On the construction of the asymmetric Chudnovsky multiplication algorithm in finite fields without derivated evaluation

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On the construction of the asymmetric Chudnovsky

multiplication algorithm in finite fields without

derivated evaluation

Alexis Bonnecaze, Stéphane Ballet, Nicolas Baudru, Mila Tukumuli

To cite this version:

Alexis Bonnecaze, Stéphane Ballet, Nicolas Baudru, Mila Tukumuli. On the construction of the

asym-metric Chudnovsky multiplication algorithm in finite fields without derivated evaluation. Comptes

Rendus. Mathématique, Centre Mersenne (2020-..) ; Elsevier Masson (2002-2019), 2017, 355 (7),

pp.729 - 733. �10.1016/j.crma.2017.06.002�. �hal-01705865�

Contents lists available atScienceDirect

C. R.

Acad.

Sci.

Paris,

Ser. I

www.sciencedirect.com

Number theory/Computer science

On

the

construction

of

the

asymmetric

Chudnovsky

multiplication

algorithm

in

finite

fields

without

derivated

evaluation

Construction

effective

de

l’algorithme

asymétrique

de

multiplication

de

Chudnovsky

dans

les

corps

finis

Stéphane Ballet

a

,

Nicolas Baudru

b

,

Alexis Bonnecaze

a

,

Mila Tukumuli

a a_{AixMarseilleUniv,CNRS,CentraleMarseille,I2M,Marseille,France}

b_{AixMarseilleUniv,CNRS,CentraleMarseille,LIF,Marseille,France}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received21November2016 Acceptedafterrevision2June2017 Availableonline16June2017 PresentedbytheEditorialBoard

TheChudnovskyalgorithmforthemultiplicationinextensionsoffinitefieldsprovidesa bilinearcomplexityuniformlylinearwithrespecttothedegreeoftheextension.Recently, Randriambololona hasgeneralized the method,allowingasymmetry intheinterpolation procedureand leadingtonewupperboundsonthebilinearcomplexity.Inthisnote,we describethe construction ofthisasymmetricmethodwithoutderived evaluation. To do this,wetranslatethisgeneralizationintothelanguageofalgebraicfunctionfieldsandwe giveastrategyofconstructionandimplementation.

©2017Académiedessciences.PublishedbyElsevierMassonSAS.Thisisanopenaccess articleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).

r

é

s

u

m

é

L’algorithme de multiplication dans les corps finis de Chudnovsky a une complexité

bilinéaireuniformémentlinéaireenledegrédel’extension.Randriambololonaarécemment généralisé cette méthode en introduisant l’asymétrie dans la procédure d’interpolation et en obtenant ainsi de nouvelles bornes surla complexité bilinéaire. Dans cette note, nous décrivons la construction de cette méthode asymétrique sans évaluation dérivée. Pourcefaire,noustraduisonscettegénéralisationdanslelangagedescorpsdefonctions algébriques,etnousdonnonsunestratégiedeconstructionetd’implantation.

©2017Académiedessciences.PublishedbyElsevierMassonSAS.Thisisanopenaccess articleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).

E-mailaddresses:stephane.BALLET@univ-amu.fr(S. Ballet),nicolas.BAUDRU@univ-amu.fr(N. Baudru),alexis.BONNECAZE@univ-amu.fr(A. Bonnecaze),

tukumulimila@gmail.com(M. Tukumuli).

http://dx.doi.org/10.1016/j.crma.2017.06.002

1631-073X/©2017Académiedessciences.PublishedbyElsevierMassonSAS.ThisisanopenaccessarticleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).

730 S. Ballet et al. / C. R. Acad. Sci. Paris, Ser. I 355 (2017) 729–733

1. Introduction

Letq beaprimepower,

F

q thefinitefieldwithq elementsand

F

qn thedegreen extensionof

F

_q.Amongallalgorithms

ofmultiplicationsin

F

qn,thosebasedontheChudnovsky–Chudnovsky[6]methodareknowntoprovidethelowestbilinear

complexity.Thismethodisbasedoninterpolationonalgebraiccurvesdefinedoverafinitefieldandprovidesabilinear com-plexity,whichislinearinn.Theoriginalalgorithmusesonlypointsofdegree1,withmultiplicity1.BalletandRolland[4,5]

andArnaud[1]improvedthealgorithm,introducinginterpolationatpointsofhigherdegreeorhighermultiplicity.The sym-metryoftheoriginalconstructioninvolves2-torsionpointsthatrepresentanobstacletotheimprovementofupperbilinear complexity bounds. Toeliminate thisdifficulty, Randriambololona[8]allowed asymmetry inthe interpolation procedure, andthenPieltantandRandriambololona[7]derivednewbounds,uniforminq,ofthebilinearcomplexity.Unlikesymmetric constructions,noeffectiveimplementationofthisasymmetricconstructionhasbeendoneyet.When g

=

1,itisknown[3]

thatanasymmetricalgorithmcanalwaysbesymmetrized.However,forgreatervaluesofg,itmaynotbethecase.Thus,it isofinteresttoknowan effectiveconstructionofthisasymmetricalgorithm.Sofar,noeffectiveimplementationhasbeen proposedforsuchanalgorithm.

1.1. Multiplicationalgorithmandtensorrank

Themultiplicationoftwoelementsof

F

qn isan

F

_q-bilinearapplicationfrom

F

_qn

×F

_qn onto

F

_qn.Thenitcanbeconsidered

as an

F

q-linearapplication fromthe tensorproduct

F

qn

⊗

_Fq

F

_qn onto

F

_qn. Consequently, itcan also be considered asan

element Tmof

F

qn

⊗

_F q

F

qn



_⊗

Fq

F

qn where



denotesthedual.When Tmiswritten

Tm

=

r



i=1

x_i

⊗

y_i

⊗

ci

,

(1)

wherether elements x_i aswell asthe r elements y_i areinthedual

F

qn of

F

_qn whilether elementsc_i arein

F

_qn,the

followingholdsforanyx

,

y

∈ F

qn:x

·

y

=



r_i=1x

i

(

x

)

yi

(

y

)

ci.Thedecomposition(1)isnotunique.

Definition1.1.Everyexpressionx

·

y

=



r_i=1x_i

(

x

)

y_i

(

y

)

cidefinesabilinearmultiplicationalgorithm

U

ofbilinearcomplexity

μ

(U)

=

r.Suchanalgorithmissaidsymmetricifxi

=

yiforalli.

Definition1.2. Theminimalnumberofsummands ina decompositionofthetensor Tm ofthemultiplicationiscalledthe

bilinearcomplexity(resp.symmetricbilinearcomplexity)ofthemultiplicationandisdenotedby

μ

q

(

n

)

(resp.

μ

symq

(

n

)

):

μ

q

(

n

)

=

min

U

μ

(

U

)

where

U

isrunningoverallbilinearmultiplicationalgorithms(resp.allbilinearsymmetricmultiplicationalgorithms)in

F

qn

over

F

q.

1.2. Organisationofthenote

InSection2,wegiveanexplicittranslationofthegeneralizationoftheChudnovskyalgorithmgivenbyRandriambololona

[8,Theorem3.5].TheninSection3,bydefininganewdesignofthisalgorithm,wegiveastrategyofconstructionand imple-mentation.Inparticular,thankstoasuitablerepresentationoftheRiemann–Rochspaces,wepresentthefirstconstructionof asymmetriceffectivealgorithmsofmultiplicationinfinitefields.Thesealgorithmsaretailoredtohardwareimplementation andthey allow computationstobe parallelized whilemaintaininga lownumberof bilinearmultiplications. InSection 4, wegiveananalysisofthenotasymptoticalcomplexityofthisalgorithm.

2. MultiplicationalgorithmsoftypeChudnovsky:generalizationofRandriambololona

In this section, we present a generalization of Chudnovsky-type algorithms, introduced in [8, Theorem 3.5] by Ran-driambololona, which is possibly asymmetric. Since our aim is to describe explicitly the effective construction of this asymmetric algorithm, we transformthe representation ofthis algorithm,initially made inthe abstract geometrical lan-guage,inthemoreexplicitlanguageofalgebraicfunctionfields.

Let F

/

F

q be an algebraic functionfield overthefinitefield

F

q ofgenus g

(

F

)

.Wedenote by N1

(

F

/

F

q

)

thenumberof

places ofdegreeone ofF over

F

q.IfD isadivisor,

L(

D

)

denotes theRiemann–Rochspaceassociatedwith D.Wedenote

by

O

Q the valuationring oftheplace Q andby FQ its residueclassfield

O

Q

/

Q ,whichisisomorphic to

F

qdeg Q,where

deg Q isthedegreeoftheplace Q .

Intheframeworkofalgebraicfunction fields,theresult[8,Theorem3.5]ofRandriambololonacanbestatedasin The-orem 2.1.Notethat wedo nottake intoaccountderived evaluations,sincewe arenot interestedinasymptoticresults.It means thatwedescribe thisasymmetric algorithmwiththedivisor G

=

P1

+ · · · +

PN wherethe Pi arepairwisedistinct

Let usdefine the following Hadamardproduct in

F

_ql1

× F

ql2

× · · · × F

qlN, where the li’s denote positive integers, by

(

u1

,

. . . ,

uN

)

 (

v1

,

. . . ,

vN

)

= (

u1v1

,

. . . ,

uNvN

)

.

Theorem2.1.LetF

/

F

qbeanalgebraicfunctionfieldofgenusg over

F

q.Supposethatthereexistsaplace Q ofdegreen.Let

P =

{

P1

,

. . . ,

PN

}

beasetofN placesofarbitrarydegreenotcontainingtheplaceQ .Supposethatthere existtwoeffectivedivisorsD1

,

D2

ofF

/

F

qsuchthat:

(i) theplaceQ andtheplacesof

P

arenotinthesupportofthedivisorsD1andD2;

(ii) thenaturalevaluationmapsEifori

=

1

,

2 definedasfollowsaresurjective

Ei

:



L

(

Di

)

−→ F

qn



F_Q

f

−→

f

(

Q

)

(iii) thenaturalevaluationmapdefinedasfollowsisinjective

T

:



L

(

D1

+

D2

)

−→ F

_qdeg P1

× F

qdeg P2

× · · · × F

qdeg P N f

−→

(

f

(

P1

),

f

(

P2

), . . . ,

f

(

PN

))

Thenforanytwoelementsx

,

y in

F

qn,wehave:

xy

=

EQ

◦

T_|−_{Im T}1



T

◦

E−₁1

(

x

)



T

◦

E−₂1

(

y

)



,

whereEQ denotesthecanonicalprojectionfromthevaluationring

O

Q oftheplace Q initsresidueclassfieldFQ,

◦

thestandard

compositionmap,T_|−_{Im T}1 therestrictionoftheinversemapofT ontheimageofT ,E−_i1theinversemapoftherestrictionofthemapEion

thequotientgroup

L(

Di

)/

ker Eiand



theHadamardproductin

F

qdeg P1

× F

qdeg P2

× · · · × F

qdeg P N;and

μ

q

(

n

)

≤



N

i=1

μ

q

(

deg Pi

)

.

3. Effectivealgorithm

3.1. Methodandstrategyofimplementation

Theconstructionofthealgorithmisbasedonthechoiceoftheplace Q ,theeffectivedivisorsD1 andD2,thebasesof

spaces

L(

D1

)

,

L(

D2

)

and

L(

D1

+

D2

)

andthebasisoftheresidueclassfield FQ.

Inpractice, followingtheideas of [2],divisors D1 and D2 arechosen asplaces ofdegree n

+

g

−

1.Furthermore,we

requiresomeadditionalproperties,whicharedescribedbelow.

3.2. FindinggoodplacesD1,D2,andQ

Inordertoobtainthegoodplaces,weproceedasfollows:

– wedraw atrandom anirreducible polynomial

Q(

x

)

ofdegreen in

F

q

[

X

]

andcheck thatthispolynomialis primitive

andtotallydecomposedinthealgebraicfunctionfield F

/

F

q;

– wechooseaplace Q ofdegreen abovethepolynomial

Q(

x

)

;

– wechooseaplace Q ofdegreen amongtheplacesofF

/

F

q lyingabovethepolynomial

Q(

x

)

;

– wedrawatrandomaplace D1 ofdegreen

+

g

−

1 andcheckthatD1

−

Q isanon-specialdivisorofdegreeg

−

1,i.e.

dim

L(

D1

−

Q

)

=

0;

– wedrawatrandomaplace D2 ofdegreen

+

g

−

1 andcheckthatD2

−

Q isanon-specialdivisorofdegreeg

−

1,i.e.

dim

(

D2

−

Q

)

=

0.

3.3. Choosinggoodbasesofthespaces TheresiduefieldFQ.

Wechoosethecanonicalbasis

B

Q generatedbyaroot

α

ofthepolynomial

Q(

x

)

,namely

B

Q

= (

1

,

α

,

α

2

,

...,

α

n−1

)

.From

nowon,weidentify

F

qn to F_Q,astheresidueclassfield F_Q oftheplace Q isisomorphictothefinitefield

F

_qn. TheRiemann–Rochspaces

L(

D1

)

and

L(

D2

)

.

Wechooseasbasisof

L(

Di

)

thereciprocalimage

B

Di ofthebasis

B

Q

= (φ

1

,

. . . ,

φ

n

)

of FQ bytheevaluationmap Ei,

namely

B

Di

= (

E−

1

i

(φ

i

),

. . . ,

E−i1

(φ

n

))

.

Letusdenote

B

Di

= (

fi,1

,

...,

fi,n

)

with fi,1

=

1 fori

=

1

,

2. TheRiemann–Rochspace

L(

D1

+

D2

)

.

Notethatsince D1 andD2areeffectivedivisors,wehave

L(

D1

)

⊂

L(

D1

+

D2

)

and

L(

D2

)

⊂

L(

D1

+

D2

)

.

732 S. Ballet et al. / C. R. Acad. Sci. Paris, Ser. I 355 (2017) 729–733

Proposition3.1.LetD1,D2andQ beplaceshavingthepropertiesdescribedin(3.2).Considerthemap



:

L(

D1

+

D2

)

→

FQ such

that

(

f

)

=

f

(

Q

)

forf

∈

L(

D1

+

D2

)

.Thereexistsavectorspace

M

⊆

ker



ofdimensiong suchthat

L

(

D1

+

D2

)

=

L

(

D1

)

⊕

L

r

(

D2

)

⊕

M

,

where

L

r

(

D2

)

issuchthat

L(

D2

)

= F

q

⊕

L

r

(

D2

)

and

⊕

denotesthedirectsum.Inparticular,ifg

=

0,then

M

=

K er



isequal

to

{

0

}

.

We chooseasbasisof

L(

D1

+

D2

)

thebasis

B

D1+D2 definedby

B

D1+D2

= (

f1

,

. . . ,

fn

,

fn+1

,

. . . ,

f2n+g−1

)

where

B

D1

=

(

f1

,

. . . ,

fn

)

isthebasisof

L(

D1

)

,

(

fn+1

,

. . . ,

f2n−1

)

isabasisof

L

r

(

D2

)

such that fn+j

=

f2,j+1

∈

B

D2 with

B

D1 and

B

D2

definedinSection3.3and

B

_M

= (

f2n

,

. . . ,

f2n+g−1

)

isabasisof

M

.

3.4. Productoftwoelementsin

F

qn

Letx

= (

x1

,

. . . ,

xn

)

andy

= (

y1

,

. . . ,

yn

)

betwoelementsof

F

qn givenbytheircomponentsover

F

_qrelativetothechosen

basis

B

Q.Accordingtothepreviousnotation,wecanconsiderthatx andy areidentifiedasrespectively fx

=



n

i=1xif1,i

∈

L(

D1

)

and fy

=



ni=1yif2,i

∈

L(

D2

)

.

Theproduct fxfy ofthetwoelements fx and fyistheirproductinthevaluationring

O

Q.Thisproductliesin

L(

D1

+

D2

)

,since D1 andD2 areeffectivedivisors.Weconsiderthat x and y arerespectivelytheelements fx and fy embedded

intheRieman–Roch space

L(

D1

+

D2

)

,viarespectivelytheembeddings Ii

:

L(

Di

)

−→

L(

D1

+

D2

)

,definedby I1

(

fx

)

and

I2

(

fy

)

asfollows.If, fx and fyhaverespectivelycoordinates fxi and fyi in

B

D1+D2,wherei

∈ {

1

,

. . . ,

2n

+

g

−

1

}

,wehave:

I1

(

fx

)

= (

fx1

:=

x1

,

. . . ,

fxn

:=

xn

,

0

,

. . . ,

0

)

andI2

(

fy

)

= (

fx1

:=

y1

,

0

,

. . . ,

0

,

fyn+1

:=

y2

,

. . . ,

fy2n−1

:=

yn

,

0

,

. . .

0

)

.Now itis

clearthatknowingx (resp. y)or fx (resp. fy)bytheircoordinatesisthesamething.

Theorem3.2.LetP_Msbetheprojectionof

L(

D₁

+

D₂

₎

onto

M

s

=

L(

D₁

₎

⊕

L

_r

₍

D₂

₎

,andlet

_

bethemapdefinedasin

Proposi-tion 3.1.Then,foranyelementsx

,

y

∈ F

qn,theproductofx byy issuchthat

xy

=  ◦

P_Ms



T_|−_{Im T}1



T

◦

I1

◦

E−11

(

x

)



T

◦

I2

◦

E−21

(

y

)



,

where

◦

denotesthestandardcompositionmap,T−_{|Im T}1 therestrictionoftheinversemapofT ontheimageofT ,and



theHadamard productasinTheorem 2.1.

Wecannowpresentthesetupalgorithm(Algorithm 1),whichisdoneonlyonce.

Algorithm1Setupalgorithm.

INPUT: F/Fq, Q,D1,D2,P1,. . . ,PN.

OUTPUT: T and T−1_.

(i) TherepresentationofthefinitefieldFq=<a>,wherea isafixedprimitiveelement.

(ii) ThefunctionfieldF/Fq,theplaceQ ,thedivisorsD1andD2andthepointsP1,. . . ,PNaresuchthatConditions(ii)and(iii)inTheorem 2.1are

satisfied.Inaddition,werequirethat1≤i≤Ndeg Pi=2n+g−1.

(iii) RepresentFqn inthecanonicalbasisBQ= {1,α,α2,...,αn−1},whereFqn=<α>withαaprimitiveelementasinSection3.3.

(iv) Constructabasis(f1,. . . ,fn,fn+1,. . . ,f2n+g−1)ofL(D1+D2),where(f1,. . . ,fn)isthebasisofL(D1),(f1,fn+1,. . . ,f2n−1)thebasisofL(D2)

and(f2n,. . . ,f2n+g−1)thebasisofM,definedinSection3.3.

(v) ComputethematricesT andT−1_.

(vi) Computethematrix.

Themultiplicationalgorithm(Algorithm 2)ispresentedhereafter.

4. Complexityanalysis

Intermsofnumberofmultiplicationsin

F

q,thecomplexity ofthismultiplicationalgorithmisasfollows:calculationof

z andt needs2

(

2n2

+

ng

−

n

)

multiplications,calculationofu needs

(

2n

+

2g

−

2

+

r

)

sup1≤i≤r

μq(i)

i bilinearmultiplications

andcalculation of2n

−

1 firstcomponents of w needs

(

2n

+

g

−

1

)(

2n

−

1

)

multiplications (remark that,inAlgorithm 2, we justhaveto computethe 2n

−

1 first componentsof w). Thecalculation ofxy needs n

+

g multiplications.The total complexityintermsofmultiplicationsisboundedby8n2

₊

_n

₍

_4g

₋

₅

₎

_{+ (}

_2n

₊

_2g

₋

₂

₊

_r

₎

_sup

1≤i≤r

μq(i)

i .

The generalconstructionoftheset-upalgorithminvolvessome randomchoiceofdivisorshavingprescribedproperties overanexponentiallylargesetofdivisors.Togetapolynomiallyconstructiblealgorithmwithlinearcomplexity,oneneeds toconstructexplicitly(i.e.polynomially)pointsofcorrespondingdegreesn oncurvesofarbitrarygenuswithmanyrational points. Unfortunately,sofaritisunknown howtoproducesuch points(cf.[9,Section 4,Remark5] and[8,Remark 6.6]). Hence,theasymptoticcomplexityofsuchaconstructionisanopenproblem.

Algorithm2Multiplicationalgorithm. INPUT: x= (x1,. . . ,xn)and y= (y1,. . . ,yn). OUTPUT: xy. (i) Compute ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ z1,d1 . . . zn,dn zn+1,dn+1 . . . zN,dN ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ z1 . . . zn zn+1 . . . z2n+g−1 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ =T ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ x1 . . . xn 0 . . . 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ and ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ t1,d1 . . . tn,dn tn+1,dn+1 . . . tN,dN ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ t1 . . . tn tn+1 . . . t2n+g−1 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ =T ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ y1 0 y2 . . . yn 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ,

whereNi=1di=2n+g−1,(zi,j,ti,j)∈ (F_qd j)2,(zi,ti)∈ (Fq)2and0 standsforthenullvector.

(ii) ComputetheHadamardproductu= (u1,d1,. . . ,uN,dN)= (u1,. . . ,u2n+g−1),whereui,di=zi,diti,di,inFqd1× Fqd2× · · · × FqdN asinTheorem 2.1.

(iii) Computew= (w1,. . . ,w2n+g−1)=T−1(u).

(iv) Extractw= (w1,. . . ,w2n−1)(instep(iii),justthe2n−1 firstcomponentshavetobecomputed).

(v) Returnxy= (w).

References

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