brainpool

brainpool

A Go library for the Brainpool elliptic curves (RFC 5639) as used across the gematik telematics infrastructure (SMC-B, HBA, eGK, IDP-Dienst, ePA): curve definitions, X.509 certificate and key parsing that the Go standard library cannot perform for these curves, a self-contained JOSE layer (JWS/JWE/JWK), and a constant-time, formally-verified core for brainpoolP256r1 covering the secret-scalar operations (ECDSA signing and ECDH).

Security posture

The security-critical design follows the principle that only operations on a secret scalar require constant-time implementations; operations on public data do not. Accordingly:

Why a constant-time core for brainpoolP256r1

Go's generic crypto/elliptic custom-curve path (the only path available for non-NIST curves) is deprecated and documented as "not guaranteed to provide any security property." Its scalar multiplication is a variable-time big.Int double-and-add whose timing correlates with the secret scalar — the class of leak exploited by Minerva (2019) and TPM-Fail (2019) to recover ECDSA private keys via lattice attacks. brainpoolP256r1 is the curve gematik uses for software-held signing keys, so it receives the same constant-time treatment the Go standard library already gives the NIST curves.

The constant-time core is built exactly as the Go standard library builds its NIST curve support (crypto/internal/fips140/nistec):

1. Field arithmetic — machine-checked code generated by Fiat Cryptography (word-by-word Montgomery), committed verbatim. See internal/bp256/fiat/PROVENANCE.md.
2. Point arithmetic — the exception-free complete addition formulas of Renes–Costello–Batina over homogeneous projective coordinates. brainpoolP256r1 has a ≠ −3, so the general formula (Algorithm 1) is used.
3. Scalar multiplication — a constant-time Montgomery ladder with no scalar-bit-dependent branches.
4. Scalar-field arithmetic (mod n, for k⁻¹ and z + r·d) — a second Fiat-Cryptography field; see internal/bp256/fiatn.
5. Deterministic nonces — RFC 6979 (HMAC-SHA256), default for software signing, removing the weak-RNG key-recovery failure mode.
6. Low-s normalisation — signatures are canonicalised to low-s (SEC 1 §4.1.3 / BIP-0062) to remove signature malleability.

Residual notes

• brainpoolP384r1 and brainpoolP512r1 remain on the standard library's non-constant-time generic path; they are used for certificate handling and are not the software-signing curves. Hardware-token (HBA/SMC-B) signing is opaque to this library and unaffected.
• This library has not undergone an independent third-party audit. It is structured to support one: the field layer is machine-checked, the point layer is differentially tested against an independent implementation and OpenSSL, and every algorithm cites its specification.

Validation

The constant-time core is validated by:

• Differential testing of point addition, doubling, scalar multiplication, ECDSA, and ECDH against the package's independent big.Int curve implementation (the oracle) and against the Go standard library.
• Project Wycheproof — all 485 ecdsa_brainpoolP256r1_sha256 edge-case vectors (zero/overflow r,s, malformed DER, …).
• RFC 7027 §A.1 ECDH known-answer vectors (cross-implementation; the expected values are independently confirmed with OpenSSL).
• Field arithmetic checked against math/big modular arithmetic.
• OpenSSL cross-tests (build tag openssl_cross): our deterministic signatures verify under OpenSSL, and OpenSSL signatures verify under the standard-library path — bidirectional interoperability.
• RFC 6979 determinism validated by reproducibility (identical inputs → identical signature; distinct messages → distinct nonces) and by verification under the standard library.

Domain parameters (brainpoolP256r1, RFC 5639 §3.4)

p  = A9FB57DBA1EEA9BC3E660A909D838D726E3BF623D52620282013481D1F6E5377
a  = 7D5A0975FC2C3057EEF67530417AFFE7FB8055C126DC5C6CE94A4B44F330B5D9
b  = 26DC5C6CE94A4B44F330B5D9BBD77CBF958416295CF7E1CE6BCCDC18FF8C07B6
Gx = 8BD2AEB9CB7E57CB2C4B482FFC81B7AFB9DE27E1E3BD23C23A4453BD9ACE3262
Gy = 547EF835C3DAC4FD97F8461A14611DC9C27745132DED8E545C1D54C72F046997
n  = A9FB57DBA1EEA9BC3E660A909D838D718C397AA3B561A6F7901E0E82974856A7
h  = 1
OID = 1.3.36.3.3.2.8.1.1.7  (brainpoolP256r1)

References

Curves and elliptic-curve cryptography

• RFC 5639 — Elliptic Curve Cryptography (ECC) Brainpool Standard Curves and Curve Generation (M. Lochter, J. Merkle), March 2010. Domain parameters.
• RFC 7027 — Elliptic Curve Cryptography (ECC) Brainpool Curves for Transport Layer Security (TLS) (J. Merkle, M. Lochter), October 2013. ECDH test vectors (Appendix A).
• SEC 1: Elliptic Curve Cryptography, Version 2.0 (Certicom Research, 2009). Point encodings (§2.3), public-key validation (§3.2.2), signature encoding and low-s (§4.1.3).
• Renes, Costello, Batina, Complete addition formulas for prime order elliptic curves, EUROCRYPT 2016 (IACR ePrint 2015/1060). Algorithm 1 (general a) is used for point addition.

Signatures and key agreement

• FIPS 186-5 — Digital Signature Standard (DSS) (NIST, February 2023). ECDSA signing (§6.4.1) and verification (§6.4.2).
• RFC 6979 — Deterministic Usage of the DSA and ECDSA (T. Pornin), August 2013. Deterministic nonce generation (§3.2).
• BIP-0062 — low-s signature canonicalisation (malleability).

BSI Technical Guidelines

• BSI TR-03111 — Elliptic Curve Cryptography, Version 2.10 (Bundesamt für Sicherheit in der Informationstechnik). Point encoding (§3.1), public-key validation (§3.2.2), ECDH (§3.5.1), ECDSA plain signatures (§5.2.1).
• BSI TR-03110 — Advanced Security Mechanisms for Machine Readable Travel Documents and eIDAS Token (PACE / Chip Authentication context for eGK/eMRTD).

JOSE (the josebp subpackage)

• RFC 7515 — JSON Web Signature (JWS).
• RFC 7516 — JSON Web Encryption (JWE).
• RFC 7517 — JSON Web Key (JWK).
• RFC 7518 — JSON Web Algorithms (JWA); raw r‖s signatures (§3.4), ECDH-ES + Concat KDF (§4.6).
• RFC 7519 — JSON Web Token (JWT).

Formal verification and code generation

• Fiat Cryptography — A. Erbsen, J. Philipoom, J. Gross, R. Sloan, A. Chlipala, Simple High-Level Code for Cryptographic Arithmetic — With Proofs, Without Compromises, IEEE S&P 2019. https://github.com/mit-plv/fiat-crypto. The field arithmetic in internal/bp256/fiat and internal/bp256/fiatn is generated by Fiat Cryptography and is machine-checked in Coq. Provenance (pinned image, commit, exact invocation) is recorded in internal/bp256/fiat/PROVENANCE.md.
• addchain — M. McLoughlin, https://github.com/mmcloughlin/addchain. Generates the constant-time field-inversion addition chains.

Test corpora

• Project Wycheproof — https://github.com/C2SP/wycheproof. ECDSA brainpoolP256r1 edge-case vectors; see internal/bp256/testdata/wycheproof/PROVENANCE.md.

Standard library precedent

• Go crypto/internal/fips140/nistec — the NIST-curve implementation whose structure (Fiat field + projective complete formulas + constant-time ladder) this library mirrors for brainpoolP256r1.

Package layout

brainpool/                 Curves, X.509 parser, helpers, signing entry points
  josebp/                  JOSE: JWS/JWE/JWK over Brainpool
  internal/bp256/          Constant-time brainpoolP256r1 core
    fiat/                  Generated field arithmetic mod p (Fiat Cryptography)
    fiatn/                 Generated scalar arithmetic mod n (Fiat Cryptography)
    testdata/wycheproof/   Vendored Wycheproof vectors

License and provenance of generated code

Generated files (*_fiat64.go, *_invert.go) carry a DO NOT EDIT header and are reproducible from the pinned toolchain described in the PROVENANCE.md files. Do not hand-edit them; regenerate with the documented commands.