Constantine - Constant Time Elliptic Curve Cryptography

This library provides constant-time implementation of elliptic curve cryptography.

Warning ⚠️: The library is in development state and cannot be used at the moment except as a showcase or to start a discussion on modular big integers internals.

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. This can be deactivated with "-d:ConstantineASM=false":

• at a significant performance cost with GCC (~50% slower than Clang).
• at misssed opportunity on recent CPUs that support MULX/ADCX/ADOX instructions (~60% faster than Clang).
• There is a 2.4x perf ratio between using plain GCC vs GCC with inline assembly.

Target audience

The library aims to be a portable, 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

Curves supported

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

The following curves are configured:

Note: At the moment, finite field arithmetic is fully supported but elliptic curve arithmetic is work-in-progress.

ECDH / ECDSA curves

• NIST P-224
• Curve25519
• NIST P-256 / Secp256r1
• Secp256k1 (Bitcoin, Ethereum 1)

Pairing-Friendly curves

Families:

• BN: Barreto-Naerig
• BLS: Barreto-Lynn-Scott
• FKM: Fotiadis-Konstantinou-Martindale

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)
• BN446
• FKM12-447
• BLS12-461
• BN462

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 "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 Assembly (+ GCC)
nimble bench_fp_clang # Using Clang only
nimble bench_fp_gcc   # Using Clang only (very slow)
nimble bench_fp2
# ...
nimble bench_ec_g1
nimble bench_ec_g2

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, for selected benchmarks on the prime field for popular pairing-friendly curves.

Compiled with GCC
Optimization level =>
  no optimization: false
  release: true
  danger: true
  inline assembly: true
Using Constantine with 64-bit limbs
Running on Intel(R) Core(TM) i9-9980XE CPU @ 3.00GHz

⚠️ Cycles measurements are approximate and use the CPU nominal clock: Turbo-Boost and overclocking will skew them.
i.e. a 20% overclock will be about 20% off (assuming no dynamic frequency scaling)

=================================================================================================================

-------------------------------------------------------------------------------------------------------------------------------------------------
Addition                                           Fp[BN254_Snarks]     333333333.333 ops/s             3 ns/op             9 CPU cycles (approx)
Substraction                                       Fp[BN254_Snarks]     500000000.000 ops/s             2 ns/op             8 CPU cycles (approx)
Negation                                           Fp[BN254_Snarks]    1000000000.000 ops/s             1 ns/op             3 CPU cycles (approx)
Multiplication                                     Fp[BN254_Snarks]      71428571.429 ops/s            14 ns/op            44 CPU cycles (approx)
Squaring                                           Fp[BN254_Snarks]      71428571.429 ops/s            14 ns/op            44 CPU cycles (approx)
Inversion (constant-time Euclid)                   Fp[BN254_Snarks]        122579.063 ops/s          8158 ns/op         24474 CPU cycles (approx)
Inversion via exponentiation p-2 (Little Fermat)   Fp[BN254_Snarks]        153822.489 ops/s          6501 ns/op         19504 CPU cycles (approx)
Square Root + square check (constant-time)         Fp[BN254_Snarks]        153491.942 ops/s          6515 ns/op         19545 CPU cycles (approx)
Exp curve order (constant-time) - 254-bit          Fp[BN254_Snarks]        104580.632 ops/s          9562 ns/op         28687 CPU cycles (approx)
Exp curve order (Leak exponent bits) - 254-bit     Fp[BN254_Snarks]        153798.831 ops/s          6502 ns/op         19506 CPU cycles (approx)
-------------------------------------------------------------------------------------------------------------------------------------------------
Addition                                           Fp[BLS12_381]        250000000.000 ops/s             4 ns/op            14 CPU cycles (approx)
Substraction                                       Fp[BLS12_381]        250000000.000 ops/s             4 ns/op            13 CPU cycles (approx)
Negation                                           Fp[BLS12_381]       1000000000.000 ops/s             1 ns/op             4 CPU cycles (approx)
Multiplication                                     Fp[BLS12_381]         35714285.714 ops/s            28 ns/op            84 CPU cycles (approx)
Squaring                                           Fp[BLS12_381]         35714285.714 ops/s            28 ns/op            85 CPU cycles (approx)
Inversion (constant-time Euclid)                   Fp[BLS12_381]            43763.676 ops/s         22850 ns/op         68552 CPU cycles (approx)
Inversion via exponentiation p-2 (Little Fermat)   Fp[BLS12_381]            63983.620 ops/s         15629 ns/op         46889 CPU cycles (approx)
Square Root + square check (constant-time)         Fp[BLS12_381]            63856.960 ops/s         15660 ns/op         46982 CPU cycles (approx)
Exp curve order (constant-time) - 255-bit          Fp[BLS12_381]            68535.399 ops/s         14591 ns/op         43775 CPU cycles (approx)
Exp curve order (Leak exponent bits) - 255-bit     Fp[BLS12_381]            93222.709 ops/s         10727 ns/op         32181 CPU cycles (approx)
-------------------------------------------------------------------------------------------------------------------------------------------------
Notes:
  - Compilers:
    Compilers are severely limited on multiprecision arithmetic.
    Inline Assembly is used by default (nimble bench_fp).
    Bench without assembly can use "nimble bench_fp_gcc" or "nimble bench_fp_clang".
    GCC is significantly slower than Clang on multiprecision arithmetic due to catastrophic handling of carries.
  - The simplest operations might be optimized away by the compiler.
  - Fast Squaring and Fast Multiplication are possible if there are spare bits in the prime representation (i.e. the prime uses 254 bits out of 256 bits)

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

As a workaround key procedures use inline assembly.

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).

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.

Example tradeoff

Unrolling Montgomery Multiplication brings about 15% performance improvement which translate to ~15% on all operations in Constantine as field multiplication bottlenecks all cryptographic primitives. This is considered a worthwhile tradeoff on all but the most constrained CPUs with those CPUs probably being 8-bit or 16-bit.

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.