6.857 Computer and Network Security Lecture 12

Admin: Project proposals due this week. Project Ideas:

Electronic auctions See: recent CACM article

http://www.cs.ut.ee/~lipmaa/crypto/link/protocols/auctions.php (good high-level set of links)

Today:

Group theory review Diffie-Hellman key exchange Five crypto groups: Z *, Q , Z *, Q , elliptic curves

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3914:47 notes S /*pplicationslsage/sageDetected 6A0E64 flagBuilding Sage on OS Z in 64—bit mode

Sage version 4.6.2, Release Date: 2011—02-25I 1ype notebook() for the GUI, and license() for intonation.

Iaqez I some experiments with elliptic tunes with sagesage: I first define a (Said mod 101sages V — Zmot(1O1)sages VRing of integers nodule 101sage: I example of multiplication is Vsages t(l0)flhl)9sage: * define elliptic ctrve aver Vnge. P — EllipticCurve(P,f0,1J)sage: SElliptic Curve defined by y2 r3 P 1 Over Ring of integers modulo 101sage: P — E.r.ndoajoint()sage: P(96 : 49 1)sages I note coordinates are La projective ton (K i I 5) representingsage; U point x — 1/I, y — 1/5, with a — 0 for point at infinity.sage: I get another pointsage: Q — !.random_point()angus 9(29 94 : 1)sage: P+Q21 : 77 , 1.)sage: I check cammutativitysage: Q+P(21 77 1)sage: I get thira pointsage; K — E.randonpoiflt()sage; K(76 t 58 1 1)sage: I check •ssoei.ativitysage: P + (Q+R)(53 2 : 1)sage: (P+9)+R(53 : 2 : 1)sages 4 find size of this groupsage: S.order()102sage: I what are factors of 102?sage: factar(102)2 — 3 * 17 -

sage: I so possible orders of elements are 1,2,3,6,17,34,51,102Pages Porder()51sega. Q.order()51sage; R.order()Sisage: * none of P,Q, A are a generator (i.e. have order 102)sages Iiet’e find onesage; a — E.racdoia_point()

13

sage: R.order()102sage: 8 bingesage: 4 what don identity look like?sage: P-P(0 i 1 a 0)sage: a

sage. I(0 1sage: Zn(96 t 49 a 1)sage; P4.1(96 : 49 a 1)sage: .

(96 a 52 a 1)sags: I cots that innrses just negate T component, nodule 101sage: I look at nec seall powers of generator asage: for i in range(15): print 1, j*g

o (0 a 1 a 0)1 (72 a 85 a 1)2 (15 a 89 a 1)3 (9 a 86 a 1)4 (84 a 21 a 1)£ (52 a 44 a 1)6 (07 a 61 a 1)7 (90 a 65 1)$ (35 a 31 • 1)9 (10 a 71 a I)10 (18 51 a 1)11 (93 14 a 1)12 (4 a 41 1)13 (38 38 a 1)14 (76 58 a 1)sage: I find discrete log at P, base asage: R.dIscretelog(P)80sage: 80*R(96 a 49 a 1)sage: C0RPTruesage: I find discrete log of Q, base Ksage: K. di.crete_log(Q)58sager I find eleaents of each possible ordersage. R.ordet()102sage: S — 2Rsage: S.order()51sage: S — 3RSage: S.order()34sage: S — 6Ksage, S.osder()17sage: S — 17CRsage; 8.order()

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6.857 Network and Computer SecuritySpring 2014

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