1. Parallel White Noise Generation on a GPU via Cryptographic Hash
Stanley Tzeng Li-Yi Wei
Microsoft Research Asia

2. What is White Noise? Spatial domain: uniform random number
Frequency domain: white noise
spatial domain
frequency domain

3. Importance Mother of all random numbers
Commonly used, e.g. rand() in C/C++
Major algorithms sequential
e.g. xn = a xn-1 + b mod c
Processors are becoming parallel
GPU, multi-core CPU, Cell
sequential algorithms cannot leverage that

4. Contribution ☺Parallel algorithm for white noises
independent evaluation for every sample
easy implementation as a GPU pixel shader
speed faster than sequential algorithms
quality same or better
usage similar to texture mapping

5. PRNG (Pseudo Random Number Generator)
The main source of randomness in programs
Desirable properties
white noise statistics
repeatable
fast computation
low memory usage

6. borrow cryptographic hash!
Core Idea
input trivially prepared in parallel, e.g. linear ramp
feed input value into hash, independently and in parallel
output white noise
key idea:
borrow cryptographic hash!
input
hash
output

7. Hash
(however nice) input → (unrecognizable) mess

8. Cryptographic Hash A subclass of hash
Commonly used for security applications
e.g. password, digital signature
Properties
irreversible – cannot find input from hash output
decorrelating – similar inputs, dissimilar outputs
uniform probability – all outputs likely to occur

9. Cryptographic Hash - Example
irreversible, decorrelating, uniform probability
CHash ("The quick brown fox jumps over the lazy dog") = 9e107d9d372bb6826bd81d3542a419d6
CHash ("The quick brown fox jumps over the lazy eog")
= ffd93f fbaef4da268dd0e

10. Cryptographic Hash as a PRNG
White noise statistics
CHash is cryptographically secure
Repeatable
CHash is invariant with same input
Fast computation
CHash is parallel + constant cost
Low memory usage
CHash maintains no state
Order-independent i.e. Random accessible
important for parallel GPU applications
hash

11. Which Cryptographic Hash?
Many options
MD5, SHA, RIPEMD, Tiger, block cipher, etc
Desirable properties
white noise quality
fast computation
power-of-2 aligned (output & operations)
pure pixel shader, no state maintenance

12. Our Hash of Choice: MD5 [Rivest 1992]
128-bit outputs and 32-bit operation
Small number of constants fit entirely in shader
Fastest among those satisfying quality criteria
Not 100% secure [Wang and Yu 2005]
but good enough for our goal

13. (bit op, table, arithmetic)
MD5 Algorithm Overview
Scrambling
(bit op, table, arithmetic)
Input
Output
shift table
sin table
64 rounds

14. Performance Bottlenecks for Pixel Shader
Scrambling
(bit op, table, arithmetic)
Input
Output
shift table
sin table
64 rounds

15. (bit op, table, arithmetic)
Our Optimization
Scrambling
(bit op, table, arithmetic)
Input
Output
reduced
shift table
shift table
sin table
sin function
64 rounds
loop unrolling

16. Previous PRNG GPU CPU rand drand48
BBS [Blum et al. 1986, Olano 2005]
O extremely fast
X not good quality
CEICG [Entacher et al. 1998, Sussman et al. 2006]
O decent quality
X processing time varies
AES [NIST 2001, Yamanouchi ]
O invertible (not hash)
CPU
rand
O commonly used
X not good quality
drand48
O better quality
X slower
Mersenne Twister [Matsumoto and Nishimura 1998]
O high quality and fast
X not random accessible

17. Assessing Quality: DIEHARD [Marsaglia 1995]
De facto standard on measuring PRNG quality
Runs 15 different tests on the bits generated
Outputs p-val. If p == 0 || p == 1, fail.
BIRTHDAY SPACINGS TEST, M= 512 N=2**24 LAMBDA=
Results for aes.bin
For a sample of size 500: mean
aes.bin using bits 1 to
duplicate number number
spacings observed expected
6 to INF
Chisquare with 6 d.o.f. = p-value=

18. Cumulative Distribution Function
Shows how data is distributed within set
Given x in data, what % of data values are ≤ x
100%
100 %
0 %
0 %
X=0 1
X=0 1
Normal Distribution
Uniform Distribution

19. Kolmogorov-Smirnov Test
Determines how two sets of data are alike
Looks at max difference D between distribution functions
100 %
not alike
100 %
alike D D
0 %
0 %
X=0 1
X=0 1

20. Assessing Quality: DIEHARD
Run the results of the DIEHARD test (p-value) through a KS-test. Look at D-value.
100
Uniform Distribution Curve
P-value Curve
D-Value D
Smaller D is better quality!
Cumulative Distribution Function

21. Assessing Quality: Power Spectrum
Radial mean: should be uniform
Radial variance: should be low & uniform
Power spectrum density
Radial mean
Radial variance (Anisotropy)

22. Assessing Speed: Batch Rendering
Clock time to generate random bits
n2 x 128 bits image, n = 512, 1024, 2048 and 4096 n2 n2

23. Assessing Speed: Texture Subset (For random accessibility)
A huge virtual texture
clock time for access A B
measure difference
(smaller is better)
220
220 B

24. Test Results: DIEHARD Results
the higher the better
the lower the better

25. Test Results: Power Spectrum Tests
MD5
M. Twister
GPU BBS

26. Test Results: Batch Render Speed

27. Test Results: Texture Subset Speed

28. Trading Quality for Speed
Reducing # of rounds
O faster speed
X lower quality
Rounds
Time(ms)
DIEHARD tests passed
KS D-Val 64
6.3
15/15
0.2029 48
4.7
14/15
0.2042 32
3.1
13/15
0.2295 16
1.6
0.253

29. Applications Fractal terrain (vertex shader) Texture tiling
(fragment shader)

30. Future Work Implement our method in hardware Alternative hashes
very similar to texture unit but much smaller
(no need for cache)
Alternative hashes
ride with advances in cryptographic hash

31. Thank You!