Leopard-RS

MDS Reed-Solomon Erasure Correction Codes for Large Data in C

Leopard-RS is a fast library for Erasure Correction Coding. From a block of equally sized original data pieces, it generates recovery symbols that can be used to recover lost original data.

Motivation:

It gets slower as O(N Log N) in the input data size, and its inner loops are vectorized using the best approaches available on modern processors, using the fastest finite fields (8-bit or 16-bit Galois fields with Cantor basis {2}).

It sets new speed records for MDS encoding and decoding of large data, achieving over 1.2 GB/s to encode with the AVX2 instruction set on a single core.

Example applications are data recovery software and data center replication.

Encoder API:

Preconditions:

• The original and recovery data must not exceed 65536 pieces.
• The recovery_count <= original_count.
• The buffer_bytes must be a multiple of 64.
• Each buffer should have the same number of bytes.
• Even the last piece must be rounded up to the block size.

#include "leopard.h"

For full documentation please read leopard.h.

• leo_init() : Initialize library.
• leo_encode_work_count() : Calculate the number of work_data buffers to provide to leo_encode().
• leo_encode(): Generate recovery data.

Decoder API:

For full documentation please read leopard.h.

• leo_init() : Initialize library.
• leo_decode_work_count() : Calculate the number of work_data buffers to provide to leo_decode().
• leo_decode() : Recover original data.

Benchmarks:

On the the MacBook Pro 15-Inch (Mid-2015) featuring a 22 nm "Haswell/Crystalwell" 2.8 GHz Intel Core i7-4980HQ processor, compiled with Visual Studio 2017 RC:

Leopard Encoder(8.192 MB in 128 pieces, 128 losses): Input=2102.67 MB/s, Output=2102.67 MB/s
Leopard Decoder(8.192 MB in 128 pieces, 128 losses): Input=686.212 MB/s, Output=686.212 MB/s

Leopard Encoder(64 MB in 1000 pieces, 200 losses): Input=2194.94 MB/s, Output=438.988 MB/s
Leopard Decoder(64 MB in 1000 pieces, 200 losses): Input=455.633 MB/s, Output=91.1265 MB/s

Leopard Encoder(2097.15 MB in 32768 pieces, 32768 losses): Input=451.168 MB/s, Output=451.168 MB/s
Leopard Decoder(2097.15 MB in 32768 pieces, 32768 losses): Input=190.471 MB/s, Output=190.471 MB/s

2 GB of 64 KB pieces encoded in 4.6 seconds, and worst-case recovery in 11 seconds.

More benchmark results are available here: https://github.com/catid/leopard/blob/master/Benchmarks.md

Comparisons:

There is another library FastECC by Bulat-Ziganshin that should have similar performance: https://github.com/Bulat-Ziganshin/FastECC. Both libraries implement the same high-level algorithm in {3}, while Leopard implements the newer polynomial basis GF(2^r) approach outlined in {1}, and FastECC uses complex finite fields modulo special primes. There are trade-offs that may make either approach preferable based on the application:

• Older processors do not support SSSE3 and FastECC supports these processors better.
• FastECC supports data sets above 65,536 pieces as it uses 32-bit finite field math.
• Leopard does not require expanding the input or output data to make it fit in the field, so it can be more space efficient.

FFT Data Layout:

We pack the data into memory in this order:

[Recovery Data (Power of Two = M)] [Original Data] [Zero Padding out to 65536]

For encoding, the placement is implied instead of actual memory layout. For decoding, the layout is explicitly used.

Encoder algorithm:

The encoder is described in {3}. Operations are done O(K Log M), where K is the original data size, and M is up to twice the size of the recovery set.

Roughly in brief:

Recovery = FFT( IFFT(Data_0) xor IFFT(Data_1) xor ... )

It walks the original data M chunks at a time performing the IFFT. Each IFFT intermediate result is XORed together into the first M chunks of the data layout. Finally the FFT is performed.

Encoder optimizations:

• The first IFFT can be performed directly in the first M chunks.
• The zero padding can be skipped while performing the final IFFT. Unrolling is used in the code to accomplish both these optimizations.
• The final FFT can be truncated also if recovery set is not a power of 2. It is easy to truncate the FFT by ending the inner loop early.
• The decimation-in-time (DIT) FFT is employed to calculate two layers at a time, rather than writing each layer out and reading it back in for the next layer of the FFT.

Decoder algorithm:

The decoder is described in {1}. Operations are done O(N Log N), where N is up to twice the size of the original data as described below.

Roughly in brief:

Original = -ErrLocator * FFT( Derivative( IFFT( ErrLocator * ReceivedData ) ) )

Precalculations:

At startup initialization, FFTInitialize() precalculates FWT(L) as described by equation (92) in {1}, where L = Log[i] for i = 0..Order, Order = 256 or 65536 for FF8/16. This is stored in the LogWalsh vector.

It also precalculates the FFT skew factors (s_i) as described by equation (28). This is stored in the FFTSkew vector.

For memory workspace N data chunks are needed, where N is a power of two at or above M + K. K is the original data size and M is the next power of two above the recovery data size. For example for K = 200 pieces of data and 10% redundancy, there are 20 redundant pieces, which rounds up to 32 = M. M + K = 232 pieces, so N rounds up to 256.

Online calculations:

At runtime, the error locator polynomial is evaluated using the Fast Walsh-Hadamard transform as described in {1} equation (92).

At runtime the data is explicit laid out in workspace memory like this:

[Recovery Data (Power of Two = M)] [Original Data (K)] [Zero Padding out to N]

Data that was lost is replaced with zeroes. Data that was received, including recovery data, is multiplied by the error locator polynomial as it is copied into the workspace.

The IFFT is applied to the entire workspace of N chunks. Since the IFFT starts with pairs of inputs and doubles in width at each iteration, the IFFT is optimized by skipping zero padding at the end until it starts mixing with non-zero data.

The formal derivative is applied to the entire workspace of N chunks.

The FFT is applied to the entire workspace of N chunks. The FFT is optimized by only performing intermediate calculations required to recover lost data. Since it starts wide and ends up working on adjacent pairs, at some point the intermediate results are not needed for data that will not be read by the application. This optimization is implemented by the ErrorBitfield class.

Finally, only recovered data is multiplied by the negative of the error locator polynomial as it is copied into the front of the workspace for the application to retrieve.

Finite field arithmetic optimizations:

For faster finite field multiplication, large tables are precomputed and applied during encoding/decoding on 64 bytes of data at a time using SSSE3 or AVX2 vector instructions and the ALTMAP approach from Jerasure.

Addition in this finite field is XOR, and a vectorized memory XOR routine is also used.

References:

This library implements an MDS erasure code introduced in this paper:

    {1} S.-J. Lin, T. Y. Al-Naffouri, Y. S. Han, and W.-H. Chung,
    "Novel Polynomial Basis with Fast Fourier Transform
	and Its Application to Reed-Solomon Erasure Codes"
    IEEE Trans. on Information Theory, pp. 6284-6299, November, 2016.

    {2} D. G. Cantor, "On arithmetical algorithms over finite fields",
    Journal of Combinatorial Theory, Series A, vol. 50, no. 2, pp. 285-300, 1989.

    {3} Sian-Jheng Lin, Wei-Ho Chung, "An Efficient (n, k) Information
    Dispersal Algorithm for High Code Rate System over Fermat Fields,"
    IEEE Commun. Lett., vol.16, no.12, pp. 2036-2039, Dec. 2012.

    {4} Plank, J. S., Greenan, K. M., Miller, E. L., "Screaming fast Galois Field
    arithmetic using Intel SIMD instructions."  In: FAST-2013: 11th Usenix
    Conference on File and Storage Technologies, San Jose, 2013

Some papers are mirrored in the /docs/ folder.

Credits

Inspired by discussion with:

• Sian-Jhen Lin sjhenglin@gmail.com : Author of {1} {3}, basis for Leopard
• Bulat Ziganshin bulat.ziganshin@gmail.com : Author of FastECC
• Yutaka Sawada tenfon@outlook.jp : Author of MultiPar

Software by Christopher A. Taylor mrcatid@gmail.com

Please reach out if you need support or would like to collaborate on a project.