1. Efficient CRT-Based RSA Cryptosystems
Immune against the Hardware Fault Attack and
the FPGA Implementations
Yonghong Yang
Supervisors: Prof. Z. Abid & Prof. W. Wang
Department of Electrical and Computer Engineering
the University of Western Ontario, Canada
November-15-18

2. Outline Introduction Literature Review
Proposed Efficient Two-Prime RSA Cryptosystem
Proposed Efficient Multi-Prime RSA Cryptosystem
FPGA Implementations and Results
Conclusions
November-15-18

3. Introduction
Network security is needed everywhere:
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4. Introduction Wide applications need security
Electronic banking and voting
Electronic commerce, such as online bidding
, file exchange/submission
Web browsing, etc.
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5. Introduction Cryptography Cryptography guarantees the needed security
The mathematical science to secure the
confidentiality/authentication of data by
replacing them with a transformed version
Two types: secret-key and public-key
Cryptography guarantees the needed security
Privacy or confidentiality
Data integrity
Authentication
Non-repudiation
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6. Introduction Secret-Key Cryptography Disadvantages:
Traditional method of cryptography
Theoretical basis: “communication theory of secrecy
systems”
Single key is used to encrypt and decrypt texts
DES, NSA and IDEA etc.
Disadvantages:
Difficult key management
Keys need to be changed frequently
Cannot yield efficient signature mechanisms
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7. Introduction Public-Key Cryptography
Relatively new field – 1975, initiated by the
paper “New directions in cryptography ”
Different keys are used for encryption and
decryption
RSA, DSA, DSS etc.
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8. Introduction Public-Key Cryptography Advantages: Disadvantage:
Easier key management
Key can remain unchanged for longer time
Yields efficient digital signature mechanisms
Disadvantage:
Slower throughputs since keys have larger
wordlengths
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9. Introduction RSA Cryptography
One of the most widely used, simplest public-
key cryptography so far
Scheme
Alice
Bob
Encrypt using B’s public key
Decrypt using by B’s private key
Sign with A’s private key
Check signature by A’s public key
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10. Outline Introduction Literature Review
Proposed Efficient Two-Prime RSA Cryptosystem
Proposed Efficient Multi-Prime RSA Cryptosystem
FPGA Implementations and Results
Conclusions
November-15-18

11. Literature Review RSA Cryptosystem
Public quantities: n, e; secret quantities: d,
Encryption/decryption:
Encryption:
Decryption:
Signing/signature verification:
Signing:
Signature verification:
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12. Literature Review Chinese Remainder Theorem Based RSA
Chinese Remainder Theorem is often used to
speedup the operations of RSA
Attacks on the CRT-based RSA
Hardware fault attack
Timing attack
Power attack
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13. Literature Review Countermeasures to the attack Padding the message,
drawback: collision-free hash function (hard)
Checking the intermediate or final results,
drawback: double the operational time and
not secure
Revising the signature expression,
make sure no secret information is leaked
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14. Outline Introduction Literature Review
Proposed Efficient Two-Prime RSA Cryptosystem
Proposed Efficient Multi-Prime RSA Cryptosystem
FPGA Implementations and Results
Conclusions
November-15-18

15. Proposed Two-Prime RSA
Standard CRT-based two-prime RSA
To calculate:
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16. Proposed Two-Prime RSA
Standard CRT-based two-prime RSA
Vulnerable to the hardware fault attack:
When available:
and
factors the system
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17. Proposed Two-Prime RSA
CRT-2 protocol proposed by Yen et al. 1. 2. 3.
where
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18. Proposed Two-Prime RSA1. 2. 3.
where
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19. Proposed Two-Prime RSA
Block diagram of the proposed two-prime RSA
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20. Proposed Two-Prime RSA
Comparison of the operational speed
Division
Modular exponentiation
CRT-2 protocol by Yen. et al.
The proposed two-prime RSA
where ( ) ,
and
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21. Proposed Two-Prime RSA
Factorization complexity
The complexity of factoring the proposed RSA
system:
The complexity of factoring CRT-2:
Similar
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22. Outline Introduction Literature Review
Proposed Efficient Two-Prime RSA Cryptosystem
Proposed Efficient Multi-Prime RSA Cryptosystem
FPGA Implementations and Results
Conclusions
November-15-18

23. Proposed Multi-Prime RSA
Standard CRT-based multi-prime RSA
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24. Proposed Multi-Prime RSA
Immunity of CRT-based multi-prime RSA:
When (j-1) faulty signatures available, calculations
according to these (j-1) faulty signatures factors the
multi-prime RSA
Still vulnerable to the hardware fault attack
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25. Proposed Multi-Prime RSA1. 2. 3.
for
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26. Proposed Multi-Prime RSA
The proposed multi-prime RSA
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27. Proposed Two-Prime RSA
Extended CRT-2 protocol 1. 2. 3.
for
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28. Proposed Multi-Prime RSA
Comparison of the operational speed
Division
Modular exponentiation
Extended CRT-2 protocol
The proposed multi-prime RSA
where ( , and )
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29. Proposed Multi-Prime RSA
Operational speed improvement has been verified
by one example of three-prime RSA
Similar factorization complexity
Still for obtaining any factor from the
proposed multi-prime RSA
Predicted to use fewer hardware resources
Will be verified by Implementation results later
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30. Outline Introduction Literature Review
Proposed Efficient Two-Prime RSA Cryptosystem
Proposed Efficient Multi-Prime RSA Cryptosystem
FPGA Implementations and Results
Conclusions
November-15-18

31. FPGA Implementations
Design flow
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32. FPGA Implementations Structure of modular exponentiation algorithm
(to calculate )
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33. FPGA Implementations Structure of Montgomery modular multiplication
algorithm (to calculate )
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34. FPGA Implementations
Hardware structure of Montgomery modular multiplication
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35. FPGA Implementations Structure of proposed two-prime RSA
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36. FPGA Implementations Structure of standard CRT-based two-prime RSA
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37. FPGA Implementations
Structure of CRT-2 protocol
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38. FPGA Implementations Implementa-tion results: CLB usage LUT Equivalent
gates
Standard CRT-based two-prime RSA
1,226
4,775
46,324
Proposed two-prime RSA
1,431
5,615
55,913
CRT-2 protocol
1,997
6,577
85,229
Standard three-prime RSA
1,759
6,939
68,144
Proposed three-prime RSA
2,130
8,252
82,233
Extended CRT-2 protocol
2,646
9,121
109,756
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39. FPGA Implementations Implementation results
Standard
2-prime
Proposed
(CRT-2)
Resources
Usage (%)
82.6
100
152
3-prime
Resources usage (%) 82
133
Conclusion: Not many more resources than the
standard CRT-based RSA and much fewer than the
systems based on CRT-2 protocol
November-15-18

40. Outline Introduction Literature Review
Proposed Efficient Two-Prime RSA Cryptosystem
Proposed Efficient Multi-Prime RSA Cryptosystem
FPGA Implementations and Results
Conclusions
November-15-18

41. Conclusions Conclusions
The immunity of the RSA cryptosystems against the hardware
fault attack is greatly increased
The proposed RSA cryptosystems provide more efficient
operations than previous work, and they bear similar
immunity against the hardware fault attack.
The proposed RSA cryptosystems use fewer resources than
previous work in hardware implementations
The standard CRT-based RSA cryptosystems with more
factors bears more difficult for the hardware fault attack
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42. Conclusions Future work
Speed up the basic block: modular exponentiation
computation
Implement the RSA cryptosystems with enhanced immunity
against other implementation attacks
Download the RSA cryptosystems implemented in Chapter
5 to the FPGA chip
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43. Thesis Examination
Thanks !
and
Questions ?
November-15-18