Page 1
Discrete Methods in Mathematical InformaticsLecture 1: What is Elliptic Curve?
9th October 2012
Vorapong Suppakitpaisarnhttp://www-imai.is.s.u-tokyo.ac.jp/~mr_t_dtone/
[email protected] , Eng. 6 Room 363Download Slide at: https://www.dropbox.com/s/xzk4dv50f4cvs18/Lecture
%201.pptx?m
Page 2
First Section of This Course [5 lectures]
Lecture 1: What is
Elliptic Curve?
Lecture 2: Elliptic Curve
Cryptography
Lecture 3-4:
Fast Implementation
for Elliptic Curve Cryptography
Lecture 5: Factoring
and Primality Testing
L. C. Washington, “Elliptic Curves: Number Theory and Cryptography”, Chapman &
Hall/CRC, 2003.
• Lecture 1: Chapter 1, Chapter 2 (2.1, 2.2)
• Lecture 2: Chapter 6 (6.1 – 6.6)
• Lecture 5: Chapter 7
Recommended Reading
H. Cohen, G. Frey, R. Avanzi, C. Doche, T. Lange, K. Nguyen, F. Vercauteren, "Handbook of Elliptic and Hyperelliptic
Curve Cryptography", Chapman & Hall/CRC, 2005.
A. Cilardo, L. Coppolino, N. Mazzocca, L. Romano, "Elliptic Curve Cryptography Engineering", Proc. of IEEE Vol. 94,
No. 2, pp. 395-406 (2006).
In each lecture, 1-2 exercises will be given,
Choose 3 Problems out of them.
Submit to
[email protected]
before 31 Dec 2012
Grading
Page 3
First Section of This Course [5 lectures]
Lecture 1: What is
Elliptic Curve?
Lecture 2: Elliptic Curve
Cryptography
Lecture 3-4:
Fast Implementation
for Elliptic Curve Cryptography
Lecture 5: Factoring
and Primality Testing
L. C. Washington, “Elliptic Curves: Number Theory and Cryptography”, Chapman &
Hall/CRC, 2003.
• Lecture 1: Chapter 1, Chapter 2 (2.1, 2.2)
• Lecture 2: Chapter 6 (6.1 – 6.6)
• Lecture 5: Chapter 7
Recommended Reading
H. Cohen, G. Frey, R. Avanzi, C. Doche, T. Lange, K. Nguyen, F. Vercauteren, "Handbook of Elliptic and Hyperelliptic
Curve Cryptography", Chapman & Hall/CRC, 2005.
A. Cilardo, L. Coppolino, N. Mazzocca, L. Romano, "Elliptic Curve Cryptography Engineering", Proc. of IEEE Vol. 94,
No. 2, pp. 395-406 (2006).
In each lecture, 1-2 exercises will be given,
Choose 3 Problems out of them.
Submit to
[email protected]
before 31 Dec 2012
Grading
Page 4
Problem 1: The Artillerymens Dilemma (is not a) Puzzle
http://cashflowco.hubpages.com/
?
Height = 0: 0 Ball Square
Height = 1: 1 Ball Square
Height = 2: 1 + 4 = 5 Balls Not Square
Height = 3: 1 + 4 + 9 = 14 Balls Not Square
Height = 4: 1 + 4 + 9 + 16 = 30 Balls Not Square
2232222
6
1
2
1
3
1
6
121321 yxxx
)x)(x(xx...
Elliptic Curve
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Problem 1: The Artillerymens Dilemma (is not a) Puzzle (cont.)
223
6
1
2
1
3
1yxxx
(0,0)
(1,1)
y = x
223
6
1
2
1
3
1xxxx
02
1
2
3 23 xxx
0)()(
0))()((23
abcxbcacabxcbax
cxbxax
a,b,c
equation the of roots
are that Suppose
solution. another is 2
1 y ,
2
1 x thatknow We
2
12
310
c
ccba
(1/2,1/2)
Page 6
Problem 1: The Artillerymens Dilemma (is not a) Puzzle (cont.)
223
6
1
2
1
3
1yxxx
(0,0)
(1,1)
y = x
(1/2,1/2)
(1/2,-1/2)
y = 3x-2
223 )23(6
1
2
1
3
1 xxxx
0...2
51 23 xx
2
511
2
1 x
70,24 yx
2222 7024...21
70 Length Square for 24 Height Pyramid
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Problem 2: Right Triangle with Rational Sides
We want to find a right triangle with rational sides
in which area = 5
3
4
5
6
15
8
17
60
15/2
4
17/2
155
5
510
Page 8
Problem 2: Right Triangle with Rational Sides (cont.)
a
b
c
ab/2 = 5
22210 cb, aab
524
102
4
2
2
22222
ccbababa
524
102
4
2
2
22222
ccbababa
numbers rational of square are 2
c
2
c
numbers rational are
5,2
,5
2,
2,
2222
c
bacba
23 25)5()5( yxxxxx
Elliptic Curve
4
25x
num rational of square
a not is 4
5 but
curve,elliptic of
solution a is
4
45,
Note
Page 9
Problem 2: Right Triangle with Rational Sides (cont.)
23 25 yxx
(-4,6)12
23
)6(2
25)4(3
2
253
2)253(
)()25(
22
2
23
y
x
x
y
yyxx
yxx
3
41)4(
12
23)6(
12
23
c
cxy3
41
12
23 xy
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Problem 2: Right Triangle with Rational Sides (cont.)
23 )3
41
12
23(25 xxx
0...144
529 23 xx
0)()(
0))()((23
abcxbcacabxcbax
cxbxax
a,b,c
equation the of roots
are that Suppose
2
6
41
144
1681
144
52944
0)))(4())(4((
x
x
cxxx
23 25 yxx
(-4,6)
3
41
12
23 xy
(1681/144,62279/1728)
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Problem 2: Right Triangle with Rational Sides (cont.)
22
22
22
212
49
144
24015
212
31
144
9615
212
41
bax
bax
cx
2
3,
3
206
492
12
496
312
12
316
412
12
41
ba
ba
ba
c
20/3
3/2
41/6
5
23 25 yxx
(-4,6)
3
41
12
23 xy
(1681/144,62279/1728)
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Exercises
5. area withtriangle right another find to
at line tangent the Use )1728
62279,
144
1681(),( yx
Exercise 1
Exercise 2
numbers. rational of squares are
that such point a in curve the intersects
at curve this to line tangent the then , and
satisfying numbers rational are if thatShow integer. an be Let
nn,x,xx),y(x(x,y)
n,xxnxy
x, yn
11111
232 0,
Page 13
Problem 3: Fermat’s Last Theorem
http://wikipedia.com/
nnn cba
a,b,c
n
that such
integers nonzero no is there
, Given 3
• Conjectured by Pierre de Fermat in Arithmetica (1637).
“I have discovered a marvellous proof to this theorem, that this margin
is too narrow to contain”
• There are more than 1,000 attempts, but
the theorem is not proved until 1995 by
Andrew Wiles.
• One of his main tools is Elliptic Curve!!!
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Problem 3: Fermat’s Last Theorem (cont.)
nnn cba
a,b,c
n
that such integers nonzero no is there
, Given 3
• Fermat kindly provided the proof for the case when n = 4
2
22
2
22 )(4,
a
cbby
a
cbx
xxy 432 Elliptic Curve
By several elliptic curves techniques, Fermat found that all rational solutions of the elliptic curve are (0,0),
(2,0), (-2,0)
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Formal Definitions of Elliptic Curve
0274 2332 BABAxxy when
B}AxxL|yL{(x,y)}{E(L) 32
223
6
1
2
1
3
1yxxx
(0,0)
(1,1)
y = x
(1/2,1/2)
(1/2,-1/2)
Weierstrass Equation
Elliptic Curve
.
)(),(),,(
33
33
21
2211
)y,(xQP
),y(xR
QP
Q Pxx
LEyxQyxP
3.
curve. the cut line the that
point another , point Find 2.
and point pass that line aDraw 1.
:follows as define we, If
Point Addition
)2
1,
2
1()1,1()0,0(
Page 16
Formal Definitions of Elliptic Curve (cont.)
.
)(),(),,(
33
33
21
2211
)y,(xQP
),y(xR
QP
Q Pxx
LEyxQyxP
3.
curve. the cut line the that
point another , point Find 2.
and point pass that line aDraw 1.
:follows as define we, If
Point Addition
)( 11
12
12
xxmyy
xx
yym
0...
))((223
311
32
xmx
BAxxyxxm
BAxxy
212
3 xxmx
1133 )( yxxmy
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Formal Definitions of Elliptic Curve (cont.)
223
6
1
2
1
3
1yxxx
x = 1/2
(1/2,1/2)
(1/2,-1/2)
QPyyxx
LEyxQyxP
, , If 2121
2211 )(),(),,(
Point Addition
,PP
)y, (xP P PQ P
),y(xR
yyxx
LEyxQyxP
33
33
221
2211
2
)(),(),,(
3.
curve. the cut
line the that point another Find 2.
P. point at curve the touching line aDraw 1.
, If 1
Point Double
1728
62279,
144
1681)6,4()6,4()6,4(2
23 25 yxx
(-4,6)
3
41
12
23 xy
(1681/144,62279/1728)
Page 18
Formal Definitions of Elliptic Curve (cont.)
Point Double
)( 11 xxmyy
0...
))((223
311
32
xmx
BAxxyxxm
BAxxy
12
3 2xmx
1133 )( yxxmy
)y, (xP P PQ P
),y(xR
yyxx
LEyxQyxP
33
33
221
2211
2
)(),(),,(
3.
curve. the cut
line the that point another Find 2.
P. point at curve the touching line aDraw 1.
, If 1
y
Ax
x
ym
xAxyy
BAxxy
2
3
)3(22
2
32
Page 19
First Section of This Course [5 lectures]
Lecture 1: What is
Elliptic Curve?
Lecture 2: Elliptic Curve
Cryptography
Lecture 3-4:
Fast Implementation
for Elliptic Curve Cryptography
Lecture 5: Factoring
and Primality Testing
L. C. Washington, “Elliptic Curves: Number Theory and Cryptography”, Chapman &
Hall/CRC, 2003.
• Lecture 1: Chapter 1, Chapter 2 (2.1, 2.2)
• Lecture 2: Chapter 6 (6.1 – 6.6)
• Lecture 5: Chapter 7
Recommended Reading
H. Cohen, G. Frey, R. Avanzi, C. Doche, T. Lange, K. Nguyen, F. Vercauteren, "Handbook of Elliptic and Hyperelliptic
Curve Cryptography", Chapman & Hall/CRC, 2005.
A. Cilardo, L. Coppolino, N. Mazzocca, L. Romano, "Elliptic Curve Cryptography Engineering", Proc. of IEEE Vol. 94,
No. 2, pp. 395-406 (2006).
In each lecture, 1-2 exercises will be given,
Choose 3 Problems out of them.
Submit to
[email protected]
before 31 Dec 2012
Grading
Page 20
Exercises
5. area withtriangle right another find to
at line tangent the Use )1728
62279,
144
1681(),( yx
Exercise 1
Exercise 2
numbers. rational of squares are
that such point a in curve the intersects
at curve this to line tangent the then , and
satisfying numbers rational are if thatShow integer. an be Let
nn,x,xx),y(x(x,y)
n,xxnxy
x, yn
11111
232 0,
Page 21
Thank you for your attention
Please feel free to ask questions or comment.
Page 22
Scalar Multiplication• Scalar Multiplication on Elliptic Curve
S = P + P + … + P = rP
when r1 is positive integer, S,P is a member of the curve• Double-and-add method• Let r = 14 = (01110)2
Compute rP = 14P r = 14 = (0 1 1 1 0)2 Weight = 3
P 3P 7P 14P
6P2P 14P
3 – 1 = 2 Point Additions
4 – 1 = 3 Point Doubles
r times
O