1. Generating Random Numbers in Hardware

2. 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

3. Two common methods used to generate pseudorandom numbers in hardware: --LFSR (linear feedback shift register) --CA (Cellular automata)

4. LFSR (Linear feedback shift register): Based on polynomials over a finite field Simplest field: Z 2 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 Z 4 with elements 0,1,2,3 we have 2 * 2 = 0—it’s NOT a field!!!)

5. 5 Linear Feedback Shift Register (LFSR): -sequential shift register with combinational logic -feedback provided by selection of points called taps

6. 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 2 n – 1 nonzero elements of the field of polynomials of degree n-1 with coefficients in Z 2, whose elements can be represented by n-bit numbers)

7. Example: suppose we have 3-bit numbers c 3 c 2 c 1 representing c 3 x 2 + c 2 x + c 1 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, …….

8. 8 Example: N = 32: Taps 32, 22, 2, 1 For each n, there is at least one such primitive polynomial (result from math)

9. 9 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 (2 8 – 1) nonzero numbers in “random” order, e.g.: SEED=10101000 gives 10101000, 01010001, 10100011, 01000110, … Bit 8Bit 1

10. 10 How good are the random numbers generated? Reference: Shruthi Narayanan, M.S. 2005, ATI Technologies Hardware 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

11. 11 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.

12. 12 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.

13. 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

14. 14 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

15. LFSR involves global signal routing and hence causes longer delays Improvement: Cellular Automata require local routing only

16. Cellular Automata A 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

17. 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 cycles Rule 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.

18. 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