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1 ( )

2 (1) (2) (3) (ECDLP) Baby-step Giant-step ρ (ECC) ECDH ECElGamal ECDSA

3 1., ( ). 2

4 1.1 p, 0 p 1 F p = {0, 1,..., p 1}. p 0 p 1, p,. 1.1 F 7 = {0, 1, 2, 3, 4, 5, 6} x + y 1.1. F p x x + p p, F p x x + p. x y = x + p y, F p. 1.2 F 7 = {0, 1, 2, 3, 4, 5, 6} x y 1.2. (additive group). 3

5 1.1 y x F y x F G (i) G a, b, c (a b) c = a (b c), (ii) G a a e = e a = a G e, (iii) G a a a = a a = e G a,, G (group). +. ( x y = y x ), ( x + y = y + x ). 4

6 F p p,. 1.3 F 7 = {0, 1, 2, 3, 4, 5, 6} x y 1.3. F p p, x, x + p, x + 2p, x + 3p,.... x y = (x + p) y = (x + 2p) y =,. (, p, x, a, b ap + by = 1, by 1 (mod p) b y 1 (mod p), x y = x b. a, b Euclid.) 1.4 F 7 = {0, 1, 2, 3, 4, 5, 6} x y (y 0) 1.4., ( 0 ) p F p (prime field).,, ( 0 ), (finite field).,.. 5

7 1.3 y x F y x F field, field ( )., köper (, ).,, field. 6

8 1.2 F p. 1.5 ( ) E : y 2 = x 3 + ax + b (a, b F p, E = 4a b 2 0) (1.1) F p (elliptic curve). 1.6 E : y 2 = x 3 + 3x + 4 F 7. E, x, y F p (x, y) F p - (F p -rational point). (the point at infinity) O. (x, y). 1.7 F 7 E : y 2 = x 3 + 3x y 2 = x 3 + 3x + 4 F 7 - P 0 O P 1 (0, 2) P 9 (0, ) P 2 (1, 1) P 8 (1, ) P 3 (2, 2) P 7 (2, ) P 4 (5, 2) P 6 (5, ) P 5 7

9 E = 4a b 2 0, (1.1). E (discriminant). 2 F : y = x 2 + Bx + C F = B 2 4C, F 0 F. n G : y = A 0 x n + A 1 x n A n G = A 2(n 1) 0 (α 1 α 2 ) 2 (α 1 α n ) 2 (α 2 α 3 ) 2... (α 2 α n ) 2 (α n 1 α n ) 2. α 1,..., α n G., n = 3, E = 4a b 2. 8

10 1.3,. ( x, y )., ( ). 1.8 ( ) R = P + Q. F p E : y 2 = x 3 + ax + b 2 P, Q ( O) 1. 2 P, Q (P = Q P ) l. 2. E l 3 R ( R = O ). 3. R x R (R = O R = O ). R R, R = R. O, F 7 E : y 2 = x 3 + 3x + 4, 1.6 y 2 = x 3 + 3x + 4 F 7 - P 0 O P 1 (0, 2) P 6 (5, 5) P 2 (1, 1) P 7 (2, 5) P 3 (2, 2) P 8 (1, 6) P 4 (5, 2) P 9 (0, 5) P 5 (6, 0) 9

11 (elliptic curve) (ellipse).,.,., ( ).. 10

12 ( ) R = P + Q. F p E : y 2 = x 3 + ax + b 2 P, Q P = O : R = Q. Q = O : R = P. : P = (x P, y P ), Q = (x Q, y Q ) y P = y Q : R = O ( Q = P ). y P y Q : R = (x R, y R ). x R, y R x R = λ 2 x P x Q, y R = λ(x P x R ) y P, λ 2 P, Q ( P ). y P y Q x λ = P x Q 3x 2 P + a 2y P x P x Q x P = x Q F 7 E : y 2 = x 3 + 3x + 4, P 2 + P 4, 2 P 6. 11

13 2 P = (x P, y P ), Q = (x Q, y Q ) (y P y Q )/(x P x Q ). x P = x Q 0,. P, Q E : y 2 = x 3 + ax + b y P y Q = (y P y Q )(y P + y Q ) x P x Q (x P x Q )(y P + y Q ) = yp 2 yq 2 (x P x Q )(y P + y Q ) = (x3 P + ax P + b) (x 3 Q + ax Q + b) (x P x Q )(y P + y Q ) = (x P x Q )(x 2 P + x P x Q + x 2 Q + a) (x P x Q )(y P + y Q ) = x2 P + x P x Q + x 2 Q + a y P + y Q, P = Q x P = x Q, y P = y Q. y P y Q = x2 P + x P x Q + x 2 Q + a = 3x2 P + a x P x Q y P + y Q 2y P 12

14 y 2 = x 3 + 3x + 4 F y 2 = x 3 + 3x + 4 F 7 - P 0 P 1 P 2 P 3 P 4 P 5 P 6 P 7 P 8 P 9 P 0 P 0 P 1 P 2 P 3 P 4 P 5 P 6 P 7 P 8 P 9 P 1 P 1 P 8 P 9 P 6 P 7 P 4 P 5 P 3 P 2 P 0 P 2 P 2 P 9 P 1 P 4 P 6 P 3 P 7 P 5 P 0 P 8 P 3 P 3 P 6 P 4 P 1 P 9 P 2 P 8 P 0 P 5 P 7 P 4 P 4 P 7 P 6 P 9 P 8 P 1 P 0 P 2 P 3 P 5 P 5 P 5 P 4 P 3 P 2 P 1 P 0 P 9 P 8 P 7 P 6 P 6 P 6 P 5 P 7 P 8 P 0 P 9 P 2 P 1 P 4 P 3 P 7 P 7 P 3 P 5 P 0 P 2 P 8 P 1 P 9 P 6 P 4 P 8 P 8 P 2 P 0 P 5 P 3 P 7 P 4 P 6 P 9 P 1 P 9 P 9 P 0 P 8 P 7 P 5 P 6 P 3 P 4 P 1 P ,

15 , y 2 = x 3 + ax + b ( 1.4), y 2 + a 1 xy + a 3 y = x 3 + a 2 x 2 + a 4 x + a 6 (Weierstrass ). y 2 = x 3 + ax + b. 14

16 1.4,. Mordell-Weil (Mordell-Weil group). (group order) F 7 E : y 2 = x 3 + 3x + 4 #E. F p, (Hasse-Weil ) F p E #E, #E. p p #E p p Hasse-Weil ( 1.14), F p p., F ,, F 7 E : y 2 = x 3 + 3x + 4 Hasse-Weil ( 1.14). 15

17 1.8 F 7 E : y 2 = x 3 + ax + b b a Deuring Hasse-Weil ( 1.14),., Deuring. Deuring, a, b, Hasse-Weil. F 7 ( 1.8). 16

18 1.5 F p E P ( (base point) ). d, P d d P = P + + P } {{ } d (scalar multiplication)., F 7 E : y 2 = x 3 + 3x + 4, P 1. 2 P 1 = 3 P 1 = 4 P 1 = 5 P 1 = 6 P 1 = 7 P 1 =, 2, 3,..., O. (point order)., P d P = O.,, F 7 E : y 2 = x 3 + 3x + 4, P 1 17

19 x d x d, ( mod N ) RSA., d P,,. 18

20 d P, d 1., 8 P = P + P + P + P + P + P + P + P 7., 8 P = 2 (2 (2 P )), 3.,. d 2 d = 2 n 1 + d n 2 2 n d d 0 (d i {0, 1}) = (1, d n 2,..., d 1, d 0 ) : P, d = (1, d n 2,..., d 1, d 0 ) 2 : d P 1. Q P 2. i = n 2,..., 1, 0 : 2.1 Q 2 Q 2.2 d i = 1 Q Q + P 3. Q., d P ( ) 1.5 log 2 d, d , d = 3045 = (1, 0, 1, 1, 1, 1, 1, 0, 0, 1, 0, 1) 2, Q.,. i d i

21 F q E Mordell-Weil E(F q ), E(F q ) 2 C 1, C 2, E(F q ) C 1 C 2 (#C 1 #C 2, #C 1 (q 1))., E(F q ) ( E(F q ) C 1 ). 20

22 1.6 (1),.,. F p,, ( ). ( 1.10),., (projective coordinates)., 3 (X : Y : Z)., 2 (X : Y : Z), (X : Y : Z ) X = cx, Y = cy, Z = cz c F p, 2. (X : Y : Z) = (2X : 2Y : 2Z) = = (X/Z : Y/Z : 1)., ( ) (x, y), (x : y : 1)., (X : Y : Z) (X/Z, Y/Z). 21

23 Jacobian, 2 X = c 2 X, Y = c 3 Y, Z = cz Jacobian (Jacobian projective coordinate).,. 22

24 Y 2 Z = X 3 + axz 2 + bz 3 (a, b F p, E = 4a b 2 0)., y 2 = x 3 +ax+b x = X/Z, y = Y/Z., X Z 0.,.,, ( ) F p E : Y 2 Z = X 3 + axz 2 + bz 3 2 P = (X P : Y P : Z P ), Q = (X Q : Y Q : Z Q ) R = P + Q = (X R : Y R : Z R ). P Q X R = va Y R = u(v 2 X P Z Q A) v 3 Y P Z Q Z R = v 3 Z P Z Q u = Y Q Z P Y P Z Q, v = X Q Z P X P Z Q, A = u 2 Z P Z Q v 3 2v 2 X P Z Q P = Q X R = 2hs Y R = w(4b h) 8YP 2 s 2 Z R = 8s 3 w = az 2 P + 3X2 P, s = Y P Z P, B = sx P Y P, h = w 2 8B P Q : P Q :, P = (x : y : 1) d P = (X : Y : Z), (X/Z, Y/Z)., 1,. 23

25 ,, Jacobian,. P Q P = Q 3, 1 4, Jacobian , d d i, P = Q, P Q 1/2, P = Q Jacobian. 24

26 1.7 (2) 1.1 d 2., d m. m = 8 ( n 3 ). 1.2 (8 ) : P, d = (d n 1, d n 2,..., d 1, d 0 ) 2 : d P 0. i = 0, 1,..., 7 : 0.1 P i i P 1. Q P 4dn 1 +2d n 2 +d n 3 2. i = n 4, n 7,..., 2 : 2.1 Q 8 Q 2.2 Q Q + P 4di +2 i 1 +d i 2 3. Q. 1.2, 0. P 0, P 1,..., P 7, , 3. (window method). 1.21, d = 3045 = (1, 0, 1, 1, 1, 1, 1, 0, 0, 1, 0, 1) 2, Q.,. i d i d i 1 d i , /3, 8 P 3, 2.1 n., 2.2 1/3, 2.2 n/ /2, 2.2 n/2, ,. 25

27 d 160, m =

28 1.8 (3) 2, P P ( (x, y), (X : Y : Z))., d 2 d = 2 n 1 + d n 2 2 n d d 0 (d i { 1, 0, 1}) = (1, d n 2,..., d 1, d 0 ) 2,. 1.3 ( 2 ) : P, d = (d n 1, d n 2,..., d 1, d 0 ) 2 : d P 1. Q P 2. i = n 2,..., 1, 0 : 2.1 Q 2 Q 2.2 d i = 1 Q Q + P 2.3 d i = 1 Q Q P 3. Q. d 2, NAF (non-adjacent form, ). d, 3d ( ) 2 (e n+1, e n,..., e 0 ) 2 d ( ) 2 (f n+1, f n,..., f 0 ) 2. d = (3d d)/2, 3d 2 d 2 2, d 2 ( d i = e i+1 f i+1 ). NAF,, 1 1, d ±1 log 2 d/ NAF, 1.33 log 2 n. 27

29 2 2, d i = 1,. 28

30 p,., y 2 = x 3 + ax + b.,.,. [ ],.,,.,,., (d P d ),.,.,,,. 29

31 2,. 30

32 2.1 (ECDLP) F p E, P d Q = d P ( )., P, Q Q = d P d (1 d l, l P ) (elliptic curve discrete logarithm problem, ECDLP) F 7 E : y 2 = x 3 + 3x + 4, P = P 1, Q = P 2, Q = d P d. 2.1,.,.,. 2.2 F p, P, Q F p Q = d P d.. 31

33 P, Q = P d, P, Q d ( ), log P Q. y = log P Q. Q = d P d ( ).,. 32

34 2.2 2 P, Q Q = d P d (1 d l, l P ) ( ), 2 P, 3 P,..., l P (brute force method) GHz, ,

35 2 160,, ( = ).,,. Mathematica Maple,,. Risa/Asir,. 34

36 2.3 Baby-step Giant-step, Baby-step Giant-step (Baby-step Giant-step method). Q = d P, P l m = l ( x x ). d = sm + t (0 s, t < m), s, t d. s, t. R = m P, Q = d P = (sm + t) P = s (m P ) + t P = s R + t P, s, t Q t P = s R (2.1). 2 Q, Q P, Q 2 P, Q 3 P,..., Q (m 1) P O, R, 2 R, 3 R,..., (m 1) R, x. 2, s, t, d. m, 2m, 2 l. 2.4 Baby-step Giant-step, l , 2 81, , Baby-step Giant-step,. 35

37 , 2., , ,

38 2.4 ρ, ρ (ρ method), Baby-step Giant-step. Q = d P, s P + t Q = s P + t Q s, t, s, t (s s, t t ). (t t ) Q = (s s) P Q = s s t t P, (s s)/(t t ) d. mod l (l P ). 2.5 F 229 E : y 2 = x 3 + x + 44, P = (5, 116), Q = (155, 166) Q = d P. P l = 239., 26 P Q = 47 P Q = (9, 18) Q = P = P = 176 P d = 176. R i = s i P + t i Q, ρ, R i = R j R i, R j.. 37

39 ( ) 2,.., (365 ) 23 1/2 2..,. ρ,. 38

40 ρ,,. R, (Random Walk ) f. R + M 0 if x(r) 0 (mod 4) R + M 1 if x(r) 1 (mod 4) f(r) = R + M 2 if x(r) 2 (mod 4) R + M 3 if x(r) 3 (mod 4) x(r) R x. M i = u i P + v i Q , R 0 = (39, 159) = 54 P Q f,. R 9 R 21. M 0 = (135, 117) = 79 P Q M 2 = (84, 62) = 87 P Q M 1 = (96, 97) = 206 P + 19 Q M 3 = (72, 134) = 219 P + 68 Q. i R i s i t i x(r i ) mod 4 i R i s i t i x(r i ) mod 4 0 (39, 159) (197, 92) (160, 9) (211, 47) (130, 182) (194, 145) (27, 17) (0, 68) (36, 97) (223, 153) (119, 180) (9, 18) (108, 89) (167, 57) (81, 168) (75, 136) (223, 153) (57, 105) (9, 18) (159, 4) (167, 57) (185, 227) (75, 136) (158, 26) (57, 105) (197, 92) (159, 4) (211, 47) (185, 227) (194, 145) (158, 26) (0, 68)

41 Random walk f,, 4.. ρ,

42 , l., R i (distinguished point). x θ., 1/θ.,. R i = R j f(r i ) = f(r j ), f(f(r i )) = f(f(r j )), f(f(f(r i ))) = f(f(f(r j ))),...,, , x 1 0 ( θ = 10), R 1 = (160, 9), R 2 = (130, 182), R 19 = (0, 68), R 31 = (0, 68). R 19 R 31, R 19 = 227 P Q = 9 P + 37 Q = R 31 d = 176. ρ, l, Baby-step Giant-step. l/θ, Baby-step Giant-step., ρ,.,,. 41

43 Certicom Certicom Waterloo Scott Vanstone 1985, (, Waterloo RIM ). Certicom,. 42

44 2.5., ρ. (Certicom Challenge.) Certicom Challenge Certicom Challenge Certicom Challenge Certicom Challenge p = ( )/( ) a = b = #E = x P = y P = l = x Q = y Q = d = Q x (π 3) 10 34,. 112, PlayStation ,

45 Certicom Challenge Certicom,,., ,000,,,. 44

46 2.6, ρ., Menezes-Okamoto-Vanstone Waterloo Alfred Menezes, Scott Vanstone NTT Tatsuaki Okamoto, F p p + 1 (Supersingular ), F 7, Supersingular., ( ) Supersingluar Semaev-Smart-Satoh-Araki Igor Semaev, Bristol Nigel Smart, Takakazu Satoh Kiyomichi Araki, F p p (Anomalous ), F 7, Anomalous. 45

47 Menezes-Okamoto-Vanstone,,,. (pairing)., 2 e : E(F q ) E(F q ) G, P, P, Q, Q, e(p + P, Q) = e(p, Q) e(p, Q), e(p, Q + Q ) = e(p, Q) e(p, Q )., (bilinear map).,,, ID. 46

48 , P, Q Q = d P d.,, Baby-step Giant-step, ρ. ρ. [ ],,.,,.,. 47

49 3,.. 48

50 3.1 (ECC),. (elliptic curve cryptosystems, ECC) ( ).,. RSA, RSA RSA, ( d) 160., F p, E : y 2 = x 3 + ax + b. P = (x P, y P ), l.. p = = a = 3 = b = #E = x P = y P = l =

51 3.1 Diffie-Hellman (ECDH) Menezes-Qu-Vanstone (ECMQV) ElGamal (ECElGamal) DSA (ECDSA), (encryption), (cryptsystem) 2.,. (cryptsystem ),. 50

52 3.2 ECDH, ( ).,, Diffie-Hellman (ECDH ). 3.2 (ECDH ) Alice Bob, F p E P. Alice Bob, 2 K A = K B. 1. Alice d A, P A = d A P Bob. 2. Bob d B, P B = d B P Alice. 3. P B Alice K A = d A P B,. 4. P A Bob K B = d B P A, F 7 E : y 2 = x 3 + 3x + 4 P = P 1, d A = 2, d B = 3. P A = K B = P B = K A = 3.4 ECDH, Alice K A Bob K B. ECDH. Carol, Alice Bob ECDH, F p, E, P. Alice Bob P A, P B. P A, P d A, P B, P d B, Carol d A, d B. ECDH. 51

53 Diffie-Hellman (ECDH ) P A = d A P, P B = d B P, P A, P B, P K = d A d B P Diffie-Hellman (ECDH ). ECDH., P A, P d A, K = d A P B, ECDH. ECDH, d A, d B K,. ECDH,, ECDH. 52

54 3.3 ECElGamal ECDH, ECElGamal. 3.5 (ECElGamal ) Alice Bob, F p E P. Alice Bob, Bob M. 1. Bob d B, P B = d B P. P B, d B. 2. Alice a r, P A = r P. b Bob P B, K = r P B. c M, C = M + K. d Bob C P A. 3. Bob a P A d B, K = d B P A. b M = C K, M. 3.6 ECElGamal, Alice K Bob K.. 53

55 , 1985 IBM Victor Miller Washington Neal Koblitz.,. 54

56 3.4 ECDSA, ECDSA. 3.7 (ECDSA ) Alice Bob, F p E P l. Alice Bob, Bob m. 1. Alice d A (1 d A l), P A = d A P. P A, d A. 2. Alice a r, U = r P = (x U, y U ). b m H(m). c u = x U mod l, v = (H(m) + u d A )/r mod l. d Bob (u, v). 3. Bob a Alice P A, d = 1/v mod l V = d H(m) P + d u P A = (x V, y V ). b u = x V mod l.,,. 3.8 ECDSA,. 55

57 DTCP,., DTCP (Digital Transmission Content Protection),. DTCP, ECDH, ECDSA. 56

58 3.5,,. 2, Baby-step Giant-step, ρ l. l,.,. 3.9 ( ) P. F p E 1. p, p. 2. a, b F p, E(a, b) : y 2 = x 3 + ax + b. (Hasse-Weil, p p #E(a, b) p p ). 3. #E(a, b) #E(a, b) = p 2. (Anomalous ). 5. E(a, b) P Supersingular ,.. 57

59 NIST,,,. NIST. 58

60 (ECC), ( ). [ ],,.,., ANSI ( ), IEEE ( ), ISO ( ), NIST ( ), CRYPTREC ( ).,. 59

61 ,.,., ( ). 2,., ( ),, , ( ),, ,. 3,. 3,,.,,, Joseph Silverman, A Friendly Introduction to Number Theory (3rd edition), Pearson Prentice Hall, 2006 [, ( 3 ),, 2007 ] Victor Shoup, A Computational Introduction to Number Theory and Algebra (1st edition) [ PDF : Jeffrey Hoffstein, Jill Pipher, Joseph Silverman, An Introduction to Mathematical Cryptography, Springer-Verlag,

62 ,,. 3,.,,, ,, BP, ( ),,, , 2 (, ).,,, ,,,, ,. 4,. Neal Koblitz, A Course in Number Theory and Cryptography (2nd edition), Springer-Verlag, 1994 [, ( 2 ),, ] Ian Blake, Gadiel Seroussi, Nigel Smart, Elliptic Curves in Cryptography, Cambridge University Press, 2000 [,,, 2001 ] Darrel Hankerson, Alfred Menezes, Scott Vanstone, Guide to Elliptic Curve Cryptography, Springer, 2002, (, ),,, , ( : ID ). ID. Luther Martin, Introduction to Identity-Based Encryption, Artech House,

63 ,,.,,. (ISEC) [2 1 ] (JANT) [3 1 ], ISEC, 300. (SCIS) [ 1 ], (IACR) CRYPTO EUROCRYPTO ASIACRYPTO PKC,. Workshop on Elliptic Curve Cryptography (ECC) 62

30 2014.08 2 1985 Koblitz Miller 2.1 0 field Fp p prime field Fp E Fp Fp Hasse Weil 2.2 Fp 2 P Q R R P Q O P O R Q Q O R P P xp, yp Q xq, yq yp yq R=O

30 2014.08 2 1985 Koblitz Miller 2.1 0 field Fp p prime field Fp E Fp Fp Hasse Weil 2.2 Fp 2 P Q R R P Q O P O R Q Q O R P P xp, yp Q xq, yq yp yq R=O An Internet Vote Using the Elliptic Curve Cryptosystem TAKABAYASHI Shigeki Nowadays various changes are taking place in the society by the spread of the Internet, and we will vote by the Internet using

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