1. Polynomial Datapaths Optimization Using Finite Abstract Algebra(I) Presenter: 陳炳元 Graduate Institute of Electronics Engineering Graduate Institute of Electronics Engineering, NTU, MD – 526 ALCom Lab

2. Outline  Overall Optimization Problem Our Focus: Fractional Optimization of Fixed-size Fractional Arithmetic DatapathsOur Focus: Fractional Optimization of Fixed-size Fractional Arithmetic Datapaths Applications: DSP for audio, video, multimedia….Applications: DSP for audio, video, multimedia….  Problem Modeling Polynomial Functions over Finite FieldPolynomial Functions over Finite Field  Limitations of Previous Work  Theory and Approach  Algorithm  Conclusions & Future Work

3. Outline  Overall Optimization Problem Our Focus: Fractional Optimization of Fixed-size Fractional Arithmetic DatapathsOur Focus: Fractional Optimization of Fixed-size Fractional Arithmetic Datapaths Applications: DSP for audio, video, multimedia….Applications: DSP for audio, video, multimedia….  Problem Modeling Polynomial Functions over Finite FieldPolynomial Functions over Finite Field  Limitations of Previous Work  Theory and Approach  Algorithm  Conclusions & Future Work

4. 2015/9/11 4 Introduction  In recent years, complexity of arithmetic circuit design grows exponentially  In order to make it more effective, it‘s necessary to reduce the cost and increase performance  Optimization of the High Level design is necessary  In RTL level most of DSP applications (such as audio, 、 video and other multimedia applications) use polynomials for optimization  Our focus lies in the optimization of polynomials.  The mathematical methods being used belong to abstract algebra

5. Problem and Motivation  y = a0. x1 2 + a1. x1 + b0. x0 2 + b1. x0 + c. x0. x1

6. Fixed-Size Data-path: Modeling   Control the datapath size: Fixed size * * Z8Z8 Z8Z8 Z 16 Z 32 * * Z 31  Image is Z p : integer values in 0,…, p-1 Fixed-size vector arithmetic Polynomials reduced %p Algebra over the field Z p

7. Example: Anti-Aliasing Function   F = 1 = 1 = 2√a 2 + b 2 2√x   Expand into Taylor series F ≈ 1 x 6 – 9 x 5 + 115 x 4 64 32 64 – 75 x 3 + 279 x 2 – 81 x 16 64 32 + 85 64 Coefficients are fractional! MAC x = a 2 + b 2 coefficients ab x F DFF

8. Outline  Overall Optimization Problem Our Focus: Fractional Optimization of Fixed-size Fractional Arithmetic DatapathsOur Focus: Fractional Optimization of Fixed-size Fractional Arithmetic Datapaths Applications: DSP for audio, video, multimedia….Applications: DSP for audio, video, multimedia….  Problem Modeling Polynomial Functions over Finite FieldPolynomial Functions over Finite Field  Limitations of Previous Work  Theory and Approach  Algorithm  Conclusions & Future Work

9. General Datapath Model  Bit-vector operands with different word-lengths  Input variables : {x 1,…, x d } Output variables : f x 1 ,..., x d , Image(f ) , p: primex 1  Z 2 n1,..., x d  Z 2 nd, Image(f )  Z p, p: prime  Model f as polynomial function f :  …  f : Z 2 n1  … Z 2 nd  Z p

10. Outline  Overall Optimization Problem Our Focus: Fractional Optimization of Fixed-size Fractional Arithmetic DatapathsOur Focus: Fractional Optimization of Fixed-size Fractional Arithmetic Datapaths Applications: DSP for audio, video, multimedia….Applications: DSP for audio, video, multimedia….  Problem Modeling Polynomial Functions over Finite FieldPolynomial Functions over Finite Field  Limitations of Previous Work  Theory and Approach  Algorithm  Conclusions & Future Work

11. Previous 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: Z 2 m → Z 2 m   SAT, MILP, Word-level ATPG, …   Vanishing Polynomial

12. Outline  Overall Optimization Problem Our Focus: Fractional Optimization of Fixed-size Fractional Arithmetic DatapathsOur Focus: Fractional Optimization of Fixed-size Fractional Arithmetic Datapaths Applications: DSP for audio, video, multimedia….Applications: DSP for audio, video, multimedia….  Problem Modeling Polynomial Functions over Finite FieldPolynomial Functions over Finite Field  Limitations of Previous Work  Theory and Approach  Algorithm  Conclusions & Future Work

13. Polyfunctions over Z p   Polynomials over Z p [x 1, …, x d ] Represented by polyfunctions from Z p [x 1, …, x d ] to Z p   F 1 % p ≡ F 2 % p => they have the same underlying polyfunction f   Use equivalence classes of polynomials f g Equivalence classes Z p [x 1, …, x d ] Z2mZ2m F2F2 F1F1 G2G2 G1G1 1

14. Unity 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) ZpZp Set of all unity polynomials f U Z p [x 1, …, x d ] 1 h: % p

15. Cyclic group Let G be a group, G is called cyclic if there is an element a  G such that G =, a is called generator of G  For example, (Z n, +) is a cyclic group with generator 1  (Z p \{0}, *) is also a cyclic group, and has some generator (we will prove it later)

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

17. Property Let G be a group and let a  G, if o(a) = n then = { a 0, a 1, a 2,…,a n-1, a n,…} i.e. n is the smallest positive integer  a n = e

18. Unit Element   Definition: Let R be a ring and a  R, if a -1 exists, then we say that a is a unit   Theorem: Let an element a in Z n, then a is unit  gcd(a, n)=1

19. Unit Element   Proposition: Z p Let P be a prime, in Z p 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

20. Polynomial Function  Let F be a function on  n and Image(F)  U(  n ), then  K > 0 s.t. F K is a unity function, where  Theorem: Let F be a function on  n and Image(F)  U(  n ), then  K > 0 s.t. F K is a unity function, where U(  n ) = { a   n | gcd(a, n) = 1 } U(  n ) = { a   n | gcd(a, n) = 1 }   Theorem: Let f: Z n1 X Z n2 x... x Z nd  Z m be a function and f(t) =c t whenever t = (t 1, t 2, …, t d ). Then f is polynomial function  = c t (mod m) has a solution

21. Example   Let f: Z 2  Z 3  Z 4 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) 00 X 00 + (t) 01 X 01 + (t) 02 X 02 + (t) 10 X 10 + + (t) 11 X 11 + (t) 12 X 12 = f(t) (mod4),   X 00 + t 2 X 01 + t 2 (t 2 - 1)X 02 + t 1 X 10 + t 1 t 2 x 11 + + t l t 2 (t 2 - 1)x 12 = f(t 1, t 2 ) (mod4),   t 1 =0, 1; t 2 =0,1,2, then

22. Example   X 00 = 1 (mod4),   X 00 + X 01 = 3 (mod4),   X 00 + 2X 01 + 2X 02 = 1 (mod4),   X 00 + 0X 01 + 0X 02 + X 10 = 1 (mod4),   X 00 + X 01 + 0X 02 + X 10 + X 11 = 0 (mod4),   X 00 + 2X 01 + 2X 02 + X 10 + 2X 11 +2X 12 = 1 (mod4).   Then we have X 00 = 1, X 01 = 2, X 02 = 0, X 10 = 0, X 11 = 1 and X 12 = 1, thus F = 1 + 2y + xy 2

23. Polynomial Function   n  In most cases, functions on  n are not necessarily polynomial   Theorem: Let P be a prime, then f is a function on Z p iff f is a polynomial   Corollary: Every unity function on Z p is unity polynomial

24. Order of Element   Let G = be a cyclic group of order n, then o(a m ) = n/d, whenever d = gcd(m, n) Proof: (a m ) n/d = (a n ) m/d = e Let K be any positive integer such that (a m ) K = e Claim: K ≥ n/d (a m ) K = e  a mK = e  n | mK  n/d | mK/d  n/d | K  K ≥ n/d

25. Euler  function  Given any positive integer n.  (n) is the of positive integers less than or equal to which are relatively to. The is known as the Euler  function. i.e. (n) = | U(  n ) |  Given any positive integer n.  (n) is the number of positive integers less than or equal to n which are relatively prime to n. The function is known as the Euler  function. i.e. (n) = | U(  n ) |  Theorem: Let n be a positive integer, then  Definition: Let p be a prime and k|p-1, we define R(k) ={a  p * | o(a) = k} we define R(k) ={a  p * | o(a) = k}

26. Theorem   U(Z p ) is a cyclic   Proof: Let s  Z p \{0}and o(s) = k mean that s k = 1, may assume f(x) = x k – 1 in Z p. According to Fundamental Theorem of Algebra, every n-th polynomial has at most n roots   {s, s 2,..., s k } is solution space of f=0.  t 0   (s t ) k/d = (s k ) t/d = 1  o(s t ) < 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(Z p ) is a cyclic

27. Outline  Overall Optimization Problem Our Focus: Fractional Optimization of Fixed-size Fractional Arithmetic DatapathsOur Focus: Fractional Optimization of Fixed-size Fractional Arithmetic Datapaths Applications: DSP for audio, video, multimedia….Applications: DSP for audio, video, multimedia….  Problem Modeling Polynomial Functions over Finite FieldPolynomial Functions over Finite Field  Limitations of Previous Work  Theory and Approach  Algorithm  Conclusions & Future Work

28. Algorithm   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  

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

30. Outline  Overall Optimization Problem Our Focus: Fractional Optimization of Fixed-size Fractional Arithmetic DatapathsOur Focus: Fractional Optimization of Fixed-size Fractional Arithmetic Datapaths Applications: DSP for audio, video, multimedia….Applications: DSP for audio, video, multimedia….  Problem Modeling Polynomial Functions over Finite FieldPolynomial Functions over Finite Field  Limitations of Previous Work  Theory and Approach  Algorithm  Conclusions & Future Work

31. Conclusions & Future Work   Currently, partial results show good performance of our algorithm   Fixed-size arithmetic circuit is a polynomial algebra over the finite field, Z p   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 …

32. Questions?