``
| Repository: https://github.com/openssl/openssl
| Docs: https://www.openssl.org/docs/

Primitives
----------

ECDH, ECDSA on standard and custom curves.
x25519, x448 and Ed25519, Ed448.
Also SM2 specific methods.

The ladder methods have coordinate randomization and fix scalar bit-length.

Has several EC_METHODs.
 - EC_GFp_simple_method
 - EC_GFp_mont_method
 - EC_GFp_nist_method
 - EC_GFp_nistp224_method
 - EC_GFp_nistp256_method
 - EC_GFp_nistz256_method
 - EC_GFp_nistp521_method

`ossl_ec_GFp_simple_ladder_pre <https://github.com/openssl/openssl/blob/openssl-3.1.4/crypto/ec/ecp_smpl.c#L1493>`__:
 - Short-Weierstrass
 - xz
 - dbl-2002-it-2

`ossl_ec_GFp_simple_ladder_step <https://github.com/openssl/openssl/blob/openssl-3.1.4/crypto/ec/ecp_smpl.c#L1563>`__:
 - Short-Weierstrass
 - xz
 - mladd-2002-it-4

`ossl_ec_GFp_simple_ladder_post <https://github.com/openssl/openssl/blob/openssl-3.1.4/crypto/ec/ecp_smpl.c#L1651>`__:
 - Short-Weierstrass
 - xz to y-recovery

ECDH
^^^^

KeyGen:
 - Short-Weierstrass
 - ? via ``EVP_EC_gen -> EVP_PKEY_Q_keygen -> evp_pkey_keygen -> EVP_PKEY_generate -> evp_keymgmt_util_gen -> evp_keymgmt_gen -> EC_KEYMGMT.gen -> ec_gen -> EC_KEY_generate_key -> ec_method.keygen  -> ossl_ec_key_simple_generate_key -> EC_POINT_mul(k, G, NULL, NULL)`` all methods then either ec_method.mul or ossl_ec_wNAF_mul
    - EC_GFp_simple_method -> ossl_ec_wNAF_mul -> `ossl_ec_scalar_mul_ladder <https://github.com/openssl/openssl/blob/openssl-3.1.4/crypto/ec/ec_mult.c#L145>`__ (Lopez-Dahab ladder) for [k]G and [k]P. Otherwise multi-scalar wNAF with interleaving?
    - EC_GFp_mont_method -> ossl_ec_wNAF_mul -> `ossl_ec_scalar_mul_ladder <https://github.com/openssl/openssl/blob/openssl-3.1.4/crypto/ec/ec_mult.c#L145>`__ (Lopez-Dahab ladder) for [k]G and [k]P. Otherwise multi-scalar wNAF with interleaving?
    - EC_GFp_nist_method -> ossl_ec_wNAF_mul -> `ossl_ec_scalar_mul_ladder <https://github.com/openssl/openssl/blob/openssl-3.1.4/crypto/ec/ec_mult.c#L145>`__ (Lopez-Dahab ladder) for [k]G and [k]P. Otherwise multi-scalar wNAF with interleaving?
       - ec_point_ladder_pre -> ec_method.ladder_pre or EC_POINT_dbl
       - ec_point_ladder_step -> ec_method.ladder_step or EC_POINT_add + EC_POINT_dbl
       - ec_point_ladder_post -> ec_method.ladder_post
       - the methods all use ossl_ec_GFp_simple_ladder_* functions as ladder_*.
    - EC_GFp_nistp224_method -> ossl_ec_GFp_nistp224_points_mul -> Comb for generator, (signed, Booth) Fixed Window (width = 5) for other points.
    - EC_GFp_nistp256_method -> ossl_ec_GFp_nistp256_points_mul -> Comb for generator, (signed, Booth) Fixed Window (width = 5) for other points.
    - EC_GFp_nistz256_method -> ecp_nistz256_points_mul -> (signed, `Booth <https://github.com/openssl/openssl/blob/openssl-3.1.4/crypto/ec/ecp_nistputil.c#L141>`__) Fixed Window (width = 7) with full precomputation from [SG14]_.
    - EC_GFp_nistp521_method -> ossl_ec_GFp_nistp521_points_mul -> Comb for generator, (signed, Booth) Fixed Window (width = 5) for other points.
 - Jacobian (or Jacobian-3 for NIST)
 - Formulas:
    - EC_GFp_simple_method -> LibreSSL add and LibreSSL dbl
    - EC_GFp_mont_method -> LibreSSL add and LibreSSL dbl
    - EC_GFp_nist_method -> LibreSSL add and LibreSSL dbl
    - EC_GFp_nistp224_method -> BoringSSL P-224 add and dbl
    - EC_GFp_nistp256_method -> `add-2007-bl <https://github.com/openssl/openssl/blob/openssl-3.1.4/crypto/ec/ecp_nistp256.c#L1235>`__, `dbl-2001-b <https://github.com/openssl/openssl/blob/openssl-3.1.4/crypto/ec/ecp_nistp256.c#L1104>`__
    - EC_GFp_nistz256_method -> Unknown: `add-openssl-z256 <https://github.com/J08nY/pyecsca/blob/master/test/data/formulas/add-openssl-z256.op3>`__, `add-openssl-z256a <https://github.com/J08nY/pyecsca/blob/master/test/data/formulas/add-openssl-z256a.op3>`__
    - EC_GFp_nistp521_method -> `add-2007-bl <https://github.com/openssl/openssl/blob/openssl-3.1.4/crypto/ec/ecp_nistp521.c#L1205>`__, `dbl-2001-b <https://github.com/openssl/openssl/blob/openssl-3.1.4/crypto/ec/ecp_nistp521.c#L1087>`__

Derive:
 - Same as KeyGen, except for:
    - nistp{224,256,521} methods, where the Fixed Window branch of the scalar multiplier is taken,
    - nistz256 where a (signed, `Booth <https://github.com/openssl/openssl/blob/openssl-3.1.4/crypto/ec/ecp_nistputil.c#L141>`__) Fixed Window (width = 5) is taken.

ECDSA
^^^^^

KeyGen:
 - Same as ECDH.

Sign:
 - Same as KeyGen.

Verify:
 - Short-Weierstrass
 - EC_GFp_simple_method, EC_GFp_mont_method, EC_GFp_nist_method: Interleaved multi-scalar wNAF via ``ec_method.verify_sig -> ossl_ecdsa_simple_verify_sig -> EC_POINT_mul -> ossl_ec_wNAF_mul``.
 - EC_GFp_nistp224_method, EC_GFp_nistp256_method, EC_GFp_nistp521_method: Interleaved Comb for G and (signed, Booth) Fixed Window (width = 5) for other point.
 - EC_GFp_nistz256_method: Same as KeyGen for G and same as ECDH Derive for other point.

x25519
^^^^^^
Taken from ref10 of Ed25519. See BoringSSL_.

KeyGen:
 - Twisted-Edwards
 - Pippenger via ``ossl_x25519_public_from_private -> ge_scalarmult_base``.
 - Mixes coordinate models::

     ge_p2 (projective): (X:Y:Z) satisfying x=X/Z, y=Y/Z
     ge_p3 (extended): (X:Y:Z:T) satisfying x=X/Z, y=Y/Z, XY=ZT
     ge_p1p1 (completed): ((X:Z),(Y:T)) satisfying x=X/Z, y=Y/T
     ge_precomp (Duif): (y+x,y-x,2dxy)

Derive:
 - Montgomery
 - Montgomery ladder via ``ossl_x25519 -> x25519_scalar_mult``
 - xz coords
 - Unknown ladder formula: `ladd-openssl-x25519 <https://github.com/J08nY/pyecsca/blob/master/test/data/formulas/ladd-openssl-x25519.op3>`__

Ed25519
^^^^^^^
Taken from ref10 of Ed25519. See BoringSSL_.

KeyGen:
 - Same as x25519 KeyGen via ``ossl_ed25519_public_from_private -> ge_scalarmult_base``.

Sign:
 - Same as x25519 KeyGen via ``ossl_ed25519_sign -> ge_scalarmult_base``.

Verify:
 - Sliding window (signed) with interleaving? via ``ossl_ed25519_verify -> ge_double_scalarmult_vartime``.
 - Otherwise same mixed coordinates and formulas.

NSS
===

| Version: ``3.94``
| Repository: https://hg.mozilla.org/projects/nss
| Docs:


Primitives
----------

ECDH, ECDSA (only standard curves P-256, P-384, P-521), also x25519.

Two ECMethods:
 - Curve25519
    - 32-bit -> own impl
    - 64-bit -> HACL*
 - P-256 from HACL*

Several ECGroups:
 - generic ``ECGroup_consGFp``
 - Montgomery arithmetic ``ECGroup_consGFp_mont``
 - P-256
 - P-384 from ECCkiila
 - P-521 from ECCkiila

The ECMethods override the scalarmult of the ECGroups in:
 - ``ec_NewKey`` via ``ec_get_method_from_name`` and then calling the ``method.mul``.
 - ``EC_ValidatePublicKey`` via ``ec_get_method_from_name`` and then calling the ``method.validate``.
 - ``ECDH_Derive`` via ``ec_get_method_from_name`` and then calling the ``method.mul``.
 - ``ECDSA_SignDigest`` and ``ECDSA_SignDigestWithSeed`` via ``ec_SignDigestWithSeed``, then ``ec_get_method_from_name`` and then calling the ``method.mul``.


P-256 from HACL*
^^^^^^^^^^^^^^^^

KeyGen:
 - Short-Weierstrass
 - Fixed Window (width = 4)? points to https://eprint.iacr.org/2013/816.pdf? via ``ec_secp256r1_pt_mul -> (Hacl*) Hacl_P256_dh_initiator -> point_mul_g``
 - projective-3 coords.
 - `add-2015-rcb`, `dbl-2015-rcb-3`

Derive:
 - Same as KeyGen.

Sign:
 - Same as Keygen.

Verify:
 - Short-Weierstrass
 - Multi-scalar simultaneous Fixed Window
 - Same coords and formulas as KeyGen.

P-384
^^^^^

KeyGen:
 - Short-Weierstrass
 - Comb from ecckiila: ``EC_NewKeyFromSeed -> ec_NewKey -> ec_points_mul -> ECPoints_mul -> ecgroup.points_mul -> point_mul_two_secp384r1_wrap -> point_mul_g_secp384r1_wrap -> point_mul_g_secp384r1 -> fixed_smul_cmb``.
 - projective-3 coords.
 - `dbl-2015-rcb-3`, `madd-2015-rcb-3` also `add-2015-rcb` in point_add_proj.

Derive:
 - Short-Weierstrass
 - Regular Window NAF (width = 5) from ecckiila: ``ECDH_Derive -> ec_points_mul -> ECPoints_mul -> ecgroup.points_mul -> point_mul_secp384r1_wrap -> point_mul_secp384r1 -> var_smul_rwnaf``.
 - projective-3 coords.
 - `dbl-2015-rcb-3`, `add-2015-rcb`.

Sign:
 - Same as KeyGen.

Verify:
 - Short-Weierstrass
 - Interleaved multi-scalar window NAF (width = 5) with Shamir's trick from ecckiila: ``ECDSA_SignDigest -> ECDSA_SignDigestWithSeed -> ec_SignDigestWithSeed -> ec_points_mul -> ECPoints_mul -> ecgroup.points_mul -> point_mul_two_secp384r1_wrap -> point_mul_two_secp384r1 -> var_smul_wnaf_two``
 - projective-3 coords.
 - `dbl-2015-rcb-3`, `madd-2015-rcb-3` also `add-2015-rcb` in point_add_proj.

P-521
^^^^^

Same as P-384.

x25519
^^^^^^

KeyGen:
 - Montgomery
 - Montgomery ladder via ``-> ec_Curve25519_pt_mul -> ec_Curve25519_mul``.
 - xz coords
 - Unknown formulas: `ladd-hacl-x25519 <https://github.com/J08nY/pyecsca/blob/master/test/data/formulas/ladd-hacl-x25519.op3>`__,
   `dbl-hacl-x25519 <https://github.com/J08nY/pyecsca/blob/master/test/data/formulas/dbl-hacl-x25519.op3>`__

Derive:
 - Same as KeyGen.

libsecp256k1
============

| Version: ``v0.4.0``
| Repository: https://github.com/bitcoin-core/secp256k1
| Docs:

Primitives
----------

Supports ECDSA, ECDH and Schnorr signatures over secp256k1.

ECDH
^^^^

KeyGen:
 - Short-Weierstrass
 - `Fixed window with full precomputation <https://github.com/bitcoin-core/secp256k1/blob/v0.4.0/src/ecmult_gen_impl.h#L45>`__ via ``secp256k1_ec_pubkey_create -> secp256k1_ec_pubkey_create_helper -> secp256k1_ecmult_gen``. Window of size 4.
 - Uses scalar blinding.
 - `Jacobian version of add-2002-bj <https://github.com/bitcoin-core/secp256k1/blob/v0.4.0/src/group_impl.h#L670>`__  (via ``secp256k1_gej_add_ge``).
 - No doubling.


Derive:
 - Uses GLV decomposition and `interleaving with width-5 NAFs <https://github.com/bitcoin-core/secp256k1/blob/v0.4.0/src/ecmult_const_impl.h#L133>`__ via ``secp256k1_ecdh -> secp256k1_ecmult_const``.
 - Addition same as in Keygen.
 - Unknown doubling: `dbl-secp256k1-v040 <https://github.com/J08nY/pyecsca/blob/master/test/data/formulas/dbl-secp256k1-v040>`__ (via `secp256k1_gej_double <https://github.com/bitcoin-core/secp256k1/blob/v0.4.0/src/group_impl.h#L406>`__)

ECDSA
^^^^^

Keygen:
 - Same as ECDH.

Sign:
 - Same as Keygen via ``secp256k1_ecdsa_sign -> secp256k1_ecdsa_sign_inner -> secp256k1_ecdsa_sig_sign -> secp256k1_ecmult_gen``.

Verify:
 - Split both scalars using GLV and then interleaving with width-5 NAFS on 4 scalars via ``secp256k1_ecdsa_verify -> secp256k1_ecdsa_sig_verify -> secp256k1_ecmult -> secp256k1_ecmult_strauss_wnaf``.
 - DBL same as in ECDH DERIVE. Two formulas for addition are implemented. For the generator part, same addition as in Keygen is used. For public key, the following::

    assume iZ2 = 1/Z2
    az = Z_1*iZ2
    Z12 = az^2
    u1 = X1
    u2 = X2*Z12
    s1 = Y1
    s2 = Y2*Z12
    s2 = s2*az
    h = -u1
    h = h+u2
    i = -s2
    i = i+s1
    Z3 = Z1*h
    h2 = h^2
    h2 = -h2
    h3 = h2*h
    t = u1*h2
    X3 = i^2
    X3 = X3+h3
    X3 = X3+t
    X3 = X3+t
    t = t+X3
    Y3 = t*i
    h3 = h3*s1
    Y3 = Y3+h3

 - Before the addition the Jacobian coordinates are mapped to an isomorphic curve.


Nettle
======

| Version: ``3.9.1``
| Repository: https://git.lysator.liu.se/nettle/nettle
| Docs: https://www.lysator.liu.se/~nisse/nettle/nettle.html

Primitives
----------

ECDSA on P192, P224, P256, P384 and P521, also EdDSA on Curve25519, Curve448.

.. csv-table:: Pippenger parameters
    :header: "Curve", "K", "C"

    P192, 8, 6
    P224, 16, 7
    P256, 11, 6
    P384, 32, 6
    P521, 44, 6
    Curve25519, 11, 6

ECDSA
^^^^^

KeyGen:
 - Short-Weierstrass
 - `Pippenger <https://git.lysator.liu.se/nettle/nettle/-/blob/nettle_3.9.1_release_20230601/ecc-mul-g.c?ref_type=tags#L44>`__ via ``ecdsa_generate_keypair -> ecc_curve.mul_g -> ecc_mul_g``.
 - Jacobian
 - `madd-2007-bl <https://git.lysator.liu.se/nettle/nettle/-/blob/nettle_3.9.1_release_20230601/ecc-add-jja.c?ref_type=tags#L53>`__, `dbl-2001-b <https://git.lysator.liu.se/nettle/nettle/-/blob/nettle_3.9.1_release_20230601/ecc-dup-jj.c?ref_type=tags#L46>`__

Sign:
 - Short-Weierstrass
 - `Pippenger <https://git.lysator.liu.se/nettle/nettle/-/blob/nettle_3.9.1_release_20230601/ecc-mul-g.c?ref_type=tags#L44>`__ via ``ecc_ecdsa_sign -> ecc_mul_g``.
 - Same as KeyGen.


Verify:
 - Short-Weierstrass
 - `Pippenger <https://git.lysator.liu.se/nettle/nettle/-/blob/nettle_3.9.1_release_20230601/ecc-mul-g.c?ref_type=tags#L44>`__ and `4-bit Fixed Window <https://git.lysator.liu.se/nettle/nettle/-/blob/nettle_3.9.1_release_20230601/ecc-mul-a.c?ref_type=tags#L52>`__ via ``ecc_ecdsa_verify -> ecc_mul_a + ecc_mul_g``.
 - Jacobian
 - `madd-2007-bl <https://git.lysator.liu.se/nettle/nettle/-/blob/nettle_3.9.1_release_20230601/ecc-add-jja.c?ref_type=tags#L53>`__, `dbl-2001-b <https://git.lysator.liu.se/nettle/nettle/-/blob/nettle_3.9.1_release_20230601/ecc-dup-jj.c?ref_type=tags#L46>`__,
   also `add-2007-bl <https://git.lysator.liu.se/nettle/nettle/-/blob/nettle_3.9.1_release_20230601/ecc-add-jjj.c?ref_type=tags#L42>`__.

Ed25519
^^^^^^^

KeyGen:
 - Twisted Edwards
 - `Pippenger <https://git.lysator.liu.se/nettle/nettle/-/blob/nettle_3.9.1_release_20230601/ecc-mul-g-eh.c?ref_type=tags#L44>`__ via ``ed25519_sha512_public_key -> _eddsa_public_key -> ecc_curve.mul_g -> ecc_mul_g_eh``.
 - Projective
 - `madd-2008-bbjlp <https://git.lysator.liu.se/nettle/nettle/-/blob/nettle_3.9.1_release_20230601/ecc-add-th.c?ref_type=tags#L42>`__, `add-2008-bbjlp <https://git.lysator.liu.se/nettle/nettle/-/blob/nettle_3.9.1_release_20230601/ecc-add-thh.c?ref_type=tags#L41>`__ and `dup-2008-bbjlp <https://git.lysator.liu.se/nettle/nettle/-/blob/nettle_3.9.1_release_20230601/ecc-dup-th.c?ref_type=tags#L41>`__.

Sign:
 - Twisted Edwards
 - `Pippenger <https://git.lysator.liu.se/nettle/nettle/-/blob/nettle_3.9.1_release_20230601/ecc-mul-g-eh.c?ref_type=tags#L44>`__ via ``ed25519_sha512_sign -> _eddsa_sign -> ecc_curve.mul_g -> ecc_mul_g_eh``.
 - Same as KeyGen.

Verify:
 - Twisted Edwards
 - `Pippenger <https://git.lysator.liu.se/nettle/nettle/-/blob/nettle_3.9.1_release_20230601/ecc-mul-g-eh.c?ref_type=tags#L44>`__ and `4-bit Fixed Window <https://git.lysator.liu.se/nettle/nettle/-/blob/nettle_3.9.1_release_20230601/ecc-mul-a-eh.c?ref_type=tags#L116>`__ via ``ed25519_sha512_verify -> _eddsa_verify -> ecc_curve.mul + ecc_curve.mul_g``.
 - Same as KeyGen.


mbedTLS
=======

| Version: ``3.5.1``
| Repository: https://github.com/Mbed-TLS/mbedtls
| Docs: https://mbed-tls.readthedocs.io/en/latest/index.html

Primitives
----------

ECDH and ECDSA on P192, P224, P256, P384, P521 (their R and K variants) as well
as x25519 and x448.

x25519 has two implementations, and mbedTLS one (described below) and `one <https://github.com/Mbed-TLS/mbedtls/tree/v3.5.1/3rdparty/everest>`__ from
`Project Everest <https://github.com/project-everest/everest>`__.

ECDH
^^^^

KeyGen:
 - Short-Weierstrass
 - `Comb <https://github.com/Mbed-TLS/mbedtls/blob/v3.5.1/library/ecp.c#L2299>`__ via ``mbedtls_ecdh_gen_public -> ecdh_gen_public_restartable -> mbedtls_ecp_mul_restartable -> ecp_mul_restartable_internal -> ecp_mul_comb``.
   w = 5 for curves < 384 bits, then w = 6.
 - `Jacobian <https://github.com/Mbed-TLS/mbedtls/blob/v3.5.1/library/ecp.c#L1313>`__ coords with coordinate randomization.
 - `add-gecc-322 [GECC]_ algorithm 3.22 <https://github.com/Mbed-TLS/mbedtls/blob/v3.5.1/library/ecp.c#L1593>`__, `dbl-1998-cmo-2 <https://github.com/Mbed-TLS/mbedtls/blob/v3.5.1/library/ecp.c#L1496>`__. Also has alternative impl (``_ALT``).

Derive:
 - Short-Weierstrass
 - `Comb <https://github.com/Mbed-TLS/mbedtls/blob/v3.5.1/library/ecp.c#L2299>`__ via ``mbedtls_ecdh_compute_shared -> ecdh_compute_shared_restartable -> mbedtls_ecp_mul_restartable -> ecp_mul_restartable_internal -> ecp_mul_comb``.
   w = 4 for curves < 384 bits, then w = 5. The width is smaller by 1 than the case when the generator point is used (in KeyGen).
 - Same coords and formulas as KeyGen.

ECDSA
^^^^^

KeyGen:
 - Short-Weierstrass
 - `Comb <https://github.com/Mbed-TLS/mbedtls/blob/v3.5.1/library/ecp.c#L2299>`__ via ``mbedtls_ecdsa_genkey -> mbedtls_ecp_gen_keypair -> mbedtls_ecp_gen_keypair_base -> mbedtls_ecp_mul -> mbedtls_ecp_mul_restartable -> ecp_mul_restartable_internal -> ecp_mul_comb``.
 - Same as ECDH (KeyGen).

Sign:
 - Short-Weierstrass
 - `Comb <https://github.com/Mbed-TLS/mbedtls/blob/v3.5.1/library/ecp.c#L2299>`__ via ``mbedtls_ecdsa_sign -> mbedtls_ecdsa_sign_restartable -> mbedtls_ecp_mul_restartable -> ecp_mul_restartable_internal -> ecp_mul_comb``.
 - Same as ECDH (KeyGen).

Verify:
 - Short-Weierstrass
 - `Comb <https://github.com/Mbed-TLS/mbedtls/blob/v3.5.1/library/ecp.c#L2299>`__ + `Comb <https://github.com/Mbed-TLS/mbedtls/blob/v3.5.1/library/ecp.c#L2299>`__ via ``mbedtls_ecdsa_verify -> mbedtls_ecdsa_verify_restartable -> mbedtls_ecp_muladd_restartable -> mbedtls_ecp_mul_shortcuts + mbedtls_ecp_mul_shortcuts -> ecp_mul_restartable_internal -> ecp_mul_comb``.
 - Same as ECDH (KeyGen, Derive).

x25519
^^^^^^

KeyGen:
 - Montgomery
 - `Montgomery Ladder <https://github.com/Mbed-TLS/mbedtls/blob/v3.5.1/library/ecp.c#L2555>`__ via ``mbedtls_ecdh_gen_public -> ecdh_gen_public_restartable -> mbedtls_ecp_mul_restartable -> ecp_mul_restartable_internal -> ecp_mul_mxz``.
 - `xz <https://github.com/Mbed-TLS/mbedtls/blob/v3.5.1/library/ecp.c#L2555>`__ coords.
 - `mladd-1987-m <https://github.com/Mbed-TLS/mbedtls/blob/v3.5.1/library/ecp.c#L2509>`__.

Derive:
 - Montgomery
 - `Montgomery Ladder <https://github.com/Mbed-TLS/mbedtls/blob/v3.5.1/library/ecp.c#L2555>`__ via ``mbedtls_ecdh_compute_shared -> ecdh_compute_shared_restartable -> mbedtls_ecp_mul_restartable -> ecp_mul_restartable_internal -> ecp_mul_mxz``.
 - Same as KeyGen.

SunEC
=====

| Version: ``jdk-21-ga`` (JDK 21)
| Repository: https://github.com/openjdk/jdk/
| Docs:


Primitives
----------

ECDH, ECDSA, x25519, Ed25519

P-256
^^^^^

The only special thing is the generator scalarmult, ``Secp256R1GeneratorMultiplier`` which is a Comb.

ECDH
^^^^

KeyGen:
 - Short-Weierstrass
 - Fixed Window (width = 4) via ``ECKeyPairGenerator.generateKeyPair -> ECKeyPairGenerator.generateKeyPairImpl -> ECPrivateKeyImpl.calculatePublicKey -> ECOperations.multiply -> Default(PointMultiplier).pointMultiply``
 - projective-3 coords
 - RCB-based formulas: `add-sunec-v21 <https://github.com/J08nY/pyecsca/blob/master/test/data/formulas/add-sunec-v21.op3>`__,
   `dbl-sunec-v21 <https://github.com/J08nY/pyecsca/blob/master/test/data/formulas/dbl-sunec-v21.op3>`__,


Derive:
 - Same as KeyGen.

ECDSA
^^^^^

Same as ECDH.

x25519
^^^^^^

KeyGen:
 - Montgomery
 - Montgomery ladder
 - xz
 - Ladder formula from RFC 7748

Derive:
 - Same as KeyGen.

Ed25519
^^^^^^^

KeyGen:
 - Twisted-Edwards
 - Double and add always
 - Extended coords
 - Unknown formulas: `add-sunec-v21-ed25519 <https://github.com/J08nY/pyecsca/blob/master/test/data/formulas/add-sunec-v21-ed25519.op3>`__,  `dbl-sunec-v21-ed25519 <https://github.com/J08nY/pyecsca/blob/master/test/data/formulas/dbl-sunec-v21-ed25519.op3>`__

Sign:
 - Same as KeyGen.

Verify:
 - Same as KeyGen.

Go
==

| Version: ``go1.21.4``
| Repository: https://github.com/golang/go
| Docs:

Primitives
----------

ECDH, ECDSA over P-224, P-256, P-384 and P-521.
Ed25519, X25519

ECDH
^^^^

KeyGen:
 - Short-Weierstrass
 - `Fixed window (w=4) <https://github.com/golang/go/blob/go1.21.4/src/crypto/internal/nistec/p224.go#L412>`__ (link points to P-224, but others are the same) via ``privateKeyToPublicKey -> ScalarBaseMult``
 - Projective
 - `add-2015-rcb <https://github.com/golang/go/blob/go1.21.4/src/crypto/internal/nistec/p224.go#L215>`__, `dbl-2015-rcb <https://github.com/golang/go/blob/go1.21.4/src/crypto/internal/nistec/p224.go#L270>`__

Derive:
 - Short-Weierstrass
 - `Fixed window (w=4) <https://github.com/golang/go/blob/go1.21.4/src/crypto/internal/nistec/p224.go#L342>`__ via ``ecdh -> ScalarMult``.
 - Same formulas as in Keygen.

Also supports constant-time, 64-bit assembly implementation of P256 described in https://eprint.iacr.org/2013/816.pdf

ECDSA
^^^^^

KeyGen:
 - Same as ECDH KeyGen via ``ecdsa.go:GenerateKey -> generateNISTEC -> randomPoint -> ScalarBaseMult``.

Sign:
 - Same as KeyGen via ``ecdsa.go:SignASN1 -> signNISTEC -> randomPoint -> ScalarBaseMult``.

Verify:
 - Two separate scalar multiplications ``ScalarBaseMult`` (same as KeyGen) and ``ScalarMult`` (same as ECDH Derive) via ``ecdsa.go:VerifyASN1 -> verifyNISTEC``.

X25519
^^^^^^

KeyGen:
 - Montgomery
 - `Ladder <https://github.com/golang/go/blob/go1.21.4/src/crypto/ecdh/x25519.go#L54>`__ via ``privateKeyToPublicKey -> x25519ScalarMult``.
 - xz
 - Unknown formula: `ladd-go-1214 <https://github.com/J08nY/pyecsca/blob/master/test/data/formulas/ladd-go-1214.op3>`__

Derive:
 - Same as KeyGen via ``x25519.go:ecdh -> x25519ScalarMult``.

Ed25519
^^^^^^^

KeyGen:
 - Twisted-Edwards
 - Pippenger's signed 4-bit method with precomputation via ``ed25519.go:GenerateKey -> NewKeyFromSeed -> newKeyFromSeed -> ScalarBaseMult``.
 - `Extended coordinates <https://github.com/golang/go/blob/go1.21.4/src/crypto/internal/edwards25519/edwards25519.go#L28>`__ mixed with `y-x,y+x,2dxy <https://github.com/golang/go/blob/go1.21.4/src/crypto/internal/edwards25519/edwards25519.go#L52>`__ coordinates
 - `AddAffine <https://github.com/golang/go/blob/go1.21.4/src/crypto/internal/edwards25519/edwards25519.go#L312>`__ (and similar SubAffine)::

      YplusX.Add(&p.y, &p.x)
      YminusX.Subtract(&p.y, &p.x)

      PP.Multiply(&YplusX, &q.YplusX)
      MM.Multiply(&YminusX, &q.YminusX)
      TT2d.Multiply(&p.t, &q.T2d)

      Z2.Add(&p.z, &p.z)

      v.X.Subtract(&PP, &MM)
      v.Y.Add(&PP, &MM)
      v.Z.Add(&Z2, &TT2d)
      v.T.Subtract(&Z2, &TT2d)

Sign:
 - Same as Keygen via ``ed25519.go: Sign -> sign ->  ScalarBaseMult``.

Verify:
 - Bos-Coster method via ``ed25519.go: Verify -> verify -> VarTimeDoubleScalarBaseMult``.
 - Same coordinates and formulas as in Keygen.

libgcrypt
=========

| Version: ``1.10.2``
| Repository: https://git.gnupg.org/
| Docs: https://gnupg.org/documentation/manuals/gcrypt/

Primitives
----------

Supports ECDH, X25519 and EdDSA `on <https://gnupg.org/documentation/manuals/gcrypt/ECC-key-parameters.html#ECC-key-parameters>`__ C25519, X448, Ed25519, Ed448, NIST curves, Brainpool curves and secp256k1.
Also supports GOST and SM2 signatures.

ECDH
^^^^

KeyGen:
 - Short-Weierstrass
 - `Left to right double-and-add-always <https://git.gnupg.org/cgi-bin/gitweb.cgi?p=libgcrypt.git;a=blob;f=mpi/ec.c;h=c24921eea8bea8363a503d6d6071b116c176d8e5;hb=1c5cbacf3d88dded5063e959ee68678ff7d0fa56#l1824>`__ via ``gcry_pk_genkey -> _gcry_pk_genkey -> generate -> ecc_generate -> nist_generate_key -> _gcry_mpi_ec_mul_point``.
 - Jacobian coords
 - Unknown formulas: `add-libgcrypt-v1102 <https://github.com/J08nY/pyecsca/blob/master/test/data/formulas/add-libgcrypt-v1102.op3>`__,
   `dbl-libgcrypt-v1102 <https://github.com/J08nY/pyecsca/blob/master/test/data/formulas/dbl-libgcrypt-v1102.op3>`__,

Derive:
 - Same as Keygen via ``gcry_pk_encrypt -> _gcry_pk_encrypt -> generate -> ecc_encrypt_raw -> _gcry_mpi_ec_mul_point``.


ECDSA
^^^^^

Keygen:
 - Same as ECDH.

Sign:
 - Same as Keygen via ``gcry_ecc_ecdsa_sign -> _gcry_ecc_ecdsa_sign -> _gcry_mpi_ec_mul_point``.

Verify:
 - Two separate scalar multiplications via ``gcry_ecc_ecdsa_verify -> _gcry_ecc_ecdsa_verify``.

EdDSA
^^^^^

Keygen:
 - Twisted-Edwards
 - `Left to right double-and-add-always <https://git.gnupg.org/cgi-bin/gitweb.cgi?p=libgcrypt.git;a=blob;f=mpi/ec.c;h=c24921eea8bea8363a503d6d6071b116c176d8e5;hb=1c5cbacf3d88dded5063e959ee68678ff7d0fa56#l1824>`__ via ``gcry_pk_genkey -> _gcry_pk_genkey -> generate -> ecc_generate -> _gcry_ecc_eddsa_genkey -> _gcry_mpi_ec_mul_point``.
 - Projective, `dbl-2008-bbjlp <https://git.gnupg.org/cgi-bin/gitweb.cgi?p=libgcrypt.git;a=blob;f=mpi/ec.c;h=c24921eea8bea8363a503d6d6071b116c176d8e5;hb=1c5cbacf3d88dded5063e959ee68678ff7d0fa56#l1314>`__ and `add-2008-bbjlp <https://git.gnupg.org/cgi-bin/gitweb.cgi?p=libgcrypt.git;a=blob;f=mpi/ec.c;h=c24921eea8bea8363a503d6d6071b116c176d8e5;hb=1c5cbacf3d88dded5063e959ee68678ff7d0fa56#l1563>`__

Sign:
 - Same as Keygen via ``gcry_ecc_eddsa_sign -> _gcry_ecc_eddsa_sign -> _gcry_mpi_ec_mul_point``.

Verify:
 - Two separate scalar multiplications via ``gcry_ecc_eddsa_verify -> _gcry_ecc_eddsa_verify``.


X25519
^^^^^^

KeyGen:
 - Montgomery
 - `Montgomery ladder <https://git.gnupg.org/cgi-bin/gitweb.cgi?p=libgcrypt.git;a=blob;f=mpi/ec.c;h=c24921eea8bea8363a503d6d6071b116c176d8e5;hb=1c5cbacf3d88dded5063e959ee68678ff7d0fa56#l1858>`__ via ``gcry_pk_genkey -> _gcry_pk_genkey -> generate -> ecc_generate -> nist_generate_key -> _gcry_mpi_ec_mul_point``.
 - xz coordinates with a shuffled version of `ladd-1987-m-3 <https://git.gnupg.org/cgi-bin/gitweb.cgi?p=libgcrypt.git;a=blob;f=mpi/ec.c;h=c24921eea8bea8363a503d6d6071b116c176d8e5;hb=1c5cbacf3d88dded5063e959ee68678ff7d0fa56#l1661>`__


Derive:
 - Same as Keygen via ``gcry_pk_encrypt -> _gcry_pk_encrypt -> generate -> ecc_encrypt_raw -> _gcry_mpi_ec_mul_point``.

BearSSL
=======

| Version: ``v0.6``
| Repository: https://bearssl.org/gitweb/?p=BearSSL;a=summary
| Docs: https://bearssl.org/index.html

Primitives
----------

Supports SECG prime field curves, as well as Brainpool and Curve25519, Curve448.
Has API functions for ECDSA, but does ECDH only implicitly in its TLS implementation (no public API exposed).
Unclear whether Ed25519 is supported.

ECDH
^^^^

KeyGen:
 - Short-Weierstrass
 - (width=2) Fixed Window via ``br_ec_compute_pub -> impl.mulgen -> impl.mul``, but (width=4) Fixed Window via ``br_ec_compute_pub -> impl.mulgen`` for special (P-256) curves.
 - Jacobian coordinates
 - Unknown formulas: `add-bearssl-v06 <https://github.com/J08nY/pyecsca/blob/master/test/data/formulas/add-bearssl-v06.op3>`__,
   `dbl-bearssl-v06 <https://github.com/J08nY/pyecsca/blob/master/test/data/formulas/dbl-bearssl-v06.op3>`__,

Derive:
 - Short-Weierstrass
 - (width=2) Fixed Window via ``impl.mul``.
 - Coordinates and formulas same as in KeyGen.

ECDSA
^^^^^

KeyGen:
 - Same as ECDH.

Sign:
 - Short-Weierstrass
 - (width=2) Fixed Window via ``br_ecdsa_*_sign_raw -> impl.mulgen -> impl.mul``, but (width=4) Fixed Window via ``br_ecdsa_*_sign_raw -> impl.mulgen`` for special (P-256) curves.
 - Coordinates and formulas same as in KeyGen.

Verify:
 - Short-Weierstrass
 - Simple scalarmult then add via ``br_ecdsa_*_verify_raw -> impl.muladd -> impl.mul + add``
 - Coordinates and formulas same as in KeyGen.

x25519
^^^^^^

KeyGen:
 - Montgomery
 - Montgomery ladder via ``br_ec_compute_pub -> impl.mulgen -> impl.mul``.
 - xz coordinates
 - ladd-rfc7748

Derive:
 - Same as KeyGen.