Group multiplication of logical elements by networks having limited inputs and unit delay in computing output functio

Spira, P. M.

English

NASA Technical Reports Server

3 
5 
SEL-68-003 
ON THE TIME REQUIRED FOR GROUP MULTIPLICATION 
by 
P .  M .  Sp ira  
January 1968 
S c i e n t i f i c  Report No. 24 
Prepared under 
Nat iona l  Aeronaut ics  and Space Admin i s tra t ion  
Grant N s G  377 
Radiosc i ence  Laboratory 
Stanford U n i v e r s i t y  S tanford ,  C a l i f o r n i a  
Stanford E l e c t r o n i c s  Laborator ie s  
https://ntrs.nasa.gov/search.jsp?R=19680006982 2020-03-12T09:22:51+00:00Z
ON THE TIME RFQUIRED FOR GROUP MULTIPLICATION 
by P. M. Spira 
1. Introdurt ion 
This paper is concerned with the time required to perform group 
multiplication by means of networks of logical elements--each having a 
limited number of inputs and unit delay in computing their output function. 
Previously, Winograd [l] has considered the problem and has given a lower 
bound on the time required to multiply in a finite group as well as a meth- 
od valid for abelian groups. Here we give a new lower bound which is, in 
general, higher than Winograd's but which reduces to his bound if the group 
of interest is abelian. In addition, a scheme to realize multiplication in 
any group, abelian or  not, is given. This new circuit has computation time 
either equal to or  one time unit greater than our lower bound. Also, if 
the group of interest is abelian, it computes at least as rapidly as 
Winograd's network to do the same multiplication. 
2. Basic Concepts 
Let C be a logical circuit composed of elements having at most r 
input lines, one splittable output line, and with unit delay in computing 
their outputs. Each line carries values from the set Jd = (0, 1, ..., 
d-11. Let the input lines of C be partitioned into two sets with I 
the set of possible configurations on the j (j = 1,2). 0 is the set 
of output configurations. Following Winograd, such a circuit is called a 
(d, r )  circuit. 
c, j th 
C 
Due to the unit delay in the elements comprising the circuit, there 
will be a certain delay between the time the inputs are presented and the 
time the output is available. 
Let G be a finite group. 
Definition. A circuit C is said to compute multiplication in G in time 
7 if there are maps Z : G + I (j = 1,2) and a 1-1 function h: 
G + Oc 
time 0 through time T-1 the output at time will be h(ab). 
j c, j 
such that if C receives constant input [z (a), z2(b) 1 from 1 
Definition. Let h.(x) be the value on the jth output line of c when 
J 
the overall output configuration is h(x). 
1 SEL-68-003 
D e f i n i t i o n .  L e t  A C G. Then A i s  s a i d  t o  be r i g h t  j - s e p a r a b l e  by C - 
i f ,  g iven  a a E A wi th  a a2 t h e n  3 bcG f o r  which 1' 2 1 
L e f t  s e p a r a b i l i t y  i s  def ined  d u a l l y .  
D e f i n i t i o n .  For any set  S l e t  I S 1  be i t s  c a r d i n a l i t y .  L e t  rx? be t h e  
l e a s t  i n t e g e r  2 x, and l e t  1xJ be t h e  g r e a t e s t  i n t e g e r  5 x .  
Now l e t  G be a f i n i t e  group. L a t e r  on we w i l l  u t i l i z e  t h e  fo l lowing  
d e f i n i t i o n s  and lemma. 
D e f i n i t i o n .  L e t  H be a subgroup of G. Say P(a,H) ho lds  i f  3 acH\{e) 
ho lds  i f  conta ined  i n  any n o n - t r i v i a l  subgroup of H and say  
P(a,H) ho lds  f o r  any a€H\(e).  L e t  a ( G )  be t h e  o r d e r  of t h e  l a r g e s t  
H 5 G such  t h a t  P(H) holds .  
P(H) 
D e f i n i t i o n .  I f  P(G) ho lds  or G = (e)  l e t  B ( G )  = 1. If not f o r  any 
ccG\{e)  l e t  @ ( c )  be t h e  maximum o r d e r  of any subgroup of G not 
c o n t a i n i n g  c and let  @ ( G )  = min [ @ ( c ) ) .  
-\E e 3 
Lemma 2 . 1 .  For any f i n i t e  group G, / G I  2 a ( G )  B ( G ) .  
Proof . 
True t r i v i a l l y  i f  P(G) holds  or i f  G = { e ) .  So assume n o t .  L e t  
H < G and acH wi th  P(a ,H)  hold ing  and IHI = a ( G ) .  L e t  K S G wi th  
1K1 5 @ ( G )  and aCK. Then, s i n c e  H fl K is a subgroup of H ,  H f l  K = { e } .  
Say H gEG such t h a t  
2 SEL-6 8 - 003 
(I' 
-1 h2g = k Thus klk2 E H  n K.  hlg = k l ;  2 '  Then 3 hl,h2EH f o r  which 
Hence k = k2 and there can  be a t  most one element of K i n  any r i g h t  1 
c o s e t  of H. So 
3. The Lower Bound 
W e  now g i v e  o u r  lower bound a f t e r  some p re l imina ry  lemmas. 
Lemma 3.1. I n  a ( d , r )  c i r c u i t  t h e  ou tpu t  of an element a t  t i m e  1 
can depend upon a t  most r7 input  l i n e s ,  
Proof .  
J u s t  cons ide r  t h e  fan- in  wi th  modules having r inpu t  l i n e s  each t o  
t h e  h e i g h t  of 7. I 
D e f i n i t i o n .  For a ( d , r )  c i r c u i t  C which m u l t i p l i e s  i n  a f i n i t e  group 
G le t  
R. = (xcG:h.(xa) = h . ( a )  VaEG) 
J J J 
From t h i s  w e  o b t a i n  
Lemma 3.2. R and L are subgroups of G. A maximal s i z e  r i g h t  j- 
j j 
s e p a r a b l e  subse t  of G c o n t a i n s  e x a c t l y  one element from each l e f t  
c o s e t  of R i n  G ;  a maximal s ize  l e f t  j - s e p a r a b l e  subse t  of 
G c o n t a i n s  e x a c t l y  one element from each r i g h t  cose t  of L i n  G. 
j 
j 
3 SEL-68-003 
Proof .  
I f  x , y € R  t h e n ,  g iven  any acG 
j 
Thus x-'ycRj , SO it is a subgroup. If h j ( x a )  = hj(Ya) VaeG 
Thus Y' lXERj  SO t h a t  XRj = YRj. The rest fo l lows  by d u a l i t y .  I 
t h e n  h.(y-'xa) = h j ( a )  VaEG. 
J 
The under ly ing  lemma used i n  d e r i v i n g  t h e  lower bound is  
Lemma 3.3. L e t  C be a ( d , r )  c i r c u i t  t o  m u l t i p l y  f o r  a group G i n  
t i m e  'T. Then 
Proof .  
I G l  
The jth output  a t  t i m e  T must depend upon a t  l e a s t  
input  l i n e s  of I and upon a t  l e a s t  
s i n c e  t h e r e  a r e  r i g h t  and l e f t  j - s e p a r a b l e  sets of s i z e  
I 
c ,  1 
I r e s p e c t i v e l y .  Thus, from lemma 3.1 
W e  have enough t o  prove 
Theorem 3 . 4 .  L e t  G be a f i n i t e  group. Then i f  C i s  a ( d , r )  c i r c u i t  
t o  m u l t i p l y  i n  G i n  t i m e  T 
4 SEL-68-003 
Proof .  
4 
Let acG be  such t h a t  B(a) = P ( G ) .  Now 3 j f o r  which h j ( a )  
h j ( e ) .  Thus a{Rj(e) and a{Lj(e) .  Hence 
lRj 1 B ( G )  ; ( L j /  5 B ( G )  
and t h e  r e s u l t  fo l lows  from lemma 3.3. 1 
C o r o l l a r y  3.5. Also 
Thus Winograd’s r e s u l t  i s  a c o r o l l a r y  of our  lower bound. 
I n  h i s  paper  [ l]  Winograd i n d i c a t e s  t h a t ,  i f  G i s  a b e l i a n ,  then  
Q(G) is t h e  o r d e r  of t h e  l a r g e s t  c y c l i c  p-subgroup of G. A complete 
c h a r a c t e r i z a t i o n  of a( G) i s  provided below. F i r s t  
‘n 9 
D e f i n i t i o n .  
an w i t h  two 
- 1 
- a  
t h e  genera l ized  q u a t e r n i o n  group, i s  t h e  group of o r d e r  
g e n e r a t o r s  a and b s a t i s f y i n g  
1 
2 2n-2 
b = a  ba = a-’b 
Theorem 3.6. A p-group conta ins  a unique subgroup of o r d e r  p i f f  i t  i s  
c y c l i c  or  a genera l ized  q u a t e r n i o n  group. ( I t  must be c y c l i c  i f  
p i s  odd) .  
Proof .  
See H a l l  [ 2 )  p 189. 1 
C o r o l l a r y  3.7. L e t  G be any f i n i t e  group. Then Q(G)  i s  e i t h e r  t h e  
o r d e r  of t h e  l a r g e s t  c y c l i c  p-subgroup of G o r  t h e  o r d e r  of t h e  
5 SEL-68-003 
l a r g e s t  gene ra l i zed  qua te rn ion  group conta ined  i n  G, whichever 
i s  l a r g e r .  
P roof .  
Let H be any subgroup of G. A s  noted above, if P(H) ho lds ,  then  
H i s  a p-group wi th  IHI = pn f o r  some prime p and i n t e g e r  n .  But 
every  subgroup of H c o n t a i n s  a subgroup of o r d e r  p .  Hence P(H) w i l l  
ho ld  i f f  H c o n t a i n s  a unique subgroup of o r d e r  p. But then  H i s  
e i t h e r  cycl ic  of o r d e r  pn o r  a gene ra l i zed  q u a t e r n i o n . (  
W e  c l o s e  t h i s  s e c t i o n  wi th  an example of a group G wi th  
0) B ( G )  < I G l .  
Example. 
Let p be  a prime > 2 .  Then t h e r e  i s  a group genera ted  by t h r e e  
g e n e r a t o r s  a ,  b ,  and c with d e f i n i n g  r e l a t i o n s  [ 2 ]  (page 52) 
ab  = bac ca  = ac  c b  = bc P P  a P = b  = c  = e  
2 2 
which has no element of o r d e r  p . L e t  H be a subgroup of o r d e r  p . 
Then i t  i s  easy  t o  show t h a t  CEH. Hence 
But a ( G )  w i l l  a l s o  be p s i n c e ,  i f  2 does not d i v i d e  t h e  o r d e r  of a 
group G, t hen  a ( G )  i s  t h e  s i z e  of t h e  l a r g e s t  cyc l ic  p-subgroup of 
G. Hence 
Th i s  shows t h a t  our  lower bound is ,  i n  g e n e r a l ,  h i g h e r  t han  Winograd's. 
The fo l lowing  lemma shows t h a t  t h e  two bounds a r e  equa l  i f  G i s  a b e l i a n .  
6 SEL-68-003 
Lemma 3 . 6 .  Let G be a b e l i a n .  Then [ G I  = a ( G )  B ( G ) .  
Proof.  
G is a b e l i a n  so G zz J = J1 X . . . X Js, where each is a c y c l i c  
r .  Ji r. r 
p-group w i t h  lJil = Pi 1 and, w i t h  no loss of g e n e r a l i t y  Pi 1 I p .  j J 
if i < j .  Assume s > 1 or else t h e  theorem is  t r i v i a l .  Let b .  gen- 
e r a t e  J i ; l  6 i S ,  and w r i t e  a g e n e r i c  element of J a s  (bl  1, ..., 
t r S i m i l a r l y  blf J2 X bs s )  . Then P(bl ,  J ) holds and a( J) = p1 1. 
... x Js but is i n  any subgroup which i n t e r s e c t s  J1 n o n - t r i v i a l l y .  
Thus 
1 t 
1 
t t 
j )  and 
1 L e t  b = (bl , ... bs ”) be any element of J \ ( e / .  Then 3 j with 
t j  .$ 0 (mod p r j 
and t h e  theorem is t r u e  by isomorphism.! 
4 .  The M u l t i p l i c a t i o n  Scheme f o r  F i n i t e  Groups 
I n  t h i s  s e c t i o n  w e  d e s c r i b e  a scheme t o  m u l t i p l y  w i t h  a ( d , r )  
c i r c u i t  v a l i d  i n  any f i n i t e  group G--abelian or not--and show t h a t  our 
computation t i m e  is never  more--and o f t e n  less - - than  t h a t  of Winograd’s 
c i r c u i t  i n  t h e  c a s e  t h a t  G i s  a b e l i a n .  I n  a d d i t i o n ,  our scheme w i l l  be  
v a l i d  for r 2 2 ,  d L 2 whereas h i s  i s  no t  v a l i d  u n l e s s  r 2 3. 
7 SEL-6 8 -003 
Lemma 4 .1 .  L e t  K be any subgroup of G. Then H a ( d , r )  c i r c u i t  t o  
compute $ : G x G + [0,1) i n  t i m e  
where 
Proof .  
. Pick  a cose t  r e p r e s e n t a t i v e  vi E K V ~  f o r  each r i g h t  I4 Fr 
w i l l  be a set  of l e f t  c o s e t  represen-  
Let M = 
cose t  of K i n  G. Then {v;'} 
-1 -1 -1 -1 -1 
t a t i v e s ,  f o r  v .  K = v .  K i f f  v v EK i f f  v = v Define maps 
1 J i j  J i *  
z and z from G t o  t h e  space of Fogd  -ary v e c t o r s  over  J such 
1 2 d 
t h a t  
Thus t h e  c i r c u i t  computes 
7 = h o g  r 
(p i n  
log M 
d 
- 
where 0 is t h e  a l l  z e r o  v e c t o r  and @ is  componentwise a d d i t i o n  modulo 
d .  Note t h a t  z maps any two elements i n  t h e  same l e f t  cose t  t o  t h e  
same v e c t o r .  The f i r s t  l e v e l  of t h e  c i r c u i t  c o n s i s t s  of rlog M1 modulo 
d adders .  I f  ab  i s  be ing  computed t h e s e  adders  sum z , ( a )  and z ( b )  
componentwise mod d.  Thus a l l  o u t p u t s  a r e  0 i f f  H j such t h a t  acKv. J 
and bcv .  K, i .e .  i f f  abcK. The rest of t h e  c i r c u i t  i s  a f an - in  of r 
input  elements having ou tpu t  0 i f f  a l l  i n p u t s  a r e  0 and output  1 if 
a t  l e a s t  one input  is nonzero. Th i s  f an - in  has depth  
2 
d 
2 
-1 
J 
[logrflogdM1l. 
t i m e  
1 + 1  I 
8 SEL-68-003 
, 
Corollary 4 . 2 .  There  i s  a ( d , r )  c i r c u i t  t o  t e l l  i f  abEKu f o r  any 
uEG i n  t h e  same time. 
n 
Lemma 4.3. Say G has  subgroups K1, ... , K such t h a t  n K = ( e ) .  
n j =1 j 
Then knowing t h e  r i g h t  cose t s  c o n t a i n i n g  any agG 
s u f f i c e s  t o  de te rmine  a .  
n n ' K.a = jElKja2 . 
Proof .  
Say 3a  , a  E G  such t h a t  K.a = K.a VJ. Then 
1 2  J 1  5 2  j=1 J 1 
Thus a l  = a2. I 
C o r o l l a r y  4.4 .  
c i r c u i t  t o  
If K1, ... , K 
compute m u l t i p l i c a t i o n  i n  G i n  t i m e  
is such a set of subgroups t h e n  3 a ( d , r )  
n 
W e  now immediately ob ta in  
Theorem 4.5. For any d L 2 and any r L 2 t h e r e  is a ( d , r )  c i r c u i t  
t o  m u l t i p l y  i n  a f i n i t e  group G i n  t i m e  
Furthermore t h i s  c i r c u i t  is w i t h i n  one t i m e  u n i t  of t h e  f a s t e s t  
o b t a i n a b l e .  
P roof .  
The f i r s t  p a r t  fo l lows  from lemma 4.3 and t h e  d e f i n i t i o n  of B ( G ) ;  
t h e  rest fo l lows  from t h e  f a c t  t h a t  
9 SEL-68-003 
From lemma 3 . 5  i t  fo l lows  t h a t  i f  G i s  a b e l i a n  there i s  a ( d , r )  
c i r c u i t  t o  m u l t i p l y  i n  G i n  t i m e  
I n  h i s  paper cl] Winograd d e r i v e s  a ( d , r )  
m u l t i p l i c a t i o n  i n  such an a b e l i a n  group i n  t i m e  
c i r c u i t  which performs 
From t h e  f a c t  t h a t  t h i s  number is always a t  l e a s t  a s  g r e a t  a s  
and i n  a d d i t i o n  
it fo l lows  t h a t  h i s  computation t i m e  i s  never less than  ou r s  and o f t e n  
g r e a t e r  . 
Example. 
Say r = 4 and rlogdCX(G) 1 = 22k for some k 2 1 
Then Winograd's t i m e  is  1 + 2k and our  computation t i m e  is 1 + k,  
. 
i . e . ,  h i s  c i r c u i t  r e q u i r e s  about t w i c e  a s  long. 
The r e a d e r  can e a s i l y  c o n s t r u c t  a myriad of s i m i l a r  examples. 
It has been poin ted  ou t  t o  t h e  a u t h o r  by Winograd [31 t h a t  our 
c o n s t r u c t i o n  can be a l t e r e d  a s  f o l l o w s .  I f  there i s  an  i n t e g e r  k for 
which 2k 5 r,  t h e n  each element i n  t h e  f i rs t  l e v e l  of t h e  c i r c u i t  can 
, 
10 SEL-68-003 
a c t u a l l y  t es t  k components of t h e  inpu t  v e c t o r s .  T h i s  r e s u l t s  i n  a 
computation t i m e  of 
Thus a computation t i m e  of 
can be achieved.  T h i s  means t h a t  there a r e  c a s e s  i n  which o u r  o r i g i n a l  
c o n s t r u c t i o n  h a s  t i m e  one g r e a t e r  t h a n  t h e  lower bound but t h i s  a l t e r e d  
v e r s i o n  is  a s  r a p i d  a s  p o s s i b l e .  T h i s  is t r u e ,  e.g. ,  if 
[logd 
= 5 and r = 4 
11 SEL-68-003 
REFERENCES 
1. Winograd, "On the Time Required To Perform Addition," J. of the ACM, 
Vol. 12, N o .  2, April, 1965, pp. 277-285. -
2. Marshall Hall, Jr., The Theory of Groups, The Macmillan Company, New 
York, 1959. 
3. S. Winograd, private communication. 
. 
12 SEL-68-003 


On the time required for group multiplication

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