Theory and Applications of GF(2p) Cellular Automata

P. Pal Chaudhuri

Department of CST

Bengal Engineering College (DU)

Shibpur, Howrah India

Review of GF(2) CA• This computation model was

introduced by Von Nuemann• A CA cell is essentially a

memory element (D Flip flop) with some combinatorial logic {an XOR gate (linear) or XNOR Gate (additive) for our case}

• The cell is updated at every clock cycle

• The state of the cell is dictated by the immediate neighbours of the cell

• SIMPLE STRUCTURE • 2 STATE 3 NEIGHBOURHOOD

Clock

Combinational logic

CLK

D

Flip - Flop Q

From

left neighbour

A typical CA cell

Truth TableNeighboodhood

StateNextState

Next State

000 0 1001 1 0010 0 0011 1 1100 1 0101 0 1110 1 1111 0 1

Rule 90 Rule 150

After Minimization,

XOR Logic XNOR Logic

Rule 60 : qI(t+1) = qI-1(t) qI(t) Rule 195 : qI(t+1) = qI-1(t) qI(t)

Rule 90 : qI(t+1) = qI-1(t) qI+1(t) Rule 165 : qI(t+1) = qI-1(t) qI+1(t)

Rule 102 : qI(t+1) = qI(t) qI-1(t) Rule 153 : qI(t+1) = qI(t) qI-1(t)

Rule 150 : qI(t+1) = qI-1(t) qI(t) qI-1(t) Rule 105 : qI(t+1) = qI-1(t) qI(t) qI-1(t)

Rule 170 : qI(t+1) = qI-1(t) Rule 85 : qI(t+1) = qI-1(t)

Rule 204 : qI(t+1) = qI (t) Rule 51 : qI(t+1) = qI (t)

Rule 240 : qI(t+1) = qI+1(t Rule 240 : qI(t+1) = qI+1(t)

Rule 90 : qI(t+1) = qI-1(t) qI+1(t)

Rule 150 : qI(t+1) = qI-1(t) qI(t) qI-1(t)

CA Rules

Types of CA

• LINEAR Uses only XOR Logic• COMPLEMENT Uses XNOR logic• ADDITIVE Uses Both XOR and XNOR logic• UNIFORM All Cells obey same rule• HYBRID Different rules apply to different cells

X

X

X

X

XX

X

X

X

X

XOR Logic XNOR Logic

Characterization of GF(2) CA

• The CA is characterized by a T matrix which is essentially the dependency matrix

1 0 0 0 0

0 1 1 0 0

T = 0 1 1 1 0

0 0 0 1 1

0 0 0 0 1

• The elements are in GF(2)

• For 3-neighborhood CA, we have a band matrix as shown above

• Null Boundary CA

X

X

X

X

X

Cell 0 1 2 n-3 n-2 n-1Null Boundary

Cell 0 1 2 n-3 n-2 n-1Periodic Boundary

Cell 0 1 2 n-3 n-2 n-1Intermediate Boundary

CA with Different Boundary Conditions

Block Diagram of 2-D CA(5 Neighborhood)

Characterization Of State Transition Behavior

• Group CA All states lie on same Cycle

• Non Group CA Non cyclic and cyclic states

• Complemented CA Employs both XOR/XNOR logic

• The characterization of complemented CA is based on the corresponding non complement (XOR) CA

CHARACTERISTIC POLYNOMIAL X)

x 1 0 01 1+x 1 10 1 x 10 0 1 1+x

T + x [1] = x) = x4 + x +1

PRIMITIVE : SMALLEST INTEGER k FOR WHICH IRREDUCABLE(x) DIVIDES xk + 1, k =2n-1=15

STATE TRANSITION BEHAVIOUR: MAXIMUM LENGTH GROUPCA CYCLIC STRUCTURE:[ 1(1), 1(k) ], k = 2n-1, N CELL CA (EXHAUSIVE CA)

121163 135

12 15 14 4 810 7

90 f t+m (x) = Tm.f1(x)

0 1 0 01 0 1 00 1 0 10 0 1 0

T =

x 1 0 01 x 1 00 1 x 10 0 1 x

T + IX =

1 0 0 00 1 0 00 0 1 00 0 0 (x2 + x +1)

CHARACTERISTIC POLYNOMIAL = MINIMAL POLYNOMIAL= (x2 + x +1)2

GROUP CA

NON MAXIMUM LENGTH GROUP CA < 90, 90, 90, 90 >

(x) =

0

9 15

613

7 12

3 14

11

5

2 8

1 4

10

State Transition Behavior

Non Group CA

N = 4

Characteristic Polynomial = x 2(x+1)2

Minimal Polynomial = x 2(x+1)2

r = rank = 3, depth = 2

No of Predecessors = 2 n-r

Nonreachable states = 2,3,4,5,10,11,12,13

Cyclic states = (0,1,8,9)

Attractors = 0,1,8,9

1- basin = (1,4,11,14)

5

15

10

0

4

14

11

1

2

7

13

8

3

6

12

9

Zero tree

1 1 0 00 1 1 00 1 1 00 0 1 1

T =

Det = 0

2943

1112110

0151413

5678

Complemented CA

Rule Vector < 153,153,153,153>

From the T matrix, we can obtain the following which characterize the given CA

• Group properties (For Group CA)

• Non Group CA properties• Characteristic Polynomial• Minimal Polynomial• Cycle structure

All Results are currently available in Additive Cellular Automata

Theory and Applications Volume 1P Pal Chaudhuri, D Roy Chowdhury, S Nandi and S

ChattopadhyayIEEE Computer Society Press

Book ISBN 0-8186-7717-1

Application Areas

• Pseudo random Pattern Generation• Pseudo Exhaustive Test Pattern Generation• Deterministic Test Pattern Generation • Signature Analysis• Synthesis of testable FSM• CA - Based Cipher System• Low cost Associative memory• Perfect Hashing• 2D CA based Parallel PRPG

GF (2P) CA

D D D x

This is treated as a single cell of GF(2P) CA

Characteristic Polynomial = f(x)f(x) is a primitive polynomial

X X X X

GF (2P) CA

• P number of GF(2) cells are treated as a unit cell of GF(2P)• The characteristic polynomial of the unit GF(2P) cell is a primitive polynomial in GF(2)• The T matrix is constructed as in the case of GF(2), the elements are in { 0 .. 2P - 1 }

Why GF(2P) CA

• Ability to handle np number of

GF(2) cells using ‘n’ number

GF(2p) cells.

• Based on the theoretical found-

dation of Extension Field.

• Evidently increases the Neighborhood of GF(2) CA.

• If we use 3 Neighborhood GF(2p) CA, we are in effect using

3 p Neighborhood GF(2) CA.

• Extends the computing power of GF(2) CA

0 0 0 0 1

T =

N = 3Neighborhood = 3

0 0 0 1 0 00 0 1 1 0 00 1 0 0 0 11 1 0 0 1 10 0 0 1 1 00 0 1 1 0 1

3 x 3

6 x 6

0 11 1 =

• Provides Abstraction.• Provides hierarchy to look at the Problem from higher levels

• A bit

• A set of p bits

(pixel)

• A set p x q bits

( pixel block)

GF(2)

GF(2p)

GF(2p)q

0/1

1 2 p

0 to 2p-1

1 2 qk

1 2 p

0 to 2pq -1

Simple Analogy

Transistor net List

Gate Level Net List

RTL Net List

GF(2)

GF(2p)

GF(2p)q

Reduced Computation Overhead

• Multiplication and Addition are avoided by Star and Plus

table

• Substituted by Simple Table Lookup

• Size of the Matrices to be handled is Reduced by a factor of

p2 we work in GF(2p) np x np replaced by n x n matrix

• Overall Execution Time of algorithms coded in GF(2P) will

decrease by the order of p2

• The overhead is the memory space to store the Star Table

and Plus Table

….

Characterization Based on GF(2) : A Finite Field

Finite Field

Number of elements in { F } is finite

• Finite Cardinality of { F } is a finite field or

GALOIS FIELD (GF)

• GF(q) means |{ F }| = q

• The Binary Field is denoted by GF(2) = {0,1}

GF (2) CA GF (2p) CA

D 1 2 pA Cell

0/1Finite FieldBit String

0 to 2p -1Its extension

Symbol String (Symbol GF(2p)

Field is a set {F} with the following properties

• F is an abelian group under addition.

• F is closed under multiplication.

• F is associative under multiplication.

• There is a multiplicative Identity.

• There is a multiplicative inverse.

Extension Field

GF(2) : { 0,1 } GF(22) :{00,01,10,11} {0,1,2,3}

n Cell machine, each Cell handling

GF(2) elements GF(22) elements

…… … ….. ….

1 0 1 …… i …. 1 M/C State 3 2 0 ….. I …. 1

i GF(2) i GF(22)

n symbols = n bits n symbols = 2n bits

Extension Field

GF (2p) is the pth Extension of GF(2)

• This has elements (0, , 2…….. 2p

-1)

• Where is the generator of the field

• is the root of the Irreducible Polynomial f(x)

• f(x) is called Generator Polynomial

• For GF(2p) CA design - Primitive Polynomial as generator

( a n degree irreducible polynomial is primitive if minimum

value of k for which it divides xk+1 where k = 2n -1 )

Extension Field Theory

Generator Polynomial Primitive polynomial

p(x) = x2 + x + 1, coefficients over GF(2)

Set of Polynomial modulo p(x) is a field denoted

GF(22) = { 0, 1, x, x+1 } = { 0, 1, , 2}

where is the root of p(x).

GF(2) elements GF(2p) elements1 3 …i ...1, i GF(2p)n symbols = pn bits

….. ….. …. …

…..

A Set of p bits form a cluster that is viewed as a symbol at higher level What is the Utility? - Abstraction & Hierarchy

Transistor - Gate Level - RTL Level

p21

ni21

Further Extension

• A set of p bits form a symbol in GF(2p).

• A set of q symbols form a (super symbol in GF(2p)

q

field properties are valid) .

• N cell GF(2p)

q machine.

…. …. npq bits

… …. pq bits

……… p bits

• Formalism of a hierarchical structure

np cell GF (2) CA n cell GF(2p) CA

1 2 i n

1 2 j q

1 2 p

Structure of GF (2P) CA

OUTIN OUT OUT………..

1 2 I - 1 i i + 1

W i-1

Wi

W i-1

Structure of GF(2P) CA

CELL 0 CELL 1 CELL 2

D

Q Q

D

Q

D

0

0

1

T =0 0 0 0 1

A State of an n cell GF(2p) CA x1,x2..

Xn, xi GF(2p)Next State = T, Present State(weighted sum)

GF (2P) CA

• Use of Extension Field Theory to extend the basic

Structure of CA• Each Cell No Longer Stores 0/1

• It Stores any of (0 - 2p -1) Values

• Essentially n Cell GF(2P) CA np Cell GF(2) CA

Characterization of GF(2P) CABased on the extension field theory, we have

characterized GF(2p) CA using the T matrix.

Till date, we have identified methods to evaluate

• Determinant

• Characteristic Polynomial

• Cycle structure

• Group Properties of Group CA

• Properties of Non - Group CA

Partial Transition Diagram of a 3 Cell GF (22) CA

103

112

210

232

033

023

203212

100

010001

021322

321

333

123

111

330

101

310

T Matrix of the CA

• Matrix is the T 0 0

matrix in GF(2P) T = 0 0 1

0 0 1 1 0 0

• Matrix is in 0 0 0 1 0 0

GF(2) T = 1 1 0 0 1 1

0 1 0 0 0 1

0 0 1 1 1 0

0 0 0 1 0 1

• Both have Equivalent representations for implementation.

1 10 1

is the generatorPolynomial

For every p(x) as Generator Polynomial

We have an Equivalent Binary Companion Matrix T whose Characteristic Polynomial is p(x).

T =

Characteristic Polynomial = x2 + x + 1

0 1 1 1

GF (22)

Generator Polynomial p(x) = x2 + x + 1.

Binary Companion Matrix = T = 0 1

1 1Representation

Matrix Vector

< 0 1>

< 1 1 > =

=

3=

0 11 1

1 11 0

1 00 1

< 0 0 >

< 0 1 >

< 1 1 >

< 1 0 >

0

2

3

1

0 0 0 0 =

The Elements of GF(2p) are { 0,,2 ….. 2

p -1 }

The Matrix Representation of i = Ti

The vector Representation of i = col1[Ti]

0 11 1

i

Ti =

Col1 Ti = COL1

0 11 1

i

The star Table and Plus Table for GF(22)

+ 0 1 2 30 0 1 2 31 1 0 3 22 2 3 0 13 3 2 1 0

* 0 1 2 30 0 0 0 01 0 1 2 32 0 2 3 13 0 3 1 2

Matrix Vector Multiplication

Y = TX

= =

= =

0 2 02 0 20 2 1

222

0*2 + 2*2 + 0*22*2 + 0*2 + 2*20*2 + 2*2 + 1*2

3 3 + 3 3 + 1

302

If the Generator Polynomial for all Cells of the GF(2p) CA are same

Uniform CA

If we use Multiple Generator Polynomials we obtain

Hybrid CA

GF (2P) Results

• Factorization algorithm in GF(2P)

• Vector space Theoretic Specification of an Extended

Linear Machine

• For the T Matrix

• Characteristic Polynomial

x3 + x2 + 2 = 0• Minimal Polynomial

x3 + x2 + 2 = 0• Cycle Structure

- [ 1(1), 1(3), 3(5), 3(15)]

T = 0 0 0 1

Group CA 4 Cell GF (24) CA

5 2 0 0

T = 0 1 3 0

0 4 8 2

0 0 9 6

• All non zero lie in some cycle

• det T = 0

• Characteristic Polynomial x4 + 13 x3 + 15 x2 + 5 x + 6

• Factors (x+2)(x+5)(x+8)(x+9)

• Cycle Structure [ 1(1), 4369(15) ]

Group CA 4 Cell GF(24) CA

5 2 0 0

T = 0 1 3 0

0 4 8 2

0 0 9 6

Characteristic Polynomial = (x) = x4 + 13 x3 + 15 x2 + 5 x + 6

= factors

Primary Cycle Structure is formed by factors of (x).

1(x) (x+2) 15

2 (x) (x+5) 15

3(x) (x+5) 15

4 (x) (x+9) 15

GF (2P) Results

• (x) = 1(x)r1 2(x)r2 …………..m(x)rn

• Primary Cycle Structure

• This corresponds to the vector subspace

formed by the factors of (x)

• This therefore contributes m such vector Subspaces

Secondary Cycle StructureFormed by taking j terms of the m Vector Subspaces at a time for j = 2 to m

15*15=225

15*15=225

15*15=225 etc

15*15*15=3375 etc

ull Space (NS) of each (Tki + I) gives the No. of components,

NS(T15 + I) = 65535

NS(T225 + I) = 65535

NS(T3375 + 1) = 65535

Cycle structure = 1(1) 4369(15)

GF (2P) Results

• Secondary Cycle Structure.

• This Vector subspace is formed by taking j terms of the

m vector subspaces at a time for j = 2 to m.

• Null Space of each (Tk + I) for all possible cycle lengths

gives the number of component subspaces.

Non Group CA

9 711

5

8 410

6

13 115

3

12 214

0

D1 MACA

MACA

8 10

4

6

9 11

7

5 01100101

1110

12 14

2

0

13 15

1

30000 0011

00 01

11100100

Non Group CA

2 3 0

T = 3 3 2

2 2 1

• det T = 0• Rank = 2 Height = 1 Depth = 21

• Predecessors of Reachable States = 2 2(3-2) = 22 = 4• Characteristic Polynomial x(x+1)2

• Cycle Structure [4(1),6(2)]

Non Group CA

• Characteristic Polynomial : xkf(x)• Minimal Polynomial : xdg(x)• Height = d• No of Predecessors of reachable states = 2 p(n-r)

where r = rank of matrix• p = extension• No of Attractors : 2 p(n-d)

Complemented CA

Individual cells use XNOR / XOR Logic

X t+1 = TXt + F where F is Inversion Vector

X t+p = [ I + T + T2 +……….. T P-1] F + TP X

If cycle structure of XOR CA is (k)

XNOR CA

T =

F =

1111

Cycle Structure of XOR CA = 4(1) 20(3) 32(6)Cycle Structure of XNOR CA = 4(1) 20(3) 32(6)

2 1 0 0 0 3 2 0 0 0 3 1 0 0 0 1

X

X

X

X2

12

3

1

3 2

1 1 1 1

XNOR CA configured with all Permutations of Rule 60,

102, 204 and all weights give cycle Lengths which are of

even length - The only Constraint is that atleast one Row

of the T Matrix must contain a single one

Cycle Structure of XOR CA = 0(1) 1(k1) (k3)….. n(kn)

[1 + T + T2 + T3 + ………..+ Tri ][F] = 0 CASE I

= 0 CASE II

CASE I

XNOR CA Cycle Structure 0(1) 1(k1) (k3)….. n(kn)

CASE II

XNOR CA Cycle Structure 0(2kn)

1 2 0 0 0 00 3 0 0 0 00 0 3 2 0 00 0 2 0 0 00 0 0 0 1 20 0 0 0 0 3

T =

1 2 0 00 2 3 00 0 3 10 0 0 1

T =

0121

F =

XOR CA Cycle Structure 4(1) 6(2) 20(3) 30(6)XNOR CA Cycle Structure 4(1) 6(2) 20(3) 30(6)

XOR CA 4(1) 340(3) 512(6)XNOR CA 2(2) 512(6)(1 + T + T2) F 0(1 + T + T2 + T3 + T4 + T5) F = 0

Image Compression

Image Sizes 352 x 240

Each Pixel 8 bits wide in GF(28)

8 x 8 blocks comprise 64 cell GF(28) CA

Instead of handling 512 bits (8 x 8 x 8)

we work with 64 GF(28) Symbols

Thanks