Credit: This is developed on top of a modified blst library. Check the original library at blst library.

The arithmetic of computing multiple scalar multiplications in an elliptic curve group then adding them together is called multi-scalar multiplication (MSM). MSM over fixed points dominates the time consumption in the pairing-based trusted setup zero-knowledge succinct non-interactive argument of knowledge (zkSNARK), thus for practical applications we would appreciate fast algorithms to compute it. This paper proposes a bucket set construction that can be utilized in the context of Pippenger’s bucket method to speed up MSM over fixed points with the help of precomputation. If instantiating the proposed construction over BLS12-381 curve, when computing n-scalar multiplications for n = 2e (10 ≤ e ≤ 21), theoretical analysis indicates that the proposed construction saves more than 21% computational cost compared to Pippenger’s bucket method, and that it saves 2.6% to 9.6% computational cost compared to the most popular variant of Pippenger’s bucket method. Finally, our experimental result demonstrates the feasibility of accelerating the computation of MSM over fixed points using large precomputation tables as well as the effectiveness of our new construction.

The code is tested on M1 Mac OS.

The implementation relies on SHA256 hash function in openssl library to generate the random scalars. Before running the subsequent code, one should first install openssl (For example, by running brew install openssl), then modify openssl_include_directory and openssl_library_directory variables in makefile into the correct paths on your computer.

In the terminal under MSM_blst folder, one directly runs ./run.sh group=1 or ./run.sh group=2 to build the modified blst library, complie and run the corresponding test in \mathbb{G}_1 or \mathbb{G}_2 respectively. The number of points $n$ is $2^{10}$ by default. Check the next part for setting configuration.

Note MSM_blst is not compatible with the original blst library since some of the source code in blst has been modified.

group: Compile the corresponding group over in $\mathbb{G}_1$ or $\mathbb{G}_2$ respectively.

config: Select a list of config files to be included in the algorithm.

Examples:

The following command

./run.sh group=1 config=10,11,12 

execute the test for $n$-scalar multiplications in $\mathbb{G}_1$, and it will run through $n = 2^{10}, 2^{11}, 2^{12}$ sequentially.

The following command

./run.sh group=2 config=15,17

execute the test for $n$-scalar multiplications in $\mathbb{G}_2$, and it will run through $n = 2^{15}, 2^{17}$ sequentially.

We have config=10 by defualt.

main_p1.cpp and main_p2.cpp have 4 methods to compute $n$-scalar multiplications. They are

1. Our Construction (CHES 'nh+ 0.21*q'),
2. Our Construction with integral scalar conversion,
3. Pippenger Variant (pippenger_variant_BGMW95),
4. Pippenger (pippenger_blst_built_in).

In main_p1.cpp and main_p2.cpp there is a

/***----***
Configuration
***----***/

snippet. One can decide whether to run the test for a specific algorithm by assigning bool values to TEST_PIPPENGER_Q_OVER_5_CHES and TEST_PIPPENGER_BGMW95.