Fast Multi-Scalar Multiplication Methods on Elliptic Curves with Precomputation Strategy using Montgomery Tri

ck

Hitachi Ltd.Katsuyuki Okeya

Kouichi Sakurai Kyushu Univ.

2/22

Abstract

The use of multi-scalar multiplication

in the verification of ECDSA

Speeding up the multi-scalar multiplication

The transformation from scalar multiplication to multi-scalar

multiplication

Motivation

Problem

Result Efficient Precomputation provides speedup for multi-scalar multiplication

3 times faster

[GLV01]

[ANSI]

3/22

Contents

Multi-Scalar Multiplication

Target of Speedup

Proposed Method

Comparison

4/22

What isMulti-Scalar Multiplication?

Scalar multiplication

P

lQkP Multi-scalar multiplication

Multi-scalar multiplication

k an integeran elliptic

point

timesk

PPPkP Scalar

multiplication

QP,lk, integer

selliptic points

timesk

PPP

timesl

QQQ

5/22Two Computation Methodsfor Multi-Scalar Multiplication

QPQP ,,

Compute separatelytwo scalar multiplications

Separate Method

Compute simultaneously two scalar multiplications

Pk,

Ql,

kP

lQlQkP

Scalar multiplicati

onScalar multiplicati

on

addition

Pk,

Ql,

lQkP Multi-scalar multiplication

Simultaneous Method

[Aki01, Moe01]

[Elg85, HHM00]

Shamir’s trick

Improvement

Window method

Comb method

[Knu81, CMO98]

[LL94]

tQsP addition

6/22

Computation Process

Precomputa-tion stage

Evaluation stageInpu

tOutpu

tPreparation of a table

Actual computation

Pk,Ql,

lQkP QPQPQPQ

QPQPQPQ

QPQPQPQ

PPPO

333233

232222

32

32

tQsP

QP 32

Table

addition

Precomputation table

7/22

Target of Speedup

Precomputation Stage

Evaluation Stage

Separate Method

Simultaneous Method

Slow

SlowFast

Fast

Many researches exist

Controversial !

Not so much studies as evaluation stage

[Aki01, Moe01, Sol01]

[CC87, MO90, LL94, CMO98, …]

[CMO98]

8/22

What are Obstacles to Speed upthe Precomputation Stage?

Many precomputation points

Inversions are required (1 per

point)

Some points are not used in evaluation

stage

Obstacles

Multi-scalar multiplication only

9/22

What are Obstacles to Speed upthe Precomputation Stage?

Points are computed in affine coordinates

Reason

Table should be saved points in affine

coordinates for speeding up evaluation stage

The operation in affine coordinates requires

inversion

Inversions are required (1 per

point)

Obstacles

Many precomputation points

Some points are not used in evaluation

stageMulti-scalar multiplication

only

10/22

What are Obstacles to Speed upthe Precomputation Stage?

Reason

2 dimensions

vQuP ,1,0, vu

QPQPQPQ

QPQPQPQ

QPQPQPQ

PPPO

333233

232222

32

32

Inversions are required (1 per

point)

Obstacles

Multi-scalar multiplication only

Some points are not used in evaluation

stage

Many precomputation points

11/22

What are Obstacles to Speed upthe Precomputation Stage?

Reason

Precomputation stagePoints to compute: 64 points

Evaluation stagePoints to use: 54 points

160 bits, window width 3

Inversions are required (1 per

point)

Obstacles

Multi-scalar multiplication only

Many precomputation points

Some points are not used in evaluation

stage

12/22

Contents

Multi-Scalar Multiplication

Target of Speedup

Proposed Method

Comparison

13/22

Simple Improvements

has same x-coordinates

),(),( yxPyxP

QP

QPQP

Simultaneous inversion

Negate the y-coordinateOmit computation

Q

14/22

MM

MM

I

MM

MM

M M

Montgomery Trickof Simultaneous Inversions

naaaa ,,,, 321

21aa 321 aaa

1321aaa

2a 3a

1a

121aa1

1a

12a 1

3a

113

12

11 ,,,,

naaaa

IMn 13Input

Output

Cost

M M

I

M: multiplication

I: inversion

[Coh93]

It speeds up the ECM of

factorization

[Coh93]

15/22

Use of Montgomery Trick(Scalar Multiplication)

Montgomery trick reduces from plural inversions to 1 inversion

P2 P3P doubling

doubling

Compute inversion using Montgomery trick

P4additio

n

Preparation of precomputation table

addition

addition

addition

[CMO98]

P7

P5

P8doubling

Use of Montgomery trick

16/22

Use of Montgomery Trick(Multi-Scalar Multiplication)

QP

P2 QP 3

Q2

P

Q

doubling

doubling

addition

Compute inversion using Montgomery trick

QP 22

addition

Preparation of precomputation table

addition

addition

addition

Complicated because of 2 dimensions

Montgomery trick reduces from plural inversions to 1 inversion

17/22

Preparation of Precomputation Table

P

Q

Q2

Q3

P2 P3

QP

QP 2

QP 3

QP 2 QP 3

QP 22

QP 32

QP 23

QP 33

O Step 0

Step 1

Step 2

Step 3

Precomputation Table

Each step uses Montgomery trick of simultaneous inversion

18/22

They cannot be computed in Step 2

Some Points Do Not Needto be Computed

P

Q

Q2

Q3

P2 P3

QP

QP 2

QP 3

QP 2 QP 3

QP 22

QP 32

QP 23

QP 33

O

Precomputation Table

Consider how the points are computed!

Step 0

Step 1

Step 2

Step 3

19/22

Proposed Method

P

Q

Q2

Q3

P2 P3

QP

QP 2

QP 3

QP 2 QP 3

QP 22

QP 32

QP 23

QP 33

O Step 0

Step 1

Step 2

Step 3

Precomputation Table

are first, the middles are lastvQuP,

20/22

Some Points Do Not Needto be Computed

P

Q

Q2

Q3

P2 P3

QP

QP 2

QP 3

QP 2 QP 3

QP 22

QP 32

QP 23

QP 33

O Step 0

Step 1

Step 2

Step 3

Precomputation Table

It does not affect the computationfor the other points

21/22

Comparison

Precompu-tation stage

Evaluation stage

Separate Method

Simultaneous Method

336.8Ｍ

Total

2809.8Ｍ 3146.6Ｍ

279.2Ｍ 1655.5Ｍ 1934.7ＭProposed method

160 bits

[CMO98]

1011.6Ｍ 1655.5Ｍ 2667.1ＭConventional Method[HHM00]

[Moe01]

22/22

Conclusion

Speeding up the verification of ECDSA

Speeding up the Multi-scalar multiplication

Speeding up the scalar multiplication using multi-scalar multiplication

Application

Problem

Result

Points Montgomery trick of simultaneous inversions

Simplification of precomputation procedures

Efficient Precomputation provides speedup for multi-scalar multiplication3 times faster