Sweet B is a library which implements public key elliptic curve cryptography (ECC) using the NIST P-256 and SECG secp256k1 curves. Sweet B is:
- Safe: known attack vectors have been accounted for, design decisions have been documented, and the API has been designed to eliminate the possibility of catastrophic misuse when possible.
- Clear: the library is thoroughly commented and unit tested, and is designed to be easy to read and review.
- Compact: the library is compact in code size, uses a minimal 512-byte working context, and does not assume that keys and other intermediary products can be allocated on the stack.
- Audited: a third-party review of the library was carried out prior to public release, and the full report and status of remediations are available publicly.
You should consider using Sweet B if you need to implement elliptic curve Diffie-Hellman shared-secret generation (ECDH) or elliptic curve digital signature generation and verification (ECDSA) in a memory-constrained environment. For instance, the P-256 curve is used in Bluetooth Low Energy Security, and is often implemented on memory-constrained devices for this purpose.
Sweet B is a pun on both the Short Weierstrass form of elliptic curves and on the NSA's Suite B set of cryptographic algorithms.
Sweet B was developed by Western Digital.
Western Digital engaged the security research firm Trail of Bits to review Sweet B prior to its public release. The resulting report and the status of remediations for specific findings are available publicly.
Sweet B provides mitigation for several classes of known faults and attacks:
- Timing analyses reveal secret information by measuring the time that it takes to perform cryptographic operations. Sweet B prevents this by ensuring that all operations run in constant time with respect to the input data (though different curves have different performance characteristics).
- Power analyses reveal secret information by measuring the amount of power consumed during cryptographic operations. Sweet B addresses this by using randomized projective coordinates, also called Z blinding. The special case of zero value analysis has been addressed by representing reduced integers modulo 𝑝 as integers within the range [1, 𝑝], ensuring that the points (0, ±√𝐵 ∙ 𝑍³, 𝑍) on applicable curves do not cause observable multiplications by low-Hamming-weight field elements.
- Safe-error analyses reveal secret information by causing hardware faults during cryptographic operations and observing whether the fault affects the output. Sweet B mitigates these attacks through the use of a regular Montgomery ladder with no dummy computations prior to the final bit.
- Per-message secret reuse causes the private key to be revealed to anyone receiving more than one signature with the same secret. Sweet B prevents this by providing an internal implementation of a deterministic random-bit generator (DRBG) using HMAC-SHA256 for per-message secret generation in ECDSA signing. When an externally seeded instance of the DRBG is provided, the private key and message are provided as additional input to the DRBG, ensuring that even in cases of entropy source failure, per-message secrets are never re-used. When no externally seeded instance is provided, RFC6979 deterministic signing is used. The internal HMAC-DRBG is also used for projective-coordinate randomization when no external entropy source is available.
It is impossible to guarantee that side-channel mitigations in a portable C implementation will perform correctly with all compilers and with all target platforms. Please analyze Sweet B and your use case carefully if using it on a platform where assembly support is not available.
Sweet B is designed to be simple, safe, compact, and embeddable. In order to be as portable as possible, any word size from 8 to 64 bits may be used; you should choose the word size that corresponds to the size of your hardware multiplier. Sweet B does not assume that it's possible to store large amounts of working state on the stack; instead, a separately allocated 512-byte working context is required, which may be placed on the stack, heap allocated, or statically allocated per the user's needs.
Simple, compact implementations of SHA256, HMAC-SHA256, and HMAC-DRBG are provided both for internal use and for use in producing digests of data to be signed or verified. You are also encouraged to use the HMAC-DRBG implementation for random number generation in your system, assuming you have access to a sufficient source of hardware entropy.
Sweet B uses Montgomery multiplication, which eliminates the need for separate reduction steps. This makes it easier to produce a constant-time library supporting multiple primes, and also makes Sweet B fast compared with other embeddable implementations in C. However, there are faster implementations of ECC if you have more working memory or more code storage available.
Sweet B has been carefully designed to avoid side channel attacks, including timing and power analyses. All field operations and elliptic curve operations are designed to run in constant time, and projective coordinate randomization consistently used. All functions take an optional DRBG parameter, and you are strongly encouraged to supply a properly-seeded DRBG whenever possible to mitigate power-based side channel attacks.
sb_sw_lib.h is the main entry point for ECC operations on
short Weierstrass curves (P-256 and secp256k1). For hashing and random number
generation, see sb_sha256.h and
sb_hmac_drbg.h. Each file contains a number of test
cases; if you compile Sweet B with -DSB_TEST, you can run them using the
main routine in sb_test.c.
You can set the word size used in Sweet B with the SB_WORD_SIZE preprocessor
macro. By default, this is set to 4, meaning that 32-bit multiplies producing
64-bit results will be used. On 8- or 16-bit microcontrollers, or on 32-bit
microcontrollers without full 64-bit multiply output (such as the Cortex-M0+),
you should set this to 1 or 2. On 64-bit x86 systems, you may want to set the
multiplication size to 8 to use 128-bit multiplication output.
You can disable either of the short Weierstrass curves Sweet B supports by
setting the preprocessor defines SB_SW_P256_SUPPORT or
SB_SW_SECP256K1_SUPPORT to 0. If you have a little more program memory
available, you may want to set SB_UNROLL to a value between 1 and 3
(inclusive); on Cortex-M4, SB_UNROLL=2 provides the best balance between
size and speed.
If you have ARM support for your processor (see sb_fe_armv7.s
for an example of this); define SB_FE_ASM to 1 when compiling the code, and
supply a separate ARM assembly implementation for the core field-element
arithmetic routines listed in sb_fe.h as being supported by
assembly. The supplied example implementation targets 32-bit ARM Thumb
processors with DSP extensions; examples of this include the Cortex-M4, M7, and
A5.
CMake build support is provided; to use it, create a
directory for your build, run cmake with the path to the Sweet B sources, and
then run make to build. To run the unit tests with the clang undefined
behavior and address sanitizers, pass -DCMAKE_C_COMPILER=clang to cmake if
clang is not your default compiler.
Neal Koblitz. A Course in Number Theory and Cryptography. Springer-Verlag, 1994.
This is a rather old text, and the section on elliptic curves is dated. However, it remains an outstanding reference for any discussion of finite fields.
Alfred J. Menezes, Paul C. van Oorschot, and Scott A. Vanstone. Handbook of Applied Cryptography. CRC Press, 1996.
Another older text, but the chapter on efficient implementation remains a worthwhile reference for basic field arithmetic algorithms.
Jean-Sébastien Coron. Resistance Against Differential Power Analysis For Elliptic Curve Cryptosystems. In Cryptographic Hardware and Embedded Systems (CHES) 1999.
Introduces several countermeasures against power analyses, the third of which is the randomized projective coordinate technique used in Sweet B (often described as "Coron's third countermeasure").
Tetsuya Izu, Bodo Möller, and Tsuyoshi Takagi. Improved Elliptic Curve Multiplication Methods Resistant against Side Channel Attacks. In Progress in Cryptology — INDOCRYPT 2002.
Discusses the SPA and DPA-resistance of the Montgomery ladder for elliptic curves.
Raveen R. Goundar, Marc Joye, Atsuko Miyaji, Matthieu Rivain, and Alexandre Venelli. Scalar multiplication on Weierstraß elliptic curves from Co-Z arithmetic. In Journal of Cryptographic Engineering, Vol. 1, 161 (2011).
Introduces the co-Z Montgomery ladder on Weierstrass curves, and discusses its derivation.
Matthieu Rivain. Fast and Regular Algorithms for Scalar Multiplication over Elliptic Curves. IACR Cryptology ePrint Archive, Report 2011/338.
The main reference for Sweet B. Describes the co-Z addition and initial affine-to-Jacobian point doubling formulae implemented in the library.
Shay Gueron and Vlad Krasnov. Fast prime field elliptic-curve cryptography with 256-bit primes. In Journal of Cryptographic Engineering, Vol. 5, 141 (2011).
Discusses the use of Montgomery multiplication with the P-256 field prime, specifically due to its "Montgomery friendly" property.