A lower bound for the multiplication of polynomials modulo a polynomial

Information Processing North-Holland

Letters

17 April 1992

41 (1992) 321-326

A lower bound for the multiplication of polynomials modulo a polynomial Nader

H. Bshouty

Department of Computer Science, UniL~ersityof Calgary, Calgary, Alberta, Canada T2N IN4 Communicated by D. Cries Received 13 June 1991 Revised 23 December 1991

Abstract Bshouty, N.H., A lower bound (1992) 321-326.

for the multiplication

of polynomials

module

a polynomial,

Lempel, Seroussi and Winograd proved the lower bound (2 + l/(q - l)h -o(n) multiplication of two polynomials of degree n - 1 modulo an irreducible polynomial q elements. We prove this lower bound holds for any polynomial p of degree n. Keywords: Computational

complexity,

multiplicative

complexity,

1. Introduction

Let F be a field and let B = {B,, . . . , Elk} be a set of n x m-matrices with entries from F. Let x =(x0,. ..,x,_])~ and y = (yO,.. ., Y,_,)~ be vectors of indeterminates. A quadratic algorithm over F that computes the bilinear forms xTBy = {xTB, y, . . . , xTBk y} is a straightline algorithm over F for xTBy such that its nonscalar multiplications are of the form l(x, y) x I’(x, y), where I(x, y) and f’(x, y) are linear forms of x and y. The complexity L,(B) of B is the minimal number of nonscalar multiplications needed to compute xTBy by quadratic algorithms over F. When F is an infinite field, then L,(B) is the minimal number of nonscalar multiplications/ divisions needed to compute xTBy by straightline algorithms [13]. When F is finite, then L,(B) is the minimal

Correspondence to: N.H. Bshouty, Department of Computer Science, University of Calgary, Calgary, Alberta, Canada T2N lN4.

0

-

Science

B.V.

quadratic

Information

Processing

Letters

41

for the multiplicative complexity of the p of degree n over a finite field F with

algorithms,

linear

codes

number of nonscalar multiplications needed to compute xTBy by a straightline algorithm without divisions [HI. For the vectors x =(x0,. .., x,_,) and y = x(a)=x,+x,cz+ ... + (Y,, . . .;y,_,)wedefine and y(a) = y, + y,a + . .. + X n-la Y,z_,Cd-‘. Let p(a) be a polynomial of degree n over F and define B(p) = {B,, . . . , B, _ J, where n-1

c

(~~B;y)a

=x(a)y(cr)

mod p(cr).

i=O

That is, xTBiy is the i + 1 coefficient of x(cr)y(a) mod p(a). In ill], Lempel, Seroussi and Winograd used coding theory to prove the lower bound

when p is an irreducible polynomial of degree n. rights

321

Volume 41. Number 6

INFORMATION

In [7], Chudnovsky linear upper bound &(B(P))

G

(

and Chudnovsky

we generalize

2 ( 2+ A)”

the result

in [ll]

of de-

-o(n).

The method we use involves a combination of the coding method, which is used in [ill, and the substitution method used in 141.

2. Preliminary

results

This section surveys some basic concepts ployed throughout the paper.

em-

Definition 1. Let B = {B,, _. . , B,} be an n-set of n x n-matrices and M, N and K = (Ki,j) be n X n-matrices. We define NBM

= {N&M,.

. . , NB,M}

and B[K]

=

17 April 1992

We also define

where

(j++

Theorem. Let p E F[a] be any polynomial gree n over a finite field F. Then L,(B(p))

the

LETTERS

B@C={Bi@Cj\i=l

2+ 0

In this paper, as follows:

proved

PROCESSING

2 Kl,jBj,..., i j=l

2

.

Kn,jBj

j=l

1

For an n-set C of n X n-matrices we write B = C if there exist nonsingular n X n-matrices N, M and K such that NB[ KIM = C.

,...,

@ is the Kronecker

m},

n,j=l,..., product

of matrices.

Let n be an integer. A linear code over F of length n is a linear subspace C of f”. If dim C = k, then C is called an [n, k] code. For c E C the weight of c, denoted by wt(c>, is the number of nonzero components of c. The minimal weight of c is min{wt(c>l c E C - (O}}. We say that C is an [n, k, d] code if C c F” is a code of dimension k and minimal weight d. Let N,(k, d) be the smallest integer n such that there exists an [n, k, dl code. The connection between the linear codes and the complexity of bilinear forms over F is given in the following lemma: Lemma 3 [4,12]. Let B = {B, ,..., B,] be a set of n x m-matrices and let G = Span,(B) be the linear space over F spanned by the elements of B. Let d = min B,EG_taJ rank B’ and k = dim Span.(B). Then L,(B) > N,(k, d). Lemma 3 is proved in [4,12] for the bilinear algorithm model of computation. The proof is based on the fact that each bilinear form xTBy is the sum of at least rank B rational functions of the form l(x) x l’(y), where l(x) and l’(y) are linear forms of x and y, respectively. This fact is also true for the quadratic algorithm model [9]. The next lemma gives a lower bound for N,(k, d). Lemma 4 (Griesmer

bound [14, p. 591). We have

Definition 2. Let B = {B,, . . . , B,] be an n-set of n x n-matrices and let C = {C,, . . . , C,} be an mset of m x m-matrices. We define 1

1 B@C=

(

IZj ,,...,

fi,,c

,,...,

cl>?

l+1FI_l-

[O,“:,

and O,,, 322

IFI-1)

if k
where &=

IFI~-‘(

);I],

Cjj= [ ;;;I

is the s x r zero matrix.

Oy:]

I+

1 IFI_l

i if k > log,.,d.

d+k-log,.,

d-3

d

Volume

41, Number

INFORMATION

6

PROCESSING

17 April 1992

LETTERS

Other lower bound techniques known from the literature for the complexity of quadratic algorithms are the following.

Proof . Let B(‘) = (Bi’), . . . , B,“), BIT,, . . . , Bf)) for i=l , . . . , s, such that, for any B ~6SpanJlBj?i, . . . )Bf’}) I ’

Lemma 5 [4]. Let B = {B, ,..., Bk,) and C = {C,, . . . , C,J be sets of n X m-matrices. Then

rank

Badi.

(*)

Consider

the following

two sets

L&C U B) > dim Span,(C) V, = {diag( Bjl’, Bj2’, . . . , Bj”‘) I i = 1,. . . , I}

k, + min A,,,rFLF

Bi + C

‘l,jcjJ

and

j=l

i( . . ..Bk.f

C

j=l

.

hk,,jCj

Ii

Lemma 6. Let B and C be as in Lemma 5 with k,=k,=n=m. then L.(B) < (1) If Span,(B) c Span.(C), L,(C). (2) If B = C, then L,(B) = L,(C). (See Definition

1 for

= .)

3. Proof of the lower hound In this section we prove the theorem stated in Section 1. Let B be a linearly independent set of matrices. We say that B is a (k, I, d)-set if IB I = k and there exists k -1 matrices B1,..., B,_,E B such that for any B E Span,B, B e Span,({B,, . . . )Bk_IJ) we have

@j . . . $ (B(s) _ D’“‘),

(B (1) _ I)“‘)

V,=

k,

where Dci) = {Bi”, . . . , B,“‘). is the block matrix Here, diag( B!” I , . . . , BcS’) I I

\

B?’ I

B!“’

\

I

Obviously, Span,(V, u V,> is Span,(B(‘) $ . . . CBB(“)) . Therefore, and Lemma 5, L,(B”’

a subset of by Lemma 6

@ . . - @ B’“‘)

2 LOI

” V2)

> dim Span,(V,)

+ min, E Fkl+ +k,LF( S,) (1)

where, for A=(A ,,,,..., Aik, A,,, ,..., As,l,. . . , As,k,) E Fkl+ “. fks, hk have

A,,, .g””

Bad.

rank

We remind the reader that Span,(B) is the linear space spanned by the elements of B. If B is a (k, I, d)-set, then B is a (k, I’, d)-set for any 1’ < 1. This follows from the fact that B @ Span,(B,, . . . , B,_,,} implies that B 4 Span,@,, . . . , B,_,}. The following lemma will be used to prove the theorem. Lemma 7. Let B(‘) be a (k,, Zi, d,)-set for i = 1,. ..,s and let I= min,,iss 1;. Then L,(B”’

@ . . . @ B’“‘)

2 kki-sl+N, i=l

k,

c

Al,jBi(l), . . . ,

j=l+l

BJ”)+

2

As,jB;S)

j=l+l

Every form

nonzero

element

is,Bj’)+ i=l

5 6,Bj”‘+ i=l

in Span,(S,)

k, c

is of the

A;,jBj(l),...,

j=ltl

5 j=l+l

323

Volume

INFORMATION

41. Number 6

where not all 6, are zero. zero, we have

Since

not

PROCESSING

all 6, are

LETTERS

over F, then we have

17 April 1992

for any nonzero

C E Span,(B(p))

rank C = deg p. G, = i

S,Bj”‘+

5

i=l

j=l+l

$5 Span,({Bjq)r

,..,,

Bi:)))

for h=l,...,

by ( * >, for any P E Span,(S,)

Therefore,

(5)

h’,,iBj(h) s. we have

Another important property that will be used for the proof of the theorem is the following. Let Hi be nonsingular IZ X n-matrices for i = 1,.*., j a’ The matrix C$ ,Z,$/’ 8 H, is of the form ‘0

rank

P=

krank i=l

G,>,

ka,. i=l

Thus, by Lemma

3, we have (2)

Now, it is obvious dim Spari.

I

0“XII

that

where

=

Combining this with (1) and (2), the result lemma follows. 0

of the

0’nxn

“’

O,,,

is the n x n zero matrix.

Therefore, (6)

rank We now prove the theorem.

Let i,(n) denote the maximum possible number of distinct irreducible factors of a polynomial of degree II over the field F. The following lemma is known from [lo].

Let p(a) be any polynomial of degree n.

Theorem. Then

( &)n

z 2+

L@(P)) Lemma i,(n)

8. For sufficiently large n we have G~

2n

loq,,n .

For integers 1
k] =

It is known B(a”)

the m X m Hankel

1

if m-i-k+2=j,

0

otherwise.

= {I,$)1 j=

= {C;, C;,.

BY

(3) p(a)

of degree

II, (4)

@ ...

@B(Pk)).

(3) and (4),

B(p) l,...,m}.

..,C;-l},

= (B( adI) @ B( p,)) @(B(adk)

where C, is the companion matrix of p. It is well known that when p is an irreducible polynomial 324

Proof. Let p=p$’ .+.p$“, where pr,~..,pk are distinct irreducible polynomials. Let deg P = n, deg pj=ri,deg p,4=sj=ridi,ands,,< .*.
that

From [2,3], for any polynomial B(p)

ma-

-o(n).

=B,

@ 1.. @B,,

where B,=((Z~~~C~h)ll~i~dh,O~j~rh-l) for h= l,...,k, and CPA is the companion matrix of ph. We now prove the following claims. The first claim shows

Volume

INFORMATION

41. Number 6

PROCESSING

that the number of polynomials p$ of degree less than 1 log, n is small (o(n)). Claim 1. Let w be an integer such that deg p;‘l ’ *. p,dw>, n’j2, deg pfl

Then w G 2n1/2/log, si=r,di=deg

(7)

< n’12.

. * . p$,l

p”~>i

F, n and

log,,,n

Proof. By (7) and Lemma large n,

P=

c f&1/+ vtw,

8, for sufficiently

c q/v. VEB,-W,

Thus, there exist A,,j,l not max(dj - mi + 1, 1) such that r.-1 d, 2

[=O

hJ

all zero

for j >

12’ 8 c;,)

j=O

d,

=

I! -

I

c

1

CA

fp3 ’

\

;=o

2n’/2 w G log,,,n

exist constants {a,,( I/ E Wi} C_F not all zero and (~V)V~Bi-WI}~F such that

P = ‘c

for i=w,...,k.

17 April 1992

LETTERS

\

.c’ r,1,1

P,

I=0

J .

Suppose jo, where max(di - mi + 1, 1) <./‘o
.

Now, since deg plf’ . . . I)$- > n1i2 and s, G s2 < . . . Gq+., we have that nV2 s,=deg

p,dw>--

>(d,-m,+

> i log,,,n. W

This proves that Bi is an (si, 1Wi 1, si - m;ri + r,)set and Claim 3 is proved. Now, by Claim 2,

Since s, G s,, , G . . . G sk we have si=degpdl>$log,F,n

fori=w,w+l,...,

l)ri=si-miri+ri.

k.

This completes the proof of Claim 1. Let

min w<,
I > log,,,

log n,

and by Lemma 7, LF(B( P))

mj=

[log”~I1og

“1

for i=w,...,k,

(8)

>LL, ~O}~w~imes~{O}~B,,,~.. ( =L,(B,+,@ ... @B,)

and

. @B,

1

k

Wi= {l~‘@C~ZImax(di-mi+ 0~1


2

1, 1) GjGdi,

C si- (k logIF, log fl) i=w

for i=w,...,k.

+NF

log,,,

log n,

Claim 2. We have

i: (Si-miri+ri) i=W

. i (9)

lwi I 3 l”g,,C, log n

for i = w, . . . , k.

If max(d, - mi + 1, 1) = di - mj + 1, then by (81, IWi I =miri 2 IOgi,l log n. If max(d, - mi + 1, 1) = 1, then by Claim 1, IW 1 = d,r, =si > log,,, n > log log,,,n. Claim 3. The set Bi is an (si, IW, 1, s, - m,r, + r,)-set for i = w,. . . , k. Proof. Obviously, I Bi I = si = rid,. If P E Span.(B,) and P @ Span,(B, - Wi>, then there

We now estimate each term in (9). By (71, we have k

xsi=deg

p$...pp>n-n’/2.

(10)

i=w

By Lemma 8. k
2n log,,,n 325

Volume

41, Number

6

INFORMATION

log,,,

log IZ =o(II).

(11)

By (8), (10) and (111, k C ( si - mrri i=w an

+ ri)

-nl/*-

i

(mi-

l)ri

i=w an-n

“*-k

log,F,

log n=n-o(11).

(12)

Combining

this with (9), (10) and (11) we get

L,(B(p))

an

+&([log,,,

log +

.-o(n))

-o(n). Now, by Lemma qpog,.,

(13)

4, we have

log +

.-o(n)) 1 1E; 1hlFl“‘g

nl _ 1F 1

x(n - o(n)) =

LETTERS

17 April 1992

References

Therefore, k

PROCESSING

(l+&)n-o(n).

Combining

this with (13), we obtain

L&0))

2 ( 2 + A)“-0’“).

the result q

[l] A. Alder and V. Strassen, On the algorithmic complexity of associative algebras, Theoret. Comput. Sci. 15 (1981) 201-211. [2] A. Averbuch, Z. Galil and S. Winograd, Classification of all minimal bilinear algorithms for computing the coefficients of the product of two polynomials in the algebra G]u]/(u”), Theoret. Comtwt. Sci. 58 (1988) 17-56. Classification of ]31 A. Averbuch, Z. Galil and S. Winograd, all minimal bilinear algorithms for computing the coefficients of the product of two polynomials in the algebra G[u]/(Q(u)‘), I > 1, Manuscript. [41 R.W. Brockett and D. Dobkin, On the number of multiplications required for a matrix multiplication, SZAM J. Comput. 5 (1976) 624-628. 151 N.H. Bshouty, A lower bound for matrix multiplication, in: Proc. 29th Ann. Symp. on Foundations of Computer Science, 1988. [61 N.H. Bshouty, On the extended direct sum conjecture, in: Proc. 21st Ann. ACM Symp. on Theory of Computing, 1989. [71 D.V. Chudnovsky and D.V. Chudnovsky, Algebraic complexities and algebraic curves over finite fields, Proc. Nat. Acad. Sci. 84 (1987) 1739-1743. complexity of a pair of 181 D.Yu. Grigorive, Multiplicative bilinear forms and of polynomial multiplication, Lecture Notes in Computer Science 46 (1978) 250-256. complexity of a set of quadratic 191J. Jaja, The computational functions, J. Comput. System Sci. 24 (1982). complex[lOI M. Kaminski and N.H. Bshouty, Multiplicative ity of polynomial multiplication over finite field, J. ACM 36 (1989) 150-170. 1111A. Lempel, G. Seroussi and S. Winograd, On the complexity of multiplication in finite fields, Theoret. Comput. Sci. 22 (1983) 285-296. [121 A. Lempel and S. Winograd, A new approach to errorcorrecting codes, IEEE Trans. Inform. Theory 23 (1977) 503-508. [131 V. Strassen, Gaussian elimination is not optimal, Numer. Math. 13 (1969) 354-356. [141 J.H. Van Lint, Introduction to Coding Theory (Springer, New York, 1980). of 2X2 matrices, Linear 1151 S. Winograd, On multiplication Algebra Appl. 4 (1971) 381-388.