Polynomial Datapaths Optimization Using Finite Abstract Algebra(I) Presenter: 陳炳元 Graduate Institute of Electronics Engineering Graduate Institute of Electronics

Polynomial Datapaths Optimization Using Polynomial Datapaths Optimization Using Finite Abstract Algebra(I) Finite Abstract Algebra(I)

Presenter: 陳炳元 Graduate Institute of Electronics EngineeringGraduate Institute of Electronics Engineering,

NTU, MD – 526 ALCom Lab

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OutlineOutline Overall Optimization ProblemOverall Optimization Problem

• Our Focus: Fractional Optimization of Fixed-size Fractional Our Focus: Fractional Optimization of Fixed-size Fractional Arithmetic DatapathsArithmetic Datapaths

• Applications: DSP for audio, video, multimedia….Applications: DSP for audio, video, multimedia…. Problem ModelingProblem Modeling

• Polynomial Functions over Finite FieldPolynomial Functions over Finite Field

Limitations of Previous WorkLimitations of Previous Work Theory and ApproachTheory and Approach AlgorithmAlgorithm Conclusions & Future WorkConclusions & Future Work

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IntroductionIntroduction

In recent years, complexity of arithmetic circuit In recent years, complexity of arithmetic circuit design growsdesign grows exponentially exponentially In order to make it more effective, it‘s necessary to In order to make it more effective, it‘s necessary to reduce the costreduce the cost and increase performanceand increase performanceOptimization of the High Level design is necessaryOptimization of the High Level design is necessaryIn RTL level most of DSP applications (such as audio,In RTL level most of DSP applications (such as audio, 、、video and other multimedia applications) use video and other multimedia applications) use polynomials for optimizationpolynomials for optimization Our focus lies in the optimization of polynomials.Our focus lies in the optimization of polynomials.The mathematical methods being used belong toThe mathematical methods being used belong to abstract algebraabstract algebra

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Problem and MotivationProblem and Motivation

•

y = a0 . x1

2 + a1 . x1 + b0 . x0

2 + b1 . x0 + c . x0 . x1

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Fixed-Size Data-path: ModelingFixed-Size Data-path: Modeling

Control the datapath size: Fixed size

* *

Z8

Z8

Z16

Z32 * *

Z32

Z32

Z32

Z31

Image is Zp: integer values in 0,…, p-1

Fixed-size vector

arithmetic

Polynomials reduced %p

Algebra over the field Zp

Example: Anti-Aliasing FunctionExample: Anti-Aliasing Function

F = 1 = 1 = 2√a2 + b2 2√x Expand into Taylor series

• F ≈ 1 x6 – 9 x5 + 115 x4

64 32 64 – 75 x3 + 279 x2 – 81 x

16 64 32 + 85 64

Coefficients are fractional!

MAC

x = a2 + b2

coefficients coefficients

a b

x

F

DFF

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General Datapath ModelGeneral Datapath Model

Bit-vector operands with different word-lengthsBit-vector operands with different word-lengths Input variables : Input variables : {{xx11,…, ,…, xxdd} } Output variables Output variables : : ff

• xx11 Z2n1, . . . ,, . . . , x xdd Z2

nd, Image(, Image(f )f ) Zp , p: prime, p: prime

Model Model f f as polynomial function as polynomial function

f f ::Z2n1……Z2

nd Zp

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Previous WorkPrevious Work

Boolean Representations (f: B → B): BDDs, ZDDs etc.

Moment Diagrams (f: B → Z): BMDs, K*BMDs, HDDs etc. Canonical DAGs for Polynomials (f: Z → Z)

• Taylor Expansion Diagrams (TEDs)

Required: Representation for f: Z2m → Z2m

SAT, MILP, Word-level ATPG, … Vanishing Polynomial

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Polyfunctions over Polyfunctions over ZZpp

Polynomials over Zp[x1, …, xd]

• Represented by polyfunctions from Zp[x1, …, xd] to Zp

F1 % p ≡ F2 % p => they have the same underlying polyfunction f

Use equivalence classes of polynomials

f

g

Equivalence classes

Zp[x1, …, xd] Z2m

F2

F1

G2

G1 1

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Unity polynomials: RequirementUnity polynomials: Requirement

Generate unity polynomials U(x)

• Test if f (x) = q(x)U(x) + r(x)

• f (x) = q(x) + r(x)

Required: To generate U(x) specific to given f (x)

Zp

Set of all unity

polynomials

f

U

Zp[x1, …, xd]

1

h: % p

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Cyclic groupCyclic group

Let G be a group, G is called cyclic if there is an element a G such that G = < a >, a is called generator of G

For example, (Zn , +) is a cyclic group with

generator 1(Zp\{0}, *) is also a cyclic group, and has some generator (we will prove it later)

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OrderOrder

(1) The order of a group is its cardinality, i.e., the number of its elements.

(2) If G is a group and a G, then order of a is | < a >|.

(3) We denote the order of a group G by o(G) and the order of an element a by o(a).

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PropertyProperty

Let G be a group and let a G, if o(a) = n

then < a > = { a0, a1, a2,… ,an-1, an,…}

i.e. n is the smallest positive integer an = e

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Unit ElementUnit Element

Definition:

Let R be a ring and aR, if a-1 exists, then we say that a is a unit

Theorem:

Let an element a in Zn, then a is unit gcd(a, n)=1

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Unit ElementUnit Element

Proposition:

Let P be a prime, in ZZpp all nonzero elements are unit

Let F be a field if F is a commutative ring with unity 1 and all nonzero elements are unit

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Polynomial FunctionPolynomial Function

Theorem: LetLet FF bebe aa functionfunction onon nn andand Image(F)Image(F)

U(U(nn),), thenthen KK >> 00 s.t.s.t. FFKK is a unity function, where is a unity function, where

U(U(nn) = { a ) = { a nn | gcd(a, n) = 1 } | gcd(a, n) = 1 }

Theorem: Let f: Zn1 X Zn2 x ... x Znd Zm be a function and

f(t) =ct whenever t = (t1, t2, …, td). Then

f is polynomial function =ct(mod m) has a solutionkk

k

xt)(

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ExampleExample

Let f: Z2 Z3 Z4 by f(0,0) =1, f(0,1) =3, f(0,2) =1, f(1,0) =1, f(1,1) =0, f(1,2) =1

Then (t)00X00 + (t)01X01 + (t)02X02 + (t)10X10 +

+ (t)11X11 + (t)12X12 = f(t) (mod4),

X00 + t2X01 + t2(t2 - 1)X02 + t1X10 + t1t2x11 +

+ tlt2(t2- 1)x12 = f(t1, t2) (mod4),

t1 =0, 1; t2 =0,1,2, then

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ExampleExample

X00 = 1 (mod4), X00 + X01 = 3 (mod4), X00 + 2X01 + 2X02 = 1 (mod4), X00 + 0X01 + 0X02 + X10 = 1 (mod4), X00 + X01 + 0X02 + X10 + X11 = 0 (mod4), X00 + 2X01 + 2X02 + X10 + 2X11+2X12= 1 (mod4).

Then we have X00= 1, X01 = 2, X02= 0, X10= 0, X11 = 1 and X12 = 1, thus F = 1 + 2y + xy2

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Polynomial FunctionPolynomial Function

In most cases, functions on nn are not necessarily

polynomial

Theorem:

Let P be a prime, then f is a function on Zp iff f is a

polynomial

Corollary:

Every unity function on Zp is unity polynomial

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Order of ElementOrder of Element

Let G = < a > be a cyclic group of order n, then o(am) = n/d, whenever d = gcd(m, n)

Proof:

(am)n/d = (an)m/d = e Let K be any positive integer such that (am)K = e Claim: K ≥ n/d (am)K = e amK = e n | mK n/d | mK/d n/d | K K ≥ n/d

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Euler Euler functionfunction

Given any positive integer n. Given any positive integer n. (n)(n) is the is the number of of positive integers less than or equal to positive integers less than or equal to n which are which are relativelyrelatively prime to to n. The . The function is known as the Euler is known as the Euler function. i.e. function. i.e. (n) = |(n) = |U(U(nn) |) |

Theorem: Theorem: Let n be a positive integer, thenLet n be a positive integer, then

Definition: Let p be a prime and k|p-1, Definition: Let p be a prime and k|p-1,

we define R(k) ={awe define R(k) ={a pp**| o(a) = k}| o(a) = k}

ndnd

|)(

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TheoremTheorem

U(Zp) is a cyclic Proof: Let s Zp\{0}and o(s) = k mean that sk = 1, may assume

f(x) = xk – 1 in Zp. According to Fundamental Theorem of Algebra, every n-th polynomial has at most n roots

{s, s2,..., sk } is solution space of f=0. t < k, gcd(t, k) = d>0 (st)k/d = (sk)t/d = 1 o(st ) < k | R(k) | (k) p - 1 = k|p-1 |R(k)| k|p-1(k) | R(k) | = (k), k |p-1 | R(p-1) | = (p-1) > 0 Therefore, U(Zp) is a cyclic

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AlgorithmAlgorithm 1: OPT_POLY (F, d, x, p, n) 2: F: polynomial; d: number of variables; 3: x[1. . . d]: list of input variables; p: prime 4: n[1. . . d]: list of bit-widths of input variables, x; 5: A = {1} ; 6: // search for generator 7: for i= 2 to p-1 do 8: Oi = order(i); 9: if (Oi == p-1) 10: then break; 11: end for 12: // A: set of elements of order 1 or k 13: for j= 2 to p-1 do 14: if (gcd(j, p-1) == (p-1)/k) // k divides p-1 15: then A = A { i j }; 16: end for

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AlgorithmAlgorithm 17: // construct the function g 18: for k= 0 to p-1 do 19: g(k) = Ak+1(mod r) +1k+1(mod r) +1; // r = |A| 20: end for 21: //G is polynomial of g 22: G= solver(g); 23: U = Gk; // unity polynomial 24: quo= F/U; 25: rem= F%U; 26: F= quo + rem; 27: // reduction 28: for i= 1 to d do 29: F= F%Y(0,0,..,nii,..,0); 30: end for

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Conclusions & Future WorkConclusions & Future Work

Currently, partial results show good performance of our algorithm

Fixed-size arithmetic circuit is a polynomial algebra over the finite field, Zp

We presented an efficient algorithm to optimize polynomials

Also our algorithm can deal with fractional coefficients

Future work involves extensions for optimizing

• Multiple output polynomials(II)

• Complex polynomials

• …

Questions?Questions?

Documents

Polynomial Datapaths Optimization Using Finite Abstract Algebra(I) Presenter: 陳炳元 Graduate Institute of Electronics Engineering Graduate Institute of Electronics