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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, 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
  • 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

Intel i7-4790K (4.2GHz)

MUM City64 Spooky xxHash64 SipHash24
5 bytes (1,280M strings) 7.63s 10.95s 10.88s 19.28s 21.61s
8 bytes (1,280M strings) 5.52s 10.83s 10.35s 16.22s 26.43s
16 bytes (1,280M strings) 7.38s 11.15s 18.29s 18.07s 31.11s
32 bytes (1,280M strings) 8.06s 13.15s 18.50s 19.06s 42.73s
64 bytes (1,280M strings) 11.59s 13.75s 26.59s 25.74s 63.38s
128 bytes (1,280M strings) 17.16s 18.99s 42.88s 18.99s 106.77s
1KB (100M strings) 6.61s 6.37s 8.80s 8.45s 53.20s

Power7 (3.55GHz)

MUM City64 Spooky xxHash64 SipHash24
5 bytes (1,280M strings) 17.59s 26.01s 28.52s 82.54s 48.23s
8 bytes (1,280M strings) 8.63s 26.61s 27.04s 77.30s 55.64s
16 bytes (1,280M strings) 18.30s 26.07s 41.77s 79.51s 62.53s
32 bytes (1,280M strings) 19.29s 30.03s 41.01s 76.90s 84.89s
64 bytes (1,280M strings) 21.95s 33.71s 57.68s 82.59s 122.49s
128 bytes (1,280M strings) 30.20s 57.68s 90.56s 96.09s 197.45s
1KB (100M strings) 14.31s 16.93s 16.26s 20.96s 96.78s

AARCH64 (APM X-Gene)

MUM City64 Spooky xxHash64 SipHash24
5 bytes (1,280M strings) 24.06s 27.27s 29.41s 55.08s 66.66s
8 bytes (1,280M strings) 14.43s 27.27s 27.80s 56.69s 68.44s
16 bytes (1,280M strings) 22.94s 28.87s 43.85s 62.57s 78.08s
32 bytes (1,280M strings) 29.94s 33.69s 47.06s 79.14s 98.93s
64 bytes (1,280M strings) 43.83s 40.64s 64.70s 90.38s 141.71s
128 bytes (1,280M strings) 70.05s 74.33s 98.93s 112.83s 235.83s
1KB (100M strings) 34.05s 22.91s 24.23s 33.42s 104.98s

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
  • Here is the speed of the crypto hash functions on 4.2 GHz Intel i7-4790K:
MUM512 SHA2 SHA3
10 bytes (20 M texts) 0.73s 0.65s 1.13s
100 bytes (20 M texts) 1.08s 0.65s 2.21s
1000 bytes (20 M texts) 4.01s 4.82s 15.3s
10000 bytes (5 M texts) 7.86s 11.74s 37.3s

Pseudo-random generators

  • Files mum-prgn.h and mum512-prng.h provides pseudo-random functions based on MUM and MUM512 hash functions
  • All PRNGs pass 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 passes NIST Statistical Test Suite
    • 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

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