Parallel White Noise Generation on a GPU via Cryptographic Hash - PowerPoint PPT Presentation

Parallel White Noise Generationon a GPU via Cryptographic Hash

Stanley Tzeng Li-Yi Wei

Microsoft Research Asia

• Spatial domain: uniform random number
• Frequency domain: white noise

spatial domain

frequency domain

• 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

• ☺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

• The main source of randomness in programs
• Desirable properties
• white noise statistics
• repeatable
• fast computation
• low memory usage

feed input value into hash, independently and in parallel

output white noise

key idea:

borrow cryptographic hash!

input

hash

output

• (however nice) input → (unrecognizable) mess

• 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

• 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")

= ffd93f16876049265fbaef4da268dd0e

• 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

• 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

• 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

Scrambling

(bit op, table, arithmetic)

Input

Output

shift table

sin table

64 rounds

Scrambling

(bit op, table, arithmetic)

Input

Output

shift table

sin table

64 rounds

Scrambling

(bit op, table, arithmetic)

Input

Output

reduced

shift table

shift table

sin table

sin function

64 rounds

loop unrolling

• GPU
• BBS [Blum et al. 1986, Olano 2005]
• Oextremely 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 2007]
• O invertible (not hash)
• X not good quality

• CPU
• rand
• Ocommonly 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

• 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= 2.0000

Results for aes.bin

For a sample of size 500: mean

aes.bin using bits 1 to 24 2.036

duplicate number number

spacings observed expected

0 66. 67.668

1 130. 135.335

2 148. 135.335

3 80. 90.224

4 44. 45.112

5 20. 18.045

6 to INF 12. 8.282

Chisquare with 6 d.o.f. = 4.50 p-value= .391147

• Shows how data is distributed within set
• Given x in data, what % of data values are ≤ x

100%

100 %

0 %

0 %

X=0

X=0

1

1

Normal Distribution

Uniform Distribution

• 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

• 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!

0

Cumulative Distribution Function

• Radial mean: should be uniform
• Radial variance: should be low & uniform

Power spectrum density

Radial mean

Radial variance (Anisotropy)

• Clock time to generate random bits
• n2 x 128 bits image, n = 512, 1024, 2048 and 4096

n2

n2

A

• A huge virtual texture
• clock time for access A B
• measure difference
• (smaller is better)

220

220

B

the higher the better

the lower the better

MD5

M. Twister

GPU BBS

• Reducing # of rounds
• Ofaster speed
• Xlower quality

Texture tiling

(fragment shader)

Fractal terrain

(vertex shader)

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

Thank You!