Preparation

1
Introduction
Preparation
Main result
Conclusion
A Method for Constructing A Self-Dual Normal
Basis in Odd Characteristic Extension Field
Department of Communication Network Engineering,
Faculty of Engineering, Okayama University,
Japan Hiroaki Nasu, Yasuyuki Nogami,
Ryo
Namba, and Yoshitaka Morikawa
2
Layout
Introduction
Preparation
Main result
Conclusion

• Introduction
• Background
• Motivation
• Preparation
• Self-dual normal basis (SDN)
• A special class of Gauss period normal bases
• Main result
• How to construct SDN
• Translation between GNB and SDN
• Conclusion

3
Background
Introduction
Preparation
Main result
Conclusion

• Public key cryptography
• elliptic curve cryptography
• pairing-based cryptographic applications
• ID-based cryptography, Group signature
• Finite field
• prime field
• extension field

4
Background
Introduction
Preparation
Main result
Conclusion

• Public key cryptography
• elliptic curve cryptography
• pairing-based cryptographic applications
• ID-based cryptography, Group signature
• Finite field
• prime field
• extension field

5
Background
Introduction
Preparation
Main result
Conclusion

• Public key cryptography
• elliptic curve cryptography
• pairing-based cryptographic applications

6
Background
Introduction
Preparation
Main result
Conclusion

• Public key cryptography
• elliptic curve cryptography
• pairing-based cryptographic applications
• ID-based cryptography, Group signature
• Finite field
• prime field
• extension field
• Bases
• Gauss period normal basis (GNB), optimal normal
  basis
• dual basis, self-dual normal basis (SDN)

7
Motivation
Introduction
Preparation
Main result
Conclusion

• Bases
• Gauss period normal basis (GNB), optimal normal
  basis
• dual basis, self-dual normal basis (SDN)

8
Motivation
Introduction
Preparation
Main result
Conclusion

• Gauss period normal basis (GNB)
• self-dual normal basis (SDN)

• Bases
• Gauss period normal basis (GNB), optimal normal
  basis
• dual basis, self-dual normal basis (SDN)

9
Self-dual normal basis
Preparation
Main result
Conclusion

• Self-dual normal basis in

10
A special class of GNBs
Preparation
Main result
Conclusion

• TypeII-X normal basis (TypeII-X NB)

• Normal basis (NB) in
• NB GNB

• TypeII ONB in
• TypeII-X NB in

11
Main result
Main result
Conclusion
TypeII-X NB in an SDN in
12
Property of TypeII-X NB
Main result
Conclusion
TypeII-X NB in
By the way, it is well-known that GNB is SDN
when is divisible by characteristic .
13
How to construct SDN
Main result
Conclusion
In order to satisfy
and need to satisfy
14
How to construct SDN
Main result
Conclusion
In addition, in order to satisfy
needs to satisfy
15
How to construct SDN
Main result
Conclusion
The most important is
Changing parameter such that
except for the case , it
is always found.
16
Translation between GNB and SDN
Main result
Conclusion
TypeII-X NB in an SDN in
basis translation
17
Translation between GNB and SDN
Main result
Conclusion
TypeII-X NB (GNB) SDN
18
Translation between GNB and SDN
Main result
Conclusion
TypeII-X NB (GNB) SDN
SDN GNB and GNB SDN require
several multiplications and additions in .
19
Conclusion
Conclusion

• Main result
• How to construct SDN from GNB
• Translation between GNB and SDN

Future work

• Is the obtained SDN one of GNBs in ?