path: root/docs/libraries.rst

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

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

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 <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.

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 unknown.

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 unknown.

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 unknown.

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 unknown.

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>`__.
 - `dbl-1987-m-3 <https://github.com/bcgit/bc-java/blob/r1rv76/core/src/main/java/org/bouncycastle/math/ec/rfc7748/X25519.java#L73>`__ and
   some `ladd-1987 <https://github.com/bcgit/bc-java/blob/r1rv76/core/src/main/java/org/bouncycastle/math/ec/rfc7748/X25519.java#L111>`__ formula.

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.

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 <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.

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 <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``
 - Actually seems to use xz.
 - Unknown formula (ladder).

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
 - Something like FullPrecomputation and Comb (no doublings), 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
 - Something like FullPrecomputation and Comb (no doublings), 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
 - Something like FullPrecomputation and Comb (no doublings), 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) (signed) 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 curve2551_donna.

Ed25519
^^^^^^^
Based on ref10 of Ed255119.
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).
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
 - `Montgomery ladder <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
 - `Montgomery ladder <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
 - `Montgomery ladder <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.

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
 - Ladder 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.
 - `add-1998-hnm <https://github.com/libressl/openbsd/blob/libressl-v3.8.2/src/lib/libcrypto/ec/ecp_smpl.c#L472>`__ likely, due to the division by 2.

Dbl::

    n1 = 3 * X_a^2 + a_curve * Z_a^4
    Z_r = 2 * Y_a * Z_a
    n2 = 4 * X_a * Y_a^2
    X_r = n1^2 - 2 * n2
    n3 = 8 * Y_a^4
    Y_r = n1 * (n2 - X_r) - n3

Derive:
 - Short-Weierstrass
 - Ladder 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
 - Ladder 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-exponentation 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 Ed255119.
See `BoringSSL`_. Not exactly the same.

Ed25519
^^^^^^^
Based on ref10 of Ed255119.
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
 - `Sliding window <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
=======

NSS
===

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 findow 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.
 - `DBL <https://github.com/bitcoin-core/secp256k1/blob/v0.4.0/src/group_impl.h#L406>`__ (via ``secp256k1_gej_double``)::
     
    Z3 = Y1*Z1
    S = Y1^2
    L = X1^2
    L = 3*L
    L = L/2
    T = -S
    T = T*X1
    X3 = L^2
    X3 = X3+T
    X3 = X3+T
    S = S^2
    T = T+X3
    Y3 = T*L
    Y3 = Y3+S
    Y3 = -Y3

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
======

mbedTLS
=======

SunEC
=====

Go
==

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``.
 - `ADD <https://git.gnupg.org/cgi-bin/gitweb.cgi?p=libgcrypt.git;a=blob;f=mpi/ec.c;h=c24921eea8bea8363a503d6d6071b116c176d8e5;hb=1c5cbacf3d88dded5063e959ee68678ff7d0fa56#l1406>`__  (via ``_gcry_mpi_ec_add_points``)::

     l1 = x1 z2^2  
     l2 = x2 z1^2  
     l3 = l1 - l2 
     l4 = y1 z2^3  
     l5 = y2 z1^3  
     l6 = l4 - l5  
     l7 = l1 + l2  
     l8 = l4 + l5  
     z3 = z1 z2 l3  
     x3 = l6^2 - l7 l3^2  
     l9 = l7 l3^2 - 2 x3  
     y3 = (l9 l6 - l8 l3^3)/2 

 - `DBL <https://git.gnupg.org/cgi-bin/gitweb.cgi?p=libgcrypt.git;a=blob;f=mpi/ec.c;h=c24921eea8bea8363a503d6d6071b116c176d8e5;hb=1c5cbacf3d88dded5063e959ee68678ff7d0fa56#l1219>`__ (via ``_gcry_mpi_ec_dup_point``)::
     
     L1 = 3X^2 + aZ^4 
     Z3 = 2YZ 
     L2 = 4XY^2 
     X3 = L1^2 - 2L2 
     L3 = 8Y^4 
     Y3 = L1(L2 - X3) - L3 


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

Add::

   u1 = x1 * z2^2
   u2 = x2 * z1^2
   s1 = y1 * z2^3
   s2 = y2 * z1^3
   h = u2 - u1
   r = s2 - s1
   x3 = r^2 - h^3 - 2 * u1 * h^2
   y3 = r * (u1 * h^2 - x3) - s1 * h^3
   z3 = h * z1 * z2

Dbl::

   s = 4*x*y^2
   m = 3*(x + z^2)*(x - z^2)
   x' = m^2 - 2*s
   y' = m*(s - x') - 8*y^4
   z' = 2*y*z

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
 - mladd-1987-m

Ladder::

 269                 c255_add(a, x2, z2); // a = x2 + z2
 270                 c255_mul(aa, a, a);  // aa = a^2
 271                 c255_sub(b, x2, z2); // b = x2 - z2
 272                 c255_mul(bb, b, b);  // bb = b^2
 273                 c255_sub(e, aa, bb); // e = aa * bb
 274                 c255_add(c, x3, z3); // c = x3 + z3
 275                 c255_sub(d, x3, z3); // d = x3 - z3
 276                 c255_mul(da, d, a);  // da = d * a
 277                 c255_mul(cb, c, b);  // cb = c * b

 291                 c255_add(x3, da, cb);// x3 = da + cb
 292                 c255_mul(x3, x3, x3);// x3 = x3^2
 293                 c255_sub(z3, da, cb);// z3 = da - cb
 294                 c255_mul(z3, z3, z3);// z3 = z3^2
 295                 c255_mul(z3, z3, x1);// z3 = z3 * x1
 296                 c255_mul(x2, aa, bb);// x2 = aa * bb
 297                 c255_mul(z2, C255_A24, e);// z2 = e * A24
 298                 c255_add(z2, z2, aa);// z2 = z2 + aa
 299                 c255_mul(z2, e, z2); // z2 = z2 * e

Derive:
 - Same as KeyGen.