橢圓曲線密碼技術

交通大學 資訊工程系

陳榮傑

http://www.cs.nctu.edu.tw/~rjchen/ECC2012S/

p2.

Outline

[1] Discrete Logarithm Problem [2] Algorithms for Discrete Logarithm [3] Cryptosystems Based on DLP [4] Elliptic Curves [5] Elliptic Curve DLP [6] Signature Scheme: ECDSA [7] How to find secure ECs? [8] Hyperelliptic Curves [9] ID-based Cryptosystems [10] Pairing-based Cryptography

p3.

Let G is a finite cyclic group of size n generated by generator g, i.e.

G = <g> = {g i | i = 1, 2, …, n} or {g i | i = 0, 1, …, n-1}

Given g and i, it is easy to compute gi by repeated squaring

Discrete logarithm problem

Given , find x such that

We denote

[1] Discrete Logarithm Problem

Ga ag x

)ind ( log gaorax g

p4.

Example 1G = Z19

*= { 1, 2, …, 18}

n=18, generator g = 2

then log214 = 7

log26 = 14

i123456789ig2481613714918

10 11 12 13 14 15 16 17 18

17 15 11 3 6 12 5 10 1

p5.

Example 2

G=GF*(23) with irreducible poly.

p(x)=x3+x+1

G=Zn*/p(x)={ 1, x, x2, 1+x, 1+x2, x+x2,

1+x+x2 }

n=7, generator g = x

then logx(x+1) = 3 logx(x2+x+1) = 5 logx(x2+1) = 6

i 1 2 3 4 5 6 7ig x 2x 1x xx2 12 xx 12x 1

p6.

Example 3Let p

=105354628039501697530461658293395873194887181492

5913489342608734258717883575185867300386287737705

5779373829258737624519904504306613508596826974102

5626827114728303489756321430023716636917406661590

7176472549470083113107138189921280884003892629359

NB: p = 158(2800 + 25) + 1 and has 807 bits.

Find x in Z, such that

2x = 3 mod p

p7.

[2] Algorithms for Discrete Logarithm

trivial algorithm Shanks’ algorithm (Baby-step giant-

step) Pollard rho discrete log algorithm Pohlig-Hellman algorithm The index calculus method*

p8.

The index calculus method

The index calculus method (Suitable only for G=Zp*)

base.factor in the elements theof logarithms

discrete theof knowledge theusing a,element

desired a of logarithm discrete thecompute To

step) (2nd

}.p ..., ,p ,{p

base.factor

in the primes B theof logarithms discrete thefind To

step)(1st

B21

p9.

Example

log59451 mod 10007=?

Choose B={2, 3, 5, 7}. Of course log55=1.

Use lucky exponents 4063, 5136, and 9865 54063 mod 10007 = 42 = 2 * 3 * 7 55136 mod 10007 = 54 = 2 * 33

59865 mod 10007 = 189 = 33 * 7 And we have three congruences: log52 + log53 + log57 = 4063 mod 10006

log52 + 3 log53 = 5136 mod 10006

3 log53 + log57 = 9865 mod 10006

p10.

There happens to be a unique solution modulo 10006

log52=6578, log53=6190, and log57=1301

Choose random exponent s = 7736 and try to calculate

ags = 9451*57736 mod 10007 = 8400

Since 8400 = 24*3*52*7 factors over B, we obtain

log59451 = (4 log52 + log53 + 2 log55 + log57 – s) mod 10006

= (4*6578 + 6190 + 2*1 +1301 – 7736) mod 10006

= 6057 mod 10006

p11.

Complexity of Index Calculus

For factoring and the discrete logarithm problem in finite fields Fq* there are index calculus algorithm

(implemented with Number Field Sieve technique)

These have subexponential complexity O(exp(c(lnN)1/3(lnlnN)2/3))

p12.

[3] Cryptosystems based on DL

Key Distribution Diffie-Hellman, 1976

Encryption Massey-Omura cryptosystem, 1983

Digital Signature ElGamal, 1985

p13.

Diffie-Hellman Key Exchange Algo Global Public Elements

q : prime number α: α< q and α is a primitive root of

q User A Key Generation

Select private XA : XA< q Calculate public YA : YA= αXA mod q

User B Key Generation Select private XB : XB< q Calculate public YB : YB= αXB mod q

Generation of Secret Key by User A K = (YB)XA mod q

Generation of Secret Key by User B K = (YA)XB mod q

p14.

User A User B

Generate random XA < q ;Calculate YA = αXA mod q

Calculate K = (YB)XA mod q

Generate random XB < q ;Calculate YB = αXB mod qCalculate K = (YA)XB mod q

YA

YB

Diffie-Hellman Key Exchange

p15.

Massey-Omura for message transmission

Parameters q : prime number e : a random private integer

• 0 < e < q and gcd ( e, q-1) = 1 d : an inverse of e

• d = e-1 mod q-1 , i.e., de≡1 mod q-1 M : a message to be encrypted and

decrypted User A wants to send a message M to User B

User A : eA and dA are both private User B : eB and dB are both private

p16.

User A User B

1.Encryption(1) C1 = M eA mod q

3.Encryption(3) C3 = C2

dA

= (M eAeB)dA

= M eB mod q

2.Encryption(2) C2 = C1

eB

= M eAeB mod q

4. Decryption M = C3

dB

= M eBdB mod q

Massey-Omura for message transmission

C1

C2

C3

p17.

ElGamal signature scheme

1985 ElGamal Parameters

p : a large prime α: a primitive number in GF(p) x : a private key, x [1, p-1] y : a public key , y = αx (mod p) m : a message to be signed , m [1, p-1] k : a random integer that is privately selected, k [0, p-

2] Signature

r = αk mod p m = ks + rx mod φ(p) ,where GCD( k, φ(p) ) = 1 ( m , (r,s) ) is sent to the verifier

Verification αm = rs yr mod p The signature (r,s) is accepted when the equality holds

true.

p18.

ElGamal encryption scheme

Parameters p : a large prime α: a primitive number in GF(p) a : a private key, a [1, p-1] β : a public key , β = αa (mod p) m : a message to be signed , m [1, p-1] k : a random integer that is privately

selected, k [0, p-2] K = (p, α, a, β) : public key + private key

Encryption eK(m, k)=(y1, y2)

where y1 = αk mod p and y2=mβk mod p Decryption

m = dK(y1, y2) = y2(y1a)-1 mod p

p19.

(xP+Q, yP+Q)

(xP+Q, yP+Q)

[4] Elliptic Curves

Over Fields of Characteristic p>3 Curve form

E: Y2 = X3 + aX + b where a, b Fq, q = pn

4a3+27b2≠0 Group operation

given P1(x1,y1) and P2(x2,y2)

compute P3(x3,y3) = P1+P2

PQ

P+Q

p20.

Example: 0123 b,a,p

}O{}xxy:Z)y,x{()Z(E b,a 3222323

P

-P

Q

P+Q

Example of EC over GF(p)

p21.

Addition (P1P2)

Doubling (P1=P2) 13313

212

3

12

1

)( yxxxy

xxx

xx

yy

2

13313

12

3

1

21

)(

2

2

3

yxxxy

xx

y

ax

Computational CostI + 3 M

Computational CostI + 4 M

p22.

Over Fields of Characteristic 2 Curve form

E: Y2 + XY = X3 + aX2 + b where a, b Fq, b≠0, q = 2n

Group operationgiven P1(x1,y1) and P2(x2,y2)

compute P3(x3,y3) = P1+P2

p23.

Example of EC over GF(2m)

1)(, 4 xxxp)(/][)2( 2 xpxZGF m

1 : 2432 xgxxyyE

)0001(1

)0011(0

4

g

g

p24.

Addition (P1P2)

Doubling (P1=P2)

13313

212

3

12

1

)( yxxxy

axxx

xx

yy

2

13313

23

11

1

)( yxxxy

ax

xx

y

Computational CostI + 2 M + S

Computational CostI + 2 M + S

p25.

[5] Elliptic Curve DLP

Basic computation of ECC

Q = kP = where P is a curve point, k is an integer

Strength of ECC Given curve, the point P, and kP

It is hard to recover k- Elliptic Curve Discrete Logarithm Problem (ECDLP)

timesk

P...PP

p26.

Security of ECC versus RSA/ElGamal

Elliptic curve cryptosystems give the most

security per bit of any known public-key scheme.

The ECDLP problem appears to be much more

difficult than the integer factorisation problem

and the discrete logarithm problem of Zp. (no

index calculus algo!)

The strength of elliptic curve cryptosystems

grows much faster with the key size increases

than does the strength of RSA.

p27.

Elliptic Curve Security

Symmetric Key Size

(bits)

RSA and Diffie-Hellman

Key Size (bits)

Elliptic Curve Key Size

(bits)

80 1024 160

112 2048 224

128 3072 256

192 7680 384

256 15360 521

NIST Recommended Key Sizes

p28.

ECC Benefits

ECC is particularly beneficial for application where:

computational power is limited (wireless devices, PC cards)

integrated circuit space is limited (wireless devices, PC cards)

high speed is required. intensive use of signing, verifying or authenticating is

required. signed messages are required to be stored or

transmitted (especially for short messages). bandwidth is limited (wireless communications and

some computer networks).

p29.

[6] Signature Scheme: ECDSA

Digital Signature Algorithm (DSA)

Proposed in 1991 Was adopted as a standard on December 1,

1994

Elliptic Curve DSA (ECDSA)

FIPS 186-2 in 2000

p30.

Digital Signature Algorithm (DSA)

Let p be a L-bit prime such that the DL problem in Zp* is intractable, and let q be

a 160-bit prime that divides p-1. Let α be a qth root of 1 modulo p.

Define K={ (p,q,α,a,β): β=αa mod p }

p,q,α,β are the public key, a is private

L=0 mod 64, 512≤L≤1024

p31.

For a (secret) random number k, definesig (x,k)=(γ,δ), where

γ=(αk mod p) mod q andδ=(SHA-1(x)+aγ)k-1 mod q

For a message (x,(γ,δ)), verification is done by performing the following computations:

e1=SHA-1(x)*δ-1 mod q

e2=γ*δ-1 mod q

ver(x,(γ,δ))=true iff. (αe1βe2 mod p) mod q=γ

p32.

Elliptic Curve DSA

Let p be a prime, and let E be an elliptic curve defined over Fp. Let A be a point on E having prime order q, such that DL problem in <A> is infeasible.

Define K={ (p,q,E,A,m,B): B=mA }

p,q,E,A,B are the public key, m is private

p33.

For a (secret) random number k, define sigk(x,k)=(r,s),

where kA=(u,v), r=u mod q and

s=k-1(SHA-1(x)+mr) mod q

For a message (x,(r,s)), verification is done by performing the following computations:

i=SHA-1(x)*s-1 mod q

j=r*s-1 mod q

(u,v)=iA+jB

ver(x,(r,s))=true if and only if u mod q=r

p34.

[7] How to find secure elliptic curves ?

(1) Randomly choose a, b, p and calculate #Elliptic curve (y2=x3+ax+b) until #E = a prime q, where #E is calculate by using Schoof-

Elkies-Atkin algorithm

(2) (Complex multiplication method) Given a big prime q, find p, a, b such that #Elliptic curve (y2=x3+ax+b) = q

p35.

[8] Hyperelliptic Curves

1. Definition of HC

2. Example of HC

(rational points of HC do not form a

group)

3. Divisor

4. Jacobian (Jacobian is a group)

5. HCDLP

p36.

Definitions of hyperelliptic curves

A hyperelliptic curve C of genus g over a

finite field K (g≧1)

C : y2 + h(x)y = f(x)

where

h(x) ∈ K[x] is a polynomial of degree at most g,

f(x) ∈ K[x] is a monic polynomial of degree 2g+1.

Elliptic curves are hyperelliptic curves of

genus 1.

p37.

Group law in an elliptic curve

y2=x3-x over R

？-RP+Q=

PQ

R

PQ

R

-R

p38.

Example: Hyperelliptic curve

A genus 2 hyperelliptic curve over R:C: y2 = x5 -5x3+ 4x = x(x+1)(x-1)(x+2)(x-2)

The rational points on C do not form a group.

p39.

Divisors

Definition : (divisor, degree) A divisor D is a formal sum of points in C:

The degree of D,

The set of all divisors, denoted D, forms an additive group under the addition rule:

D0(K) is the subgroup of all divisors defined over K and of degree 0.

, p pP C

D m P m

deg pP C

D m

( )p p p PP C P C P C

m P n P m n P

D1=2P1+P2-3∞D2= P1+P3

deg(D1) =2+1-3=0deg(D2) =1+1=2

D1+D2

=3P1+P2+P3-3∞

p40.

Principal divisor

Definition : (principal divisors) Let R ∈K(C) be a rational function.

The divisor of R is called a principal divisor :

In fact, degree of a principal divisor is 0.

The set of all principal divisors, denoted P(K), is a subgroup of D0(K).

( ) ( )PP C

div R ord R P

C: y2 = x5 -5x3+ 4x

•

• x-1

3x +x+(x+1)y

x y

Q1=(1, 0) on Cdiv(x-1) = 2Q1-2∞

p41.

Jacobian

Definition: (Jacobian)

The quotient group JC(K) = D0/P is called the

Jacobian of the curve C.

If D1, D2 ∈ D0 and D1-D2 ∈ P, then D1 and D2

are said to be equivalent divisors; we write

D1~D2.

p42.

Group law in HC

A genus 2 hyperelliptic curve over R:C: y2 = x5 -5x3+ 4x

y=a3x3+a2x2+a1x+a0

P1

P2

P4

5 6( 2 )P P 1 2 3 4( 2 ) ( 2 )P P P P ？

P3

P5’

P6’

P6

P5

x-x5=0

x-x6=0

p43.

HCDLP

HCDLP:

(hyperelliptic curve discrete logarithm

problem)

Let a divisor D1 in JC(Fq) with known order

N,

and D2 in <D1>

To find an integer λ s.t. D2 = λD1 is hard.

p44.

[9] ID-based Cryptosystem

Private Key Generator(PKG)

BobAlice

Authentication(IDBob)KRIDBob

(params, IDBob) KRIDBob

IDBob is arbitrary and meaningfulex: Bob@hotmail.com

or 0912345678

Setup generate params and master key

Extract generate KRIDBob by IDBob and master key

Encrypt

Verify

or

Decrypt

Sign

or

p45.

Certificate-based Cryptosystem

Certificate Authority(CA)

BobAlice

Authentication(KUBob)

Certificate(Bob, KUBob)

Certificate(Bob, KUBob)

Encrypt

KUBob

Decrypt

KRBob

KUBob is random

Verify Sign

or or

p46.

ID-based Encryption Scheme

Proposed by Boneh and Franklin (Crypto 2001) First complete and efficient scheme

Bilinear PairingG1: additive group generated by P, ord(P)=q

G2: multiplicative group with same order q

Assume that DLP in G1 and G2 are hard

Let e: G1xG1 G2 satisfies:

1. Bilinear: e(P1+P2,Q)=e(P1,Q)e(P2,Q)

e(P,Q1+Q2)=e(P,Q1)e(P,Q2)

2. Non-degenerate: P,Q G1, s.t e(P,Q)1

3. Computability Bilinear Diffie-Hellman (BDH) Assumption

Given P, aP, bP, cP G1 , compute e(P, P)abc is HARD!

p47.

ID-based Encryption Scheme

ID-based Encryption Setup:

(1) Choose P E/Fp of order q

(2) Pick a random s Zq* and set

Ppub= sP

(3) Two hash functions: H1: {0,1}* G*1 (MapToPoint)

H2: G2 {0,1}n for some n Extract:

Given a ID {0,1}*, build private key SID as follows:

QID = H1(ID)

Set dID=sQID , where s is the master key

System: k-bit prime p p=2 mod 3, p=6q-1 E: y2=x3+1 over Fp

Params: <p, q, P, Ppub, H1, H2>Master-key: s

p48.

Encrypt:Use MapToPoint to map ID to QID

choose a random r Zq*

C = < rP, M H2(e(QID, Ppub)r) > Decrypt:

Let C=< U, V > , if U is not a point of order q then rejectM = V H2(e(dID, U))

e(dID, U)= e(sQID, rP)= e(QID, P)sr= e(QID, sP)r= e(QID, Ppub)r

dID=sQI

D

Ppub=sP

ID-based Encryption Scheme

p49.

(Def) Weil pairing where is called the m-

torsion group, Um is the group of the mth roots of unity

Given P, QE [m], DP, DQDiv 0 such that

DP ~ (P) – (O) and DQ ~ (Q) – (O). Also, fP , fQ such that div (fP) = m DP and div (fQ) = m DQ.

Suppose supp (DP) supp (DQ) =

Then

Weil Pairing

mUmEmE ][][:e} PO,P|P{][ EmmE

)D(

)D()Q,P(

PQ

QP

f

fe

p50.

End-to-end security for SMS (short message service)

RSA Mechanism

User A User BPublic Key Directory

(1) Public Key Request

(3) Encrypted Short Message

(2) Public Key Response

End-to-endSecurity

Service Provider

p51.

ID-based Mechanism

SIM Card

Message

B’s phone number (0912345678)

Cipher

Private Key Generator(PKG)

ID-based Encryption

ID-based Decryption

SIM Card

Cipher

MessageID-based Encryption

ID-based Decryption

KR

SIM CardKR

ID=0912345678

(1)

(2)

(3)

(4)

(5)

(6)

(7)

User A User B (0912345678)

(8)

GSM Network

End-to-end security for SMS

p52.

[10] Pairing-based Cryptography

1. Implementation of Pairings

Bilinear paring

e: G1 x G2 GT

G1, G2 : prime-order subgroups of an elliptic curve E

over GF(qk)

GT : prime-order subgroup of GF(qk)*

k is the embedding degree of E (w.r.t. r=#E(GF(q)))

k is the smallest positive integer s.t. r | qk - 1

p53.

Various pairings:

Weil pairing

Tate pairing

Eta pairing

Ate pairing

Generalized Ate pairing

p54.

2. Use of parings in cryptography

Attack on ECDLP (MOV attack)

One-round 3-way key exchange (Joux)

IDE (Boneh-Franklin)

Short digital signature (Boneh-Lynn-

Shacham)

Other applications: Group signatures,

Bach signatures, aggregate signatures,

threshold cryptography, authenticated

encryption, broadcast encryption, etc.

p55.

3. Constructing pairing-friendly curves

Want k large enough so that DLP in GF(qk)* is

computational infeasible, but small enough so that

pairing is easy to compute.

Cock-Pinch strategy

MNT strategy

Dupon-Enge-Morain strategy

* Brezing-Weng strategy

Scott-Barreto strategy