Generating Random Numbers in Hardware

Two types of random numbers used in computing:

--”true” random numbers:

++generated from a physical source

(e.g., clock)

++sequence cannot be “repeated”

++may not pass mathematical “randomness” tests

--pseudorandom numbers

++generated from a well-defined procedure

++repeatable (good for debugging, e.g.)

++initial value usually chosen by user (“seed”)

++may not give good random behavior

Projects: we want to use pseudorandom numbers

Two common methods used to generate pseudorandom numbers in hardware:

--LFSR (linear feedback shift register)

--CA (Cellular automata)

LFSR (Linear feedback shift register):

Based on polynomials over a finite field

Simplest field: Z2

elements: 0, 1

addition: 0 + 0 = 0; 0 + 1 = 1 + 0 = 1; 1 + 1 = 0

multiplication:0 * 0 = 0; 0 * 1 = 1 * 0 = 0; 1 * 1 = 1

(note: in Z4 with elements 0,1,2,3 we have 2 * 2 = 0—it’s NOT a field!!!)

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• Linear Feedback Shift Register (LFSR):- sequential shift register with combinational logic- feedback provided by selection of points called

taps

Need to use specific LFSR configuration to get “full cycle”:

Need to use a “primitive” polynomial to generate the entire “multiplicative group” (i.e., all 2n – 1 nonzero elements of the field of polynomials of degree n-1 with coefficients in Z2, whose elements can be represented by n-bit numbers)

Example: suppose we have 3-bit numbers c3c2c1 representing

c3x2 + c2x + c1

Field elements: 000 , 001, 010, 011, 100, 101, 110, 111

Seed : 001 “taps” 3,2 (count bits as 3,2,1)

Shift left, low order bit is xor of “taps”

001, 010, 101, 011, 111, 110, 100, 001, …….

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Example:

N = 32:

Taps 32, 22, 2, 1

For each n, there is at least one such primitive polynomial (result from math)

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Example: random number generator for n = 8:

8-bit shift register (shifts left)

Load with SEED which is any nonzero number

shift in XOR of the specified bits (8, 6, 5, 4 for n = 8)

Generate all 255 (28 – 1) nonzero numbers in “random” order, e.g.:

SEED=10101000 gives 10101000, 01010001, 10100011, 01000110, …

Bit 8 Bit 1

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How good are the random numbers generated?Reference: Shruthi Narayanan, M.S. 2005, ATI TechnologiesHardware implementation of genetic algorithm modules for intelligent systems:

Conclusion: use multiple shift registers

Random numbers generated by one shift register

Random numbers generated by multiple shift registers

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• Serial Test Results

32-bit LFSR implemented by [martin]

Martin, P., An Analysis of Random Number Generators for a Hardware Implementation of Genetic Programming using FPGAs and Handel-C, Technical Report, University of Essex, 2002.

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• Multiple Linear Feedback Shift Registers:- n LFSRs of length m are implemented- one-bit from each LFSR is taken to form

n-bit random number

Martin, P., An Analysis of Random Number Generators for a Hardware Implementation of Genetic Programming using FPGAs and Handel-C, Technical Report, University of Essex, 2002.

Another method: use cellular automata to generate pseudorandom numbers

1-dimensional example: center cell changes according to the values in its

neighbors: “rule 30”, a Wolfram favorite:

current pattern 111 110 101 100 011 010 001 000

new state for center cell 0 0 0 1 1 1 1 0

Source: http://en.wikipedia.org/wiki/Rule_30

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• Cellular Automata:- groups of cells, each cell’s life depends

on its neighbors- state of the cell in each cycle given by a

set of rules

Martin, P., An Analysis of Random Number Generators for a Hardware Implementation of Genetic Programming using FPGAs and Handel-C, Technical Report, University of Essex, 2002

See also: . Harish Ramaswamy, An extended library of hardware modules for genetic algorithms, with applications to DNA sequence matching, MS, Univ. of Cincinnati, 2008

• LFSR involves global signal routing and hence causes longer delays

• Improvement: Cellular Automata require local routing only

Cellular AutomataA 1D CA consists of a string of cells with 2 neighbors, left

(West) and right (East)

• At each time step, the value of a cell is given by a rule.• A simple 1D CA based PRNG is obtained by applying

Rule 30, which is,

C(t+1) = (West(t) XOR (C(t) OR East(t)))

• A Multiple CA is obtained by combining several 1D CAs in series

Random Number Generator Contd.Results of Serial test on 1D CA* (Single and Multiple)

Hybrid CA• CA which makes use of a combination of rules is known

as Hybrid CA• Combination of Rule 90 and Rule 150 at appropriate

sites can yield maximum length cyclesRule 90 : C(i)(t+1) = C(i-1)(t) XOR C(i+1)(t)

Rule 150: C(i)(t+1) = C(i-1)(t) XOR C(i)(t) XOR C(i+1)(t)

* Martin, P., An Analysis of Random Number Generators for a Hardware Implementation of Genetic Programming using FPGAs and Handel-C, Technical Report, University of Essex, 2002.

Generating pseudorandom numbers on an altera chip:

a. Make your own generator, using “n” lfsr’s or ca’s, start each with a different seed

b.Use code from the altera “cookbook”:

http://www.altera.com/literature/manual/stx_cookbook.pdf