Constantine - Fast, compact, hardened Pairing-Based Cryptography

“A cryptographic system should be secure even if everything about the system, except the key, is public knowledge.”
— Auguste Kerckhoffs

This library provides constant-time implementation of cryptography protocols with a particular focus on pairing-based cryptography as used in blockchains and zero-knowledge protocols.

The implementations are accompanied with SAGE code used as reference implementation and test vectors generators before writing highly optimized routines implemented in the Nim language

The library is in development state and high-level wrappers or example protocols are not available yet.

Target audience

The library aims to be a fast, compact and hardened library for elliptic curve cryptography needs, in particular for blockchain protocols and zero-knowledge proofs system.

The library focuses on following properties:

• constant-time (not leaking secret data via side-channels)
• performance
• generated code size, datatype size and stack usage

in this order.

Protocols

Protocols are a set of routines, designed for specific goals or a combination thereof:

• confidentiality: only the intended receiver of a message can read it
• authentication: the other party in the communication is the expected part
• integrity: the received message has not been tampered with
• non-repudiation: the sender of a message cannot repudiated it

Protocols to address these goals, (authenticated) encryption, signature, traitor-tracing, etc are designed.
Note: some goals might be mutually exclusive, for example "plausible deniability" and "non-repudiation".

After installation, the available high-level protocols are:

• Ethereum EVM precompiles on BN254_Snarks (also called alt_bn128 or bn256 in Ethereum)

  import constantine/ethereum_evm_precompiles

• BLS signature on BLS12-381 G2 as used in Ethereum 2. Cryptographic suite: BLS_SIG_BLS12381G2_XMD:SHA-256_SSWU_RO_POP_

  This scheme is also used in the following blockchains: Algorand, Chia, Dfinity, Filecoin, Tezos, Zcash. They may have their pubkeys on G1 and signatures on G2 like Ethereum or the other way around.

  Parameter discussion:

  As Ethereum validators' pubkeys are duplicated, stored and transmitter over and over in the protocol, having them be as small as possible was important. On another hand, BLS signatures were first popularized due to their succinctness. And having signatures on G1 is useful when short signatures are desired, in embedded for example.

• SHA256 hash

• ...

Curves supported in the backend

The backend, unlike protocols, is not public. Here be dragons.

At the moment the following curves are implemented, adding a new curve only requires adding the prime modulus and its bitsize in constantine/config/curves.nim.

The following curves are configured:

• Pairing-Friendly curves
  • BN254_Nogami
  • BN254_Snarks (Zero-Knowledge Proofs, Snarks, Starks, Zcash, Ethereum 1)
  • BLS12-377 (Zexe)
  • BLS12-381 (Algorand, Chia Networks, Dfinity, Ethereum 2, Filecoin, Zcash Sapling)
  • BW6-671 (Celo, EY Blockchain) (Pairings are WIP)
    BLS12-377 is embedded in BW6-761 for one layer proof composition in zk-SNARKS.
• Embedded curves
  • Jubjub, a curve embedded in BLS12-381 scalar field to be used in zk-SNARKS circuits.
  • Bandersnatch, a more efficient curve embedded in BLS12-381 scalar field to be used in zk-SNARKS circuits.
• Other curves
  • Edwards25519, used in ed25519 and X25519 from TLS 1.3 protocol and the Signal protocol. With Ristretto, it can be used in bulletproofs.

Installation

You can install the developement version of the library through nimble with the following command

nimble install https://github.com/mratsim/constantine@#master

For speed it is recommended to prefer Clang, MSVC or ICC over GCC (see Compiler-caveats).

Further if using GCC, GCC 7 at minimum is required, previous versions generated incorrect add-with-carry code.

On x86-64, inline assembly is used to workaround compilers having issues optimizing large integer arithmetic, and also ensure constant-time code.

Security

Hardening an implementation against all existing and upcoming attack vectors is an extremely complex task. The library is provided as is, without any guarantees at least until:

• it gets audited
• formal proofs of correctness are produced
• formal verification of constant-time implementation is possible

Defense against common attack vectors are provided on a best effort basis.

Attackers may go to great lengths to retrieve secret data including:

• Timing the time taken to multiply on an elliptic curve
• Analysing the power usage of embedded devices
• Detecting cache misses when using lookup tables
• Memory attacks like page-faults, allocators, memory retention attacks

This is would be incomplete without mentioning that the hardware, OS and compiler actively hinder you by:

• Hardware: sometimes not implementing multiplication in constant-time.
• OS: not providing a way to prevent memory paging to disk, core dumps, a debugger attaching to your process or a context switch (coroutines) leaking register data.
• Compiler: optimizing away your carefully crafted branchless code and leaking server secrets or optimizing away your secure erasure routine which is deemed "useless" because at the end of the function the data is not used anymore.

A growing number of attack vectors is being collected for your viewing pleasure at https://github.com/mratsim/constantine/wiki/Constant-time-arithmetics

Disclaimer

Constantine's authors do their utmost to implement a secure cryptographic library in particular against remote attack vectors like timing attacks.

Please note that Constantine is provided as-is without guarantees. Use at your own risks.

Thorough evaluation of your threat model, the security of any cryptographic library you are considering, and the secrets you put in jeopardy is strongly advised before putting data at risk. The author would like to remind users that the best code can only mitigate but not protect against human failures which are the weakest link and largest backdoors to secrets exploited today.

Security disclosure

TODO

Performance

High-performance is a sought out property. Note that security and side-channel resistance takes priority over performance.

New applications of elliptic curve cryptography like zero-knowledge proofs or proof-of-stake based blockchain protocols are bottlenecked by cryptography.

In blockchain

Ethereum 2 clients spent or use to spend anywhere between 30% to 99% of their processing time verifying the signatures of block validators on R&D testnets Assuming we want nodes to handle a thousand peers, if a cryptographic pairing takes 1ms, that represents 1s of cryptography per block to sign with a target block frequency of 1 every 6 seconds.

In zero-knowledge proofs

According to https://medium.com/loopring-protocol/zksnark-prover-optimizations-3e9a3e5578c0 a 16-core CPU can prove 20 transfers/second or 10 transactions/second. The previous implementation was 15x slower and one of the key optimizations was changing the elliptic curve cryptography backend. It had a direct implication on hardware cost and/or cloud computing resources required.

Measuring performance

To measure the performance of Constantine

git clone https://github.com/mratsim/constantine
nimble bench_fp             # Using default compiler + Assembly
nimble bench_fp_clang       # Using Clang + Assembly (recommended)
nimble bench_fp_gcc         # Using GCC + Assembly (decent)
nimble bench_fp_clang_noasm # Using Clang only (acceptable)
nimble bench_fp_gcc         # Using GCC only (slowest)
nimble bench_fp2
# ...
nimble bench_ec_g1_clang
nimble bench_ec_g2_clang
nimble bench_pairing_bn254_nogami_clang
nimble bench_pairing_bn254_snarks_clang
nimble bench_pairing_bls12_377_clang
nimble bench_pairing_bls12_381_clang

# And per-curve summaries
nimble bench_summary_bn254_nogami_clang
nimble bench_summary_bn254_snarks_clang
nimble bench_summary_bls12_377_clang
nimble bench_summary_bls12_381_clang

As mentioned in the Compiler caveats section, GCC is up to 2x slower than Clang due to mishandling of carries and register usage.

On my machine i9-11980HK (8 cores 2.6GHz, turbo 5GHz), for Clang + Assembly, all being constant-time (including scalar multiplication, square root and inversion).

BLS12_381 (Clang + inline Assembly)

--------------------------------------------------------------------------------------------------------------------------------------------------------
EC ScalarMul 255-bit G1             ECP_ShortW_Prj[Fp[BLS12_381]]                  16086.740 ops/s         62163 ns/op        205288 CPU cycles (approx)
EC ScalarMul 255-bit G1             ECP_ShortW_Jac[Fp[BLS12_381]]                  16670.834 ops/s         59985 ns/op        198097 CPU cycles (approx)
EC ScalarMul 255-bit G2             ECP_ShortW_Prj[Fp2[BLS12_381]]                  8333.403 ops/s        119999 ns/op        396284 CPU cycles (approx)
EC ScalarMul 255-bit G2             ECP_ShortW_Jac[Fp2[BLS12_381]]                  9300.682 ops/s        107519 ns/op        355071 CPU cycles (approx)
--------------------------------------------------------------------------------------------------------------------------------------------------------
Miller Loop BLS12                   BLS12_381                                       5102.223 ops/s        195993 ns/op        647251 CPU cycles (approx)
Final Exponentiation BLS12          BLS12_381                                       4209.109 ops/s        237580 ns/op        784588 CPU cycles (approx)
Pairing BLS12                       BLS12_381                                       2343.045 ops/s        426795 ns/op       1409453 CPU cycles (approx)
--------------------------------------------------------------------------------------------------------------------------------------------------------
Hash to G2 (Draft #11)              BLS12_381                                       6558.495 ops/s        152474 ns/op        503531 CPU cycles (approx)
--------------------------------------------------------------------------------------------------------------------------------------------------------

Why Nim

The Nim language offers the following benefits for cryptography:

• Compilation to machine code via C or C++ or alternatively compilation to Javascript. Easy FFI to those languages.
  • Obscure embedded devices with proprietary C compilers can be targeted.
  • WASM can be targeted.
• Performance reachable in C is reachable in Nim, easily.
• Rich type system: generics, dependent types, mutability-tracking and side-effect analysis, borrow-checking, compiler enforced distinct types (Miles != Meters, SecretBool != bool and SecretWord != uint64).
• Compile-time evaluation, including parsing hex string, converting them to BigInt or Finite Field elements and doing bigint operations.
• Assembly support either inline or __attribute__((naked)) or a simple {.compile: "myasm.S".} away
• No GC if no GC-ed types are used (automatic memory management is set at the type level and optimized for latency/soft-realtime by default and can be totally deactivated).
• Procedural macros working directly on AST to
  • create generic curve configuration,
  • derive constants
  • write a size-independent inline assembly code generator
• Upcoming proof system for formal verification via Z3 (DrNim, Correct-by-Construction RFC)

Compiler caveats

Unfortunately compilers and in particular GCC are not very good at optimizing big integers and/or cryptographic code even when using intrinsics like addcarry_u64.

Compilers with proper support of addcarry_u64 like Clang, MSVC and ICC may generate code up to 20~25% faster than GCC.

This is explained by the GMP team: https://gmplib.org/manual/Assembly-Carry-Propagation.html and can be reproduced with the following C code.

See https://gcc.godbolt.org/z/2h768y

#include <stdint.h>
#include <x86intrin.h>

void add256(uint64_t a[4], uint64_t b[4]){
  uint8_t carry = 0;
  for (int i = 0; i < 4; ++i)
    carry = _addcarry_u64(carry, a[i], b[i], &a[i]);
}

GCC

add256:
        movq    (%rsi), %rax
        addq    (%rdi), %rax
        setc    %dl
        movq    %rax, (%rdi)
        movq    8(%rdi), %rax
        addb    $-1, %dl
        adcq    8(%rsi), %rax
        setc    %dl
        movq    %rax, 8(%rdi)
        movq    16(%rdi), %rax
        addb    $-1, %dl
        adcq    16(%rsi), %rax
        setc    %dl
        movq    %rax, 16(%rdi)
        movq    24(%rsi), %rax
        addb    $-1, %dl
        adcq    %rax, 24(%rdi)
        ret

Clang

add256:
        movq    (%rsi), %rax
        addq    %rax, (%rdi)
        movq    8(%rsi), %rax
        adcq    %rax, 8(%rdi)
        movq    16(%rsi), %rax
        adcq    %rax, 16(%rdi)
        movq    24(%rsi), %rax
        adcq    %rax, 24(%rdi)
        retq

Inline assembly

While using intrinsics significantly improve code readability, portability, auditability and maintainability, Constantine use inline assembly on x86-64 to ensure performance portability despite poor optimization (for GCC) and also to use dedicated large integer instructions MULX, ADCX, ADOX that compilers cannot generate.

The speed improvement on finite field arithmetic is up 60% with MULX, ADCX, ADOX on BLS12-381 (6 limbs).

Finally assembly is a requirement to ensure constant-time property and to avoid compilers turning careful branchless code into branches, see Fighting the compiler (wiki)

In summary, pure C/C++/Nim implies:

• a smart compiler might unravel the constant time bit manipulation and reintroduce branches.
• a significant performance cost with GCC (~50% slower than Clang).
• missed opportunities on recent CPUs that support MULX/ADCX/ADOX instructions (~60% faster than Clang).
• 2.4x perf ratio between using plain GCC vs GCC with inline assembly.

Sizes: code size, stack usage

Thanks to 10x smaller key sizes for the same security level as RSA, elliptic curve cryptography is widely used on resource-constrained devices.

Constantine is actively optimize for code-size and stack usage. Constantine does not use heap allocation.

At the moment Constantine is optimized for 32-bit and 64-bit CPUs.

When performance and code size conflicts, a careful and informed default is chosen. In the future, a compile-time flag that goes beyond the compiler -Os might be provided.

License

Licensed and distributed under either of

• MIT license: LICENSE-MIT or http://opensource.org/licenses/MIT

or

• Apache License, Version 2.0, (LICENSE-APACHEv2 or http://www.apache.org/licenses/LICENSE-2.0)

at your option. This file may not be copied, modified, or distributed except according to those terms.

This library has no external dependencies. In particular GMP is used only for testing and differential fuzzing and is not linked in the library.