TheIronBorn / mum-hash

Hashing functions and PRNGs based on them

MUM Hash

• MUM hash is a fast non-cryptographic hash function suitable for different hash table implementations
• MUM means MUltiply and Mix
  • It is a name of the base transformation on which hashing is implemented
  • Modern processors have a fast logic to do long number multiplications
  • It is very attractable to use it for fast hashing
    • For example, 64x64-bit multiplication can do the same work as 32 shifts and additions
  • I'd like to call it Multiply and Reduce. Unfortunately, MUR (MUltiply and Rotate) is already taken for famous hashing technique designed by Austin Appleby
  • I've chosen the name also as I am releasing it on Mother's day
• MUM hash passes all SMHasher tests
  • For comparison, only 4 out of 15 non-cryptographic hash functions in SMHasher passes the tests, e.g. well known FNV, Murmur2, Lookup, and Superfast hashes fail the tests
• MUM algorithm is simpler than City64 and Spooky ones
• MUM is specifically designed for 64-bit CPUs (Sorry, I did not want to spend my time on dying architectures)
  • Still MUM will work for 32-bit CPUs and it will be sometimes faster Spooky and City
• On x86-64 MUM hash is faster than City64 and Spooky on all tests except for one test for the bulky speed
  • Starting with 240-byte strings, City uses Intel SSE4.2 crc32 instruction
  • I could use the same instruction but I don't want to complicate the algorithm
  • In typical scenario, such long strings are rare. Usually another interface (see mum_hash_step) is used for hashing big data structures
• MUM has a fast startup. It is particular good to hash small keys which are a majority of hash table applications

MUM implementation details

• Input 64-bit data are randomized by 64x64->128 bit multiplication and mixing high- and low-parts of the multiplication result by using an addition. The result is mixed with the current state by using XOR
  • Instead of the addition for mixing high- and low- parts, XOR could be used
    • Using the addition instead of XOR improves performance by about 10% on Haswell and Power7
• Prime numbers randomly generated with the equal probability of their bit values are used for the multiplication
• When all primes are used once, the state is randomized and the same prime numbers are used again for subsequent data randomization
• Major loop is transformed to be unrolled by compiler to benefit from the compiler instruction scheduling optimization and OOO instruction execution in modern CPUs
• x86-64 AVX2 insn MULX is used on processors where it is implemented
  • Using MULX increases the bulk speed upto 25%
  • Although on modern Intel processors MULQ takes 3-cylces vs. 4 for MULX, MULX permits more freedom in insn scheduling as it uses less fixed registers. That is a reason for the improvement
  • To make the code portable, GCC function multiversioning is used. The best code is chosen depending on the used processor features
• AARCH64 128-bit result multiplication is very slow as it is implemented by a GCC library function
  • To use only 2 insns for such multiplication one GCC asm extension was added

MUM benchmarking vs Spooky, City64, xxHash64, MetroHash64, and SipHash24

• Here are the results of benchmarking MUM and the fastest non-cryptographic hash functions I know:
  • Google City64 (sources are taken from SMHasher)
  • Bob Jenkins Spooky (sources are taken from SMHasher)
  • Yann Collet's xxHash64 (sources are taken from the original repository)
• Murmur hash functions are slower so I don't compare it here
• I also added J. Aumasson and D. Bernstein's SipHash24 for the comparison as it is a popular choice for hash table implementation these days
• Update:
  • A metro hash was added as people asked and as metro hash is claimed to be the fastest hash function
    • metro hash is not portable as others functions as it does not deal with unaligned accesses problem on some targets
    • metro hash will produce different hash for LE/BE targets
    • some people on hackernews pointed out that the algorithm is very close to xxHash one but still it is much faster xxHash
• Measurements were done on 3 different architecture machines:
  • 4.2 GHz Intel i7-4790K
  • 3.55 GHz Power7
  • APM X-Gene Mustang Board
• Each test was run 3 times and the minimal time was taken
  • GCC-4.8 was used for PPC and AARCH64 machines and GCC-4.9 was used for Intel one
  • -O3 was used for all compilations
  • The strings were generated by rand calls
  • The strings were aligned to see a hashing speed better
  • No constant propagation for string length is forced. Otherwise, the results for MUM hash would be even better
  • Update: Some people complaint that my comparison is unfair as most hash functions are not inlined
    • I believe that the interface is the part of the implementation. So when the interface does not provide an easy way for inlining, it is an implementation pitfall
    • Still to address the complaints I added -flto for benchmarking all hash functions excluding MUM. This option makes cross-file inlining
    • xxHash64 results became worse for small strings and better for the bulk speed test
    • All results for other functions improved, sometimes quite a lot

Intel i7-4790K (4.2GHz)

	MUM	City64	Spooky	xxHash64	Metro64	SipHash24
5 bytes (1,280M strings)	7.74s	10.80s	9.22s	9.57s	8.23s	21.56s
8 bytes (1,280M strings)	5.52s	10.80s	7.42s	9.20s	4.88s	26.67s
16 bytes (1,280M strings)	7.45s	9.42s	18.29s	11.05s	7.32s	25.89s
32 bytes (1,280M strings)	8.11s	10.84s	17.73s	16.47s	13.51s	40.43s
64 bytes (1,280M strings)	11.51s	12.01s	23.24s	18.03s	14.55s	61.75s
128 bytes (1,280M strings)	17.07s	16.55s	41.26s	21.14s	16.90s	104.97s
1KB (100M strings)	6.61s	6.53s	8.31s	6.67s	6.18s	53.86s

Power7 (3.55GHz)

	MUM	City64	Spooky	xxHash64	Metro64	SipHash24
5 bytes (1,280M strings)	17.06s	26.31s	21.44s	24.63s	23.21s	41.33s
8 bytes (1,280M strings)	8.64s	26.34s	14.41s	20.16s	10.44s	51.19s
16 bytes (1,280M strings)	17.53s	22.68s	22.68s	29.75s	15.87s	61.50s
32 bytes (1,280M strings)	18.63s	25.56s	29.76s	37.07s	26.64s	79.15s
64 bytes (1,280M strings)	24.26s	27.72s	49.15s	39.53s	28.44s	118.20s
128 bytes (1,280M strings)	32.82s	38.16s	78.92s	47.45s	31.82s	193.42s
1KB (100M strings)	9.25s	15.24s	14.15s	27.16s	11.00s	96.38s

AARCH64 (APM X-Gene)

	MUM	City64	Spooky	xxHash64	Metro64	SipHash24
5 bytes (1,280M strings)	22.40s	27.21s	21.34s	27.21s	19.73s	54.41s
8 bytes (1,280M strings)	15.47s	27.21s	17.60s	25.07s	13.87s	67.22s
16 bytes (1,280M strings)	22.40s	24.54s	32.54s	48.55s	19.20s	83.23s
32 bytes (1,280M strings)	30.41s	28.81s	33.07s	61.65s	36.27s	118.97s
64 bytes (1,280M strings)	47.48s	32.01s	56.02s	76.82s	37.88s	186.73s
128 bytes (1,280M strings)	80.56s	45.88s	91.23s	107.23s	42.14s	331.94s
1KB (100M strings)	38.39s	24.51s	22.55s	44.85s	29.59s	170.01s

Vectorization

• A major loop in function _mum_hash_aligned could be vectorized using vector multiplication, addition, and shuffle instructions
• Unfortunately, x86-64 CPUs currently does not have vector multiplication 64 x 64-bit -> 128-bit
• AVX2 CPUs only have vector multiplication 32 x 32-bit -> 64-bit
  • One such vector instruction makes 4 multiplication which is roughly equivalent what one MULQ/MULX insn does but it has bigger latency time than MULQ/MULX
  • Therefore all my vectorized code I tried was slower
• If Intel introduces a new vector insn for 64 x 64-bit -> 128-bit multiplication, potentially it could increase MUM speed up to 2 times (may be less as it requires to provide aligned data)

Using cryptographic vs. non-cryptographic hash function

• People worrying about denial attacks based on generating hash collisions started to use cryptographic hash functions in hash tables
• Cryptographic functions are very slow
  • sha1 is about 20-30 slower than MUM and City on the bulk speed tests
  • The new fastest cryptographic hash function SipHash is up to 10 times slower
• MUM is also resistant to preimage attack (finding a string with given hash)
  • To make hard moving to previous state values we use mostly 1-to-1 one way function lo(x*C) + hi(x*C) where C is a constant. Brute force solution of equation f(x) = a probably requires 2^63 tries. Another used function equation x ^ y = a has a 2^64 solutions. It complicates finding the overal solution further
• If somebody is not convinced, you can use randomly chosen multiplication constants (see function mum_hash_randomize). Finding a string with a given hash even if you know a string with such hash probably will be close to finding two or more solutions of Diophantine equations
• If somebody is still not convinced, you can implement hash tables to recognize the attack and rebuild the table using MUM function with the new multiplication constants
• Analogous approach can be used if you use weak hash function as MurMur or City. Instead of using cryptographic hash functions all the time, hash tables can be implemented to recognize the attack and rebuild the table and start using a cryptographic hash function
• This approach solves the speed problem and permits to switch easily to a new cryptographic hash function if a flaw is found in the old one, e.g. switching from SipHash to SHA2

How to use MUM

• Please just include file mum.h into your C/C++ program and use the following functions:
  • optional mum_hash_randomize for choosing multiplication constants randomly
  • mum_hash_init, mum_hash_step, and mum_hash_finish for hashing complex data structures
  • mum_hash64 for hashing a 64-bit data
  • mum_hash for hashing any continuous block of data
• To compare MUM speed with Spooky, City64, and SipHash24 on your machine go to the directory src and run a script

sh bench

• The script will compile source files and run the tests printing the results

Crypto-hash function MUM512

• MUM is not designed to be a crypto-hash
  • The key (seed) and state are only 64-bit which are not crypto-level ones
  • The result can be different for different targets (BE/LE machines, 32- and 64-bit machines) as for other hash functions, e.g. City (hash can be different on SSE4.2 nad non SSE4.2 targets) or Spooky (BE/LE machines)
    • If you need the same MUM hash independent on the target, please define macro MUM_TARGET_INDEPENDENT_HASH
• There is a variant of MUM called MUM512 which can be a candidate for a crypto-hash function and keyed crypto-hash function and might be interesting for researchers
  • The key is 256-bit
  • The state and the output are 512-bit
  • The block size is 512-bit
  • It uses 128x128->256-bit multiplication which is analogous to about 64 shifts and additions for 128-bit block word instead of 80 rounds of shifts, additions, logical operations for 512-bit block in sha2-512.
• It is only a candidate for a crypto hash function
  • I did not make any differential crypto-analysis or investigated probabilities of different attacks on the hash function (sorry, it is too big job)
    • I might be do this in the future as I am interesting in differential characteristics of the MUM512 base transformation step (128x128-bit multiplications with addition of high and low 128-bit parts)
    • I am interesting also in the right choice of the multiplication constants
    • May be somebody will do the analysis. I will be glad to hear anything. Who knows, may be it can be easily broken as Nimbus cipher.
  • The current code might be also vulnerable to timing attack on systems with varying multiplication instruction latency time. There is no code for now to prevent it
• To compare the MUM512 speed with the speed of SHA-2 (SHA512) and SHA-3 (SHA3-512) go to the directory src and run a script sh bench-crypto
  • SHA-2 and SHA-3 code is taken from RHash
• Update: Blake2 crypto-hash from github.com/BLAKE2/BLAKE2 was added for comparison. I use sse version of 64-bit Blake2 (blake2b).
• Here is the speed of the crypto hash functions on 4.2 GHz Intel i7-4790K:

	MUM512	SHA2	SHA3	Blake2B
10 bytes (20 M texts)	0.73s	0.65s	1.13s	0.80s
100 bytes (20 M texts)	1.08s	0.65s	2.21s	0.81s
1000 bytes (20 M texts)	3.90s	4.82s	15.3s	3.31s
10000 bytes (5 M texts)	7.96s	11.74s	37.3s	7.27s

Pseudo-random generators

• Files mum-prgn.h and mum512-prng.h provides pseudo-random functions based on MUM and MUM512 hash functions
• All PRNGs passed NIST Statistical Test Suite for Random and Pseudorandom Number Generators for Cryptographic Applications (version 2.2.1) with 1000 bitstreams each containing 1M bits
  • Although MUM PRNG pass the test, it is not a cryptographically secure PRNG as the hash function used for it
• To compare the PRNG speeds go to the directory src and run a script sh bench-prng
• For the comparison I wrote crypto-secured Blum Blum Shub PRNG (file bbs-prng.h) and PRNGs based on fast cryto-level hash functions in ChaCha stream cipher (file chacha-prng.h) and SipHash24 (file sip24-prng.h).
  • The additional PRNGs also pass the Statistical Test Suite
• For the comparison I also added the fastest PRNG xoroshiro128+ (it is not a crypto level PRNG)
• Update: I had no intention to tune MUM based PRNG first but after adding xoroshiro128+ and finding how fast it is, I've decided to speedup MUM PRNG
  • I added code to calculate a few PRNs at once to calculate them in parallel
  • I added AVX2 version functions to use faster MULX instruction
  • The new version also passed NIST Statistical Test Suite. It was tested even on bigger data (10K bitstreams each containing 10M bits). The test took several days on i7-4790K
  • The new version is almost 2 times faster the old one and MUM PRN speed became almost the same as xoroshiro128+ one
    • xoroshiro128+ and MUM PRNG functions are inlined in the benchmark program
    • both code without inlining will be visibly slower and the speed difference will be negligible as one PRN calculation takes only about 3.5 machine cycle for xoroshiro128+ and MUM PRN.
• Here is the speed of the PRNGs in millions generated PRNs per second on 4.2 GHz Intel i7-4790K:

	M prns/sec
BBS	0.057
ChaCha	106
SipHash24	383
MUM512	67
MUM	1116
XOROSHIRO128+	1145
GLIBC RAND	169