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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12
3914:47 notes S /*pplicationslsage/sage
Detected 6A0E64 flag
Building Sage on OS Z in 64—bit mode
Sage version 4.6.2, Release Date: 2011—02-25
I
1ype notebook() for the GUI,
Iaqez I
sage: I
sages V
sages V
Ring of
sage: I
and license() for intonation.
some experiments with elliptic tunes with sage
first define a (Said mod 101
— Zmot(1O1)
integers nodule 101
example of multiplication is V
sages t(l0)flhl)
9
sage: * define elliptic ctrve aver V
nge. P — EllipticCurve(P,f0,1J)
sage: S
Elliptic Curve defined by y2
sage:
sage:
(96 :
sages
r3 P 1 Over Ring of integers modulo 101
P — E.r.ndoajoint()
P
1)
49
I note coordinates are La projective ton (K i I
5) representing
U point x — 1/I, y — 1/5, with a — 0 for point at infinity.
sage;
sage: I get another point
sage: Q — !.random_point()
angus 9
(29
94 : 1)
sage: P+Q
21 : 77 ,
1.)
sage: I check cammutativity
sage: Q+P
(21
77
1)
sage: I get thira point
sage; K — E.randonpoiflt()
sage; K
(76 t 58 1 1)
sage: I check •ssoei.ativity
sage: P + (Q+R)
(53
2 : 1)
sage:
(P+9)+R
: 1)
sages 4 find size of this group
sage: S.order()
102
sage: I what are factors of 102?
sage: factar(102)
2 — 3 * 17
sage: I so possible orders of elements are 1,2,3,6,17,34,51,102
Pages Porder()
51
(53
: 2
-
sega. Q.order()
51
sage; R.order()
Si
sage: * none of P,Q, A are a generator (i.e. have order 102)
sages Iiet’e find one
sage; a — E.racdoia_point()
13
sage: R.order()
102
sage: 8 binge
sage: 4 what don identity look like?
sage: P-P
(0 i 1 a 0)
a
sage:
sage. I
(0
1
sage: 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 101
sage: I look at nec seall powers of generator a
sage: 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)
1)
7 (90 a 65
$ (35 a 31 • 1)
9 (10 a 71 a I)
10 (18
51 a 1)
14 a 1)
11 (93
12 (4 a 41
1)
13 (38
38 a 1)
14 (76
58 a 1)
sage: I find discrete log at P, base a
sage: R.dIscretelog(P)
80
sage: 80*R
(96 a 49 a 1)
sage: C0RP
True
sage: I find discrete log of Q, base K
sage: K. di.crete_log(Q)
58
sager I find eleaents of each possible order
sage. R.ordet()
102
sage: S — 2R
sage: S.order()
51
sage: S — 3R
Sage: S.order()
34
sage: S — 6K
sage, S.osder()
17
sage: S — 17CR
sage; 8.order()
14
15
f
0
‘
aot)
9
t
.t5 — S
npog :abgs
— 9
p
MIT OpenCourseWare
http://ocw.mit.edu
1HWZRUNDQG&RPSXWHU6HFXULW\
Spring 2014
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