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

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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

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1 Discrete Logarithm Problem

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

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• Example 1G Z19 1, 2, , 18n18,
  generator g 2
• then log214 7 log26 14

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• Example 2GGF(23) with irreducible poly.
  p(x)x3x1GZn/p(x) 1, x, x2, 1x, 1x2,
  xx2,
• 1xx2 n7,
  generator g x
• then logx(x1) 3 logx(x2x1) 5
  logx(x21) 6

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• Example 3Let p 105354628039501697530461658293395
  87319488718149259134893426087342587178835751858673
  00386287737705577937382925873762451990450430661350
  85968269741025626827114728303489756321430023716636
  91740666159071764725494700831131071381899212808840
  03892629359
• NB p 158(2800 25) 1 and has 807 bits.
• Find x in Z, such that
• 2x 3 mod p

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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

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The index calculus method

• The index calculus method (Suitable only for
  GZp)

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• Example
• log59451 mod 10007?
• Choose B2, 3, 5, 7. Of course log551.
• 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

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• There happens to be a unique solution modulo
  10006
• log526578, log536190, and log571301
• Choose random exponent s 7736 and try to
  calculate
• ags 945157736 mod 10007 8400
• Since 8400 243527 factors over B, we obtain
• log59451 (4 log52 log53 2 log55 log57
  s) mod 10006
• (46578 6190 21 1301
  7736) mod 10006
• 6057 mod 10006

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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))

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3 Cryptosystems based on DL

• Key Distribution
• Diffie-Hellman, 1976
• Encryption
• Massey-Omura cryptosystem, 1983
• Digital Signature
• ElGamal, 1985

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Diffie-Hellman Key Exchange Algo

• Global Public Elements
• q prime number
• a alt q and a is a primitive root of q
• User A Key Generation
• Select private XA XAlt q
• Calculate public YA YA aXA mod q
• User B Key Generation
• Select private XB XBlt q
• Calculate public YB YB aXB 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

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User A
User B
Generate random XA lt q Calculate YA
aXA mod q Calculate K (YB)XA mod q
Generate random XB lt q Calculate
YB aXB mod q Calculate K (YA)XB mod q
YA
YB
Diffie-Hellman Key Exchange
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Massey-Omura for message transmission

• Parameters
• q prime number
• e a random private integer
• 0 lt e lt q and gcd ( e, q-1) 1
• d an inverse of e
• d e-1 mod q-1 , i.e., de1 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

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User A
User B
1.Encryption(1) C1 M eA mod
q 3.Encryption(3) C3 C2dA (M
eAeB)dA M eB mod q
2.Encryption(2) C2 C1eB
M eAeB mod q 4. Decryption M C3dB
M eBdB mod q
C1
C2
C3
Massey-Omura for message transmission
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ElGamal signature scheme

• 1985 ElGamal
• Parameters
• p a large prime
• a a primitive number in GF(p)
• x a private key, x 1, p-1
• y a public key , y ax (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 ak mod p
• m ks rx mod f(p) ,where GCD( k, f(p) ) 1
• ( m , (r,s) ) is sent to the verifier
• Verification
• am rs yr mod p
• The signature (r,s) is accepted when the equality
  holds true.

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ElGamal encryption scheme

• Parameters
• p a large prime
• a a primitive number in GF(p)
• a a private key, a 1, p-1
• ß a public key , ß aa (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, a, a, ß) public key private key
• Encryption
• eK(m, k)(y1, y2)
• where y1 ak mod p and y2mßk mod p
• Decryption
• m dK(y1, y2) y2(y1a)-1 mod p

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4 Elliptic Curves

• Over Fields of Characteristic pgt3
• Curve form
• E Y2 X3 aX b
• where a, b ? Fq, q pn
• 4a327b2?0
• Group operation
• given P1(x1,y1) and P2(x2,y2)
• compute P3(x3,y3) P1P2

(xPQ, yPQ?)
(xPQ, yPQ)
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Example of EC over GF(p)

• Example

-P
PQ
P
Q
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• Addition (P1?P2)
• Doubling (P1P2)

Computational Cost I 3 M
Computational Cost I 4 M
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• Over Fields of Characteristic 2
• Curve form
• E Y2 XY X3 aX2 b
• where a, b ? Fq, b?0, q 2n
• Group operation
• given P1(x1,y1) and P2(x2,y2)
• compute P3(x3,y3) P1P2

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Example of EC over GF(2m)
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• Addition (P1?P2)
• Doubling (P1P2)

Computational Cost I 2 M S
Computational Cost I 2 M S
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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)

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• 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.

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Elliptic Curve Security
Symmetric Key Size(bits) RSA and Diffie-HellmanKey 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
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• 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).

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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

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Digital Signature Algorithm (DSA)
L0 mod 64, 512L1024

• 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 a be a qth root of 1
  modulo p.
• Define K (p,q,a,a,ß) ßaa mod p
• p,q,a,ß are the public key, a is private

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• For a (secret) random number k, define
• sig (x,k)(?,d), where
• ?(ak mod p) mod q and
• d(SHA-1(x)a?)k-1 mod q
• For a message (x,(?,d)), verification is done by
  performing the following computations
• e1SHA-1(x)d-1 mod q
• e2?d-1 mod q
• ver(x,(?,d))true iff. (ae1ße2 mod p) mod q?

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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 ltAgt is
  infeasible.
• Define K (p,q,E,A,m,B) BmA
• p,q,E,A,B are the public key, m is private

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• For a (secret) random number k, define
  sigk(x,k)(r,s),
• where kA(u,v), ru mod q and
• sk-1(SHA-1(x)mr) mod q
• For a message (x,(r,s)), verification is done by
  performing the following computations
• iSHA-1(x)s-1 mod q
• jrs-1 mod q
• (u,v)iAjB
• ver(x,(r,s))true if and only if u mod qr

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7 How to find secure elliptic curves ?

• (1) Randomly choose a, b, p and calculate
• Elliptic curve (y2x3axb)
• 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 (y2x3axb) q

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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

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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) ? Kx is a polynomial of degree at most g,
• f(x) ? Kx is a monic polynomial of degree 2g1.
• Elliptic curves are hyperelliptic curves of genus
  1.

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Group law in an elliptic curve

• y2x3-x over R

?
-R
PQ
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Example Hyperelliptic curve

• A genus 2 hyperelliptic curve over RC y2 x5
  -5x3 4x x(x1)(x-1)(x2)(x-2)
• The rational points on C do not form a group.

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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.

D12P1P2-38 D2 P1P3
deg(D1) 21-30 deg(D2) 112
D1D2 3P1P2P3-38
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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).

Q1(1, 0) on C div(x-1) 2Q1-28
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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 D1D2.

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Group law in HC

• A genus 2 hyperelliptic curve over RC y2 x5
  -5x3 4x
• ya3x3a2x2a1xa0

P4
P2
P1
P3
?
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HCDLP

• HCDLP
• (hyperelliptic curve discrete logarithm
  problem)
• Let a divisor D1 in JC(Fq) with known order N,
  and D2 in ltD1gt
• To find an integer ? s.t. D2 ?D1 is hard.

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9 ID-based Cryptosystem
Setup generate params and master key
IDBob is arbitrary and meaningful ex
Bob_at_hotmail.com or 0912345678
Private Key Generator (PKG)
Extract generate KRIDBob by IDBob and master
key
Authentication (IDBob)
KRIDBob
Bob
Alice
(params, IDBob)
KRIDBob
Encrypt
Decrypt
or
or
Verify
Sign
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Certificate-based Cryptosystem
Certificate Authority (CA)
KUBob is random
Authentication (KUBob)
Certificate(Bob, KUBob)
Certificate(Bob, KUBob)
Bob
Alice
KUBob
KRBob
Encrypt
Decrypt
or
or
Verify
Sign
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ID-based Encryption Scheme

• Proposed by Boneh and Franklin (Crypto 2001)
• First complete and efficient scheme
• Bilinear Pairing
• G1 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(P1P2,Q)e(P1,Q)e(P2,Q)
• e(P,Q1Q2)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!

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ID-based Encryption Scheme
System k-bit prime p p2 mod 3, p6q-1
E y2x31 over Fp

• 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 ? G1 (MapToPoint)
• H2 G2 ? 0,1n for some n
• Extract
• Given a ID ?0,1, build private key SID as
  follows
• QID H1(ID)
• Set dIDsQID , where s is the master key

Params ltp, q, P, Ppub, H1, H2gt Master-key s
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ID-based Encryption Scheme

• Encrypt
• Use MapToPoint to map ID to QID
• choose a random r ?Zq
• C lt rP, M ? H2(e(QID, Ppub)r) gt
• Decrypt
• Let Clt U, V gt , if U is not a point of order q
  then reject
• M V ? H2(e(dID, U))

dIDsQID
PpubsP
e(dID, U) e(sQID, rP) e(QID, P)sr e(QID, sP)r
e(QID, Ppub)r
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Weil Pairing

• (Def) Weil pairing
• where
  is called the m-torsion group,
• Um is the group of the mth roots of unity
• Given P, Q?E m, ? DP, DQ?Div 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

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

• RSA Mechanism

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End-to-end security for SMS

• ID-based Mechanism

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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.
  rE(GF(q)))
• k is the smallest positive integer s.t.
  r qk - 1

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• Various pairings
• Weil pairing
• Tate pairing
• Eta pairing
• Ate pairing
• Generalized Ate pairing

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• 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.

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• 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