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Wireless SensornetworksWireless SensornetworksConcepts, Protocolls and ApplicationsConcepts, Protocolls and Applications
Zoya Dyka/Peter Langendörfer
e-mail: dyka [ at ] ihp-microelectronics.comweb: www.tu-cottbus.de/systeme
Übungen: RSA und ECC, Teil 2
Chair Systemswww.tu-cottbus.de/systeme
Monday, 09. January 2012winter term 2011/12 – Wireless Sensor Networks
Chapter 6 – Page 2
- EC- Operationen mit EC-Punkten- kryptographische Operationen
ECC (Elliptic Curve Cryptography)
Chair Systemswww.tu-cottbus.de/systeme
Monday, 09. January 2012winter term 2011/12 – Wireless Sensor Networks
ECC
EC: y2=x3+ax+b
Kryptographie =>keine reelle Zahlen !!!
sondern:Elemente aus GF(p), z.B. GF(5)
Chair Systemswww.tu-cottbus.de/systeme
Monday, 09. January 2012winter term 2011/12 – Wireless Sensor Networks
ECC
“modulo”= n=5 { 0, 1, 2,
3, 4}7 mod 5 = 2
2+4 mod 5 = 1 2-3 mod 5 = 4
2·3 mod 5 = 3+3 mod 5 =1
(2/3) mod 5 = ? = 2·3-1 mod 5 ... 3-1 mod 5 = ?3 · 3-1 = 1 mod 53 · 0 = 0 ≠ 1 3 · 1 = 3 ≠ 13 · 2 = 1
2/3= 2·2 = 4
Chair Systemswww.tu-cottbus.de/systeme
Monday, 09. January 2012winter term 2011/12 – Wireless Sensor Networks
ECC
EC:y2=x3+x+1
x y2 y2 mod 5 y - y-y mod
5EC-Punkte
0 1 1 1 -1 4 (0,1) (0,4)
1 3 3 -- -- -- (1,∞)
2 11 1 1 -1 4 (2,1) (2,4)
3 31 1 1 -1 4 (3,1) (3,4)
4 69 4 2 -2 3 (4,2) (4,3)
x und y aus { 0, 1, 2, 3, 4}
0
1
23
4GF(5)EC über GF(5)
9 E
C-P
unkt
e:
„or
der
“ o
f E
C r
= 9
Chair Systemswww.tu-cottbus.de/systeme
Monday, 09. January 2012winter term 2011/12 – Wireless Sensor Networks
ECC
EC: y2=x3+ax+b
y2+xy=x3+ax2+b
y2=x3+x+1
reelle Zahlen !!!
Chair Systemswww.tu-cottbus.de/systeme
Monday, 09. January 2012winter term 2011/12 – Wireless Sensor Networks
ECC
EC: y2=x3+x+1x y2 y2 mod 5 y - y
-y mod 5
EC-Punkte
0 1 1 1 -1 4 (0,1) (0,4)
1 3 3 -- -- -- (1,∞)
2 11 1 1 -1 4 (2,1) (2,4)
3 31 1 1 -1 4 (3,1) (3,4)
4 69 4 2 -2 3 (4,2) (4,3)
x und y aus { 0, 1, 2, 3, 4}
0
1
23
4GF(5)
9 E
C-P
unkt
e:
„or
der
“ o
f E
C r
= 9
EC über GF(5): EC-Punkte aus { A1, A2, ... , A9}
Chair Systemswww.tu-cottbus.de/systeme
Monday, 09. January 2012winter term 2011/12 – Wireless Sensor Networks
ECC
EC: y2=x3+x+1EC-Punkte
A1=(0,1) A2=(0,4)
A9=(1,∞)=O
A3=(2,1) A4=(2,4)
A5=(3,1) A6=(3,4)
A7=(4,2) A8=(4,3)
A2 A4 A6
A8
A7
A1 A3 A5
A3+A8= ?A1+A1= 2A1=?A1+A1+A1= 3A1=?A1+A1+...+A1= kA1=?A1-A1= O
→ EC-Punkt-Addition→ EC-Punkt-Verdopplung
→ EC-Punkt-Multiplikation (kP-Operation)
Chair Systemswww.tu-cottbus.de/systeme
Monday, 09. January 2012winter term 2011/12 – Wireless Sensor Networks
ECC
3113
212
3
1
21
12
12
33
2211
y)3
)2
2
3)1
::
,:
,,,:
xxy
xxx
y
ax
xx
yy
QPwennQPwenn
QPyxSOutput
yxQyxPInput
EC-Punkt-Addition EC-Punkt-Verdopplung
EC-Punkt-Multiplikation = kP-Operation= eine Reihe von EC-Punkt- Additionen und Verdopplungen
Chair Systemswww.tu-cottbus.de/systeme
Monday, 09. January 2012winter term 2011/12 – Wireless Sensor Networks
ECC
EC:y2=x3+x+1
A2 A4 A6
A8
A7
A1 A3 A5
A5+A8= ?
15mod2321y)3
25mod432)2
25mod34
13)1
3113
221
23
12
12
xxy
xxx
xx
yy
A5+A8= (2,1)=A3
EC-Punkte
A1=(0,1) A2=(0,4)
A9=(1,∞)
A3=(2,1) A4=(2,4)
A5=(3,1) A6=(3,4)
A7=(4,2) A8=(4,3)
Chair Systemswww.tu-cottbus.de/systeme
Monday, 09. January 2012winter term 2011/12 – Wireless Sensor Networks
ECC
EC:y2=x3+x+1
EC-Punkte
A1=(0,1) A2=(0,4)
A9=(1,∞)
A3=(2,1) A4=(2,4)
A5=(3,1) A6=(3,4)
A7=(4,2) A8=(4,3)
A2 A4 A6
A8
A7
A1 A3 A5
A1+A1= ?
25mod4031y)3
45mod003)2
35mod2
15mod
12
103
2
3)1
3113
221
23
1
21
xxy
xxx
y
ax
A1+A1= (4,2)=A7
Chair Systemswww.tu-cottbus.de/systeme
Monday, 09. January 2012winter term 2011/12 – Wireless Sensor Networks