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Wireless Sensornetworks Concepts, Protocolls and Applications

Wireless Sensornetworks Concepts, Protocolls and Applications. Übungen: RSA und ECC, Teil 2. Zoya Dyka/Peter Langendörfer e-mail: dyka [ at ] ihp-microelectronics.com web: www.tu-cottbus.de/systeme. ECC (Elliptic Curve Cryptography). - EC - Operationen mit EC-Punkten

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Wireless Sensornetworks Concepts, Protocolls and Applications

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  1. Wireless SensornetworksConcepts, Protocolls and Applications Übungen: RSA und ECC, Teil 2 Zoya Dyka/Peter Langendörfer e-mail: dyka [ at ] ihp-microelectronics.com web: www.tu-cottbus.de/systeme

  2. ECC (Elliptic Curve Cryptography) - EC - Operationen mit EC-Punkten - kryptographische Operationen

  3. ECC EC: y2=x3+ax+b Kryptographie => keine reelle Zahlen !!! sondern: Elemente aus GF(p), z.B. GF(5)

  4. ECC 0 GF(5) “modulo”= n=5 { 0, 1, 2, 3, 4} 4 1 7 mod 5 = 2 2+4 mod 5 = 1 2-3 mod 5 = 4 2·3 mod 5 = 3+3 mod 5 =1 3 2 (2/3) mod 5 = ? = 2·3-1 mod 5 ... 3-1 mod 5 = ? 3 · 3-1 = 1 mod 5 3 · 0 = 0 ≠ 1 3 · 1 = 3 ≠ 1 3 · 2 = 1 2/3= 2·2 = 4

  5. x und y aus { 0, 1, 2, 3, 4} ECC 0 EC: y2=x3+x+1 GF(5) 4 1 3 2 9 EC-Punkte: „order“ of EC r = 9 EC über GF(5)

  6. ECC EC: y2=x3+ax+b y2+xy=x3+ax2+b reelle Zahlen !!! y2=x3+x+1

  7. x und y aus { 0, 1, 2, 3, 4} ECC 0 EC: y2=x3+x+1 GF(5) 4 1 3 2 EC über GF(5): EC-Punkte aus { A1, A2, ... , A9} 9 EC-Punkte: „order“ of EC r = 9

  8. ECC EC: y2=x3+x+1 A2 A4A6 A8 A7 A1 A3 A5 → EC-Punkt-Addition → EC-Punkt-Verdopplung A3+A8= ? A1+A1= 2A1=? A1+A1+A1= 3A1=? A1+A1+...+A1= kA1=? A1-A1= O → EC-Punkt-Multiplikation (kP-Operation)

  9. ECC EC-Punkt-Addition EC-Punkt-Verdopplung EC-Punkt-Multiplikation = kP-Operation= eine Reihe vonEC-Punkt-Additionen und Verdopplungen

  10. ECC EC: y2=x3+x+1 A2 A4A6 A8 A7 A1 A3 A5 A5+A8= ? A5+A8= (2,1)=A3

  11. ECC EC: y2=x3+x+1 A2 A4A6 A8 A7 A1 A3 A5 A1+A1= ? A1+A1= (4,2)=A7

  12. ECC EC: y2=x3+x+1 A2 A4A6 A8 A7 A1 A3 A5 A1+A1+A1= ? A1+A1+A1= 3∙A1=2∙A1+A1=A7+A1=A3 5∙A1=4∙A1+A1=A6+A1=A5 A1+A1+A1+A1= 4∙A1=2∙(2∙A1)=2∙A7=A6 6∙A1=4∙A1+2∙A1=A4 7∙A1=4∙A1+2∙A1+A1=A8 8∙A1=2∙(2∙(2∙A1))=A2 9∙A1=A9=O 10∙A1=A9+A1=O+A1=A1

  13. EC und Kryptographie EC-Parameter: y2=x3+x+1 order ein festgelegtes EC-Punkt G (z.B.G=A1) privat_key: Zufallszahl k public_key: k∙G=Pund alle EC-Parameter encryption:( R=d∙G;S=d∙P+M ) decryption:( R;S)→M=S-k∙R=(x,y) d ist Zufallszahl M=(x,y), x ist message

  14. ECC Hausaufgabe: 1) berechnen k: k=(MatrikelNr mod 5) wenn (MatrikelNr mod 5)=0 k=((MatrikelNr+2) mod 7) oder k=((MatrikelNr+3) mod 7) wenn (MatrikelNr mod 5)=1 k=((MatrikelNr+4) mod 7) 2) berechnen: k∙A2=? auf freie Wahl

  15. Summary • Thank you for your attention!

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