Slide 1

BI THUYT TRNHElliptic Curve Cryptography

NHM 4:H Ngc LinhNguyn c TonPhan Nguyn Nht TrngGVHD : L Ngc Luyn.Lp VT k37

Gii thiu ng cong elliptic

ng cong Elliptic trn s thc ng cong Elliptic l ng cong c dng: Y2 = x3+ax+b

PHNG TRNH TNG QUT ECC

Trong ng cong Elliptic, chng ta nh ngha thm mt im O (im v cc). Gi E(a, b) l tp cc im thuc ng cong y=x3+ax+b cng vi im O.

PHP CNG ECC

PHP CNG HAI IMP+Q=R(xr,yr)

xr=2-xp-xqyr =(xp-xr)-yp

Php cng hai im

PHP NHN I.

P+P=2Pxr=2-2xpyr =(xp-xr)-yp

ng cong elliptic trn trng Zp

ng con elliptic trn trng Zp, ng cong ny c dngY2 mod p = (x3+ax+b) mod p a,b,x,y ZpV d trong trng Z11, chn a=-1, b=0, x=4,y=4 ta c42 mod 11 = (43-4) mod 1116 mod 11= 60 mod 11 = 5

ng cong elliptic trn trng Zp

Y2X3 XX Y000001101024610039244 or 745594 or 753106163161075688 or 38999451010

ng cong elliptic trn trng Zp

ng cong Elliptic trn trng GF(2m) ng cong Elliptic trn trng GF(2m) l ng cong c cc h s thuc trng GF(2m), ng cong ny c dng hi khc so vi trn Zp:y2+xy=x3+ax=ba,b,x,y GF(2m)

ng cong y2+xy=x3+ax=b trn trng s thc

By gi chng ta s xt tp E2m(a,b) gm cc im trn ng cong Elliptic ny cng vi im v cc O. V d, xt trng GF(24) vi a thc ti gin l m(x)=x4+x+ 1. Phn t sinh g ca trng ny c iu kin g4 = g+ 1 . Bng cc ly tha ca g l:

Bng cc ly tha ca g l:

Xt v d v ng cong Elliptic trn GF(24):y2 + xy = x3 + g4x + 1(a = g4,b = 1)Bng bn di lit k cc im thuc ng cong ny

Vi 2 im P, Q bt k (P Q)php cng R=P+Q c xc nh bng cng thc:

Vi im P bt k R=P+P

ng cong Elliptic trong m ha ECCi vi m ha ng cong Elliptic, chng ta xy dng hm mt chiu nh sau: Trong nhm Abel Ep(a,b) xy dng t ng cong Elliptic Zp, xt phng trnh: Q=P+P+P+P++P=kP (im Q l tng ca k im P, k < p)Cho trc k v P, vic tnh Q thc hin d dng. Tuy nhin nu cho trc P v Q, vic tm ra k l cng vic kh khn. y chnh l hm logarit ri rc ca ng cong Elliptic.

V d:Y2 mod 17 = (x3+2x+2) mod 17a,b,x,y Z17

Cho im G =(5,1); M(7,6)

2G=(6;3)6G=(16;13)3G=(10;6)7G=(0;6)4G(3;1)8G=(13;17)5G=(9;16)9G=(7;6)

V 9G = M nn K = 9.

Trong thc t chng ta s s dng ng cong Elliptic Zp vi gi tr p ln, sao cho vic vt cn l bt kh thi. Hin nay ngi ta tm ra phng php tm k nhanh hn vt cn l phng php Pollar rho. Da vo hm mt chiu trn chng ta c 2 cch s dng ng cong Elliptic trong lnh vc m ha l trao i kha EC Diffie-Hellman v m ha EC.

Vd:EC Diffie Hellman

Vd:EC Diffie Hellman

NHNG THNG S THC T.

Alice

d