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Student Number 92522085 Author Lee-Chun Ko(¬_¤O¸s) Author's Email Address brentko@giga.net.tw Statistics This thesis had been viewed 2020 times. Download 607 times. Department Computer Science and Information Engineering Year 2004 Semester 2 Degree Master Type of Document Master's Thesis Language English Title Physical Cryptanalysis of RSA Implementations Date of Defense 2005-05-12 Page Count 79 Keyword physical attack smart card Abstract The recent communications are mostly through the electronic

channel due to the increasingly usage of the Internet. Some

applications such as micro-payment systems, on-line shopping, and

other transaction applications employ temper-proof devices such as

smart cards. These cards are embedded a cryptographic computation

function so as to providing highly security, and they usually

contain owner's identification and some secret information related

to the owner.

Since the introduction of the public-key cryptography, plenty of

digital signature schemes are then proposed. Among these schemes,

the RSA public-key cryptosystem is considered as the most popular

scheme due to its highly security and easily implementation.

Therefore, by deploying RSA or other signature schemes into smart

cards, these temper-proof devices can be used to providing

authentication and identification.

Since Kocher proposed the power analysis attacks against the

implementation of smart cards or other cryptographic hardware

devices, many of cryptosystem designers concern not only the

mathematic security of cryptography but also the implementation of

smart cards. Contrary to the previously active attack such as the

fault attack, power analysis attacks are passive attacks and more

easier to mount. Therefore, many researchers have focusing on

developing a secure and efficient countermeasure against power

analysis attacks and some other physical attacks.

In the related literatures, some of the countermeasures are still

controversial and insecure in advanced physical attacks. In this

thesis, we pointed out some of the existent countermeasures are

insecure by the proposed three new physical attacks. First of all,

by combining fault attack and simple power analysis, we proposed

an attack on Montgomery ladder which was originally proposed to

defeat simple power analysis and some fault-based attacks. Second,

we proposed a more powerful power analysis attack against a

countermeasure which was based on a randomized binary sign digit

representation to defeat differential power analysis. Third, we

extended the existent attack to develop a new type of attack

against Montgomery ladder. Three attacks are then confirmed either

by experimental result or by simulation result.Table of Content 1 Introduction 1

1.1 Motivation of the Research . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Overview of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Review of RSA Cryptosystem and Its Implementations 6

2.1 Brief Historical Review of Public-Key Cryptography . . . . . . . . . . 6

2.2 The RSA Cryptosystem . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3 Modular Exponentiation Algorithms . . . . . . . . . . . . . . . . . . 9

3 Review of Physical Cryptanalysis against RSA 12

3.1 Simple Power Analysis { SPA . . . . . . . . . . . . . . . . . . . . . . 12

3.1.1 Some countermeasures . . . . . . . . . . . . . . . . . . . . . . 13

3.2 Di®erential Power Analysis { DPA . . . . . . . . . . . . . . . . . . . 15

3.2.1 Some countermeasures . . . . . . . . . . . . . . . . . . . . . . 18

3.3 Fault Attack { FA . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.3.1 CRT-based fault attack . . . . . . . . . . . . . . . . . . . . . . 20

3.3.2 Computational safe-error attack . . . . . . . . . . . . . . . . . 22

3.3.3 Memory safe-error attack . . . . . . . . . . . . . . . . . . . . . 22

3.3.4 Some countermeasures . . . . . . . . . . . . . . . . . . . . . . 23

4 Side-Channel Security of Montgomery Ladder Revisited 27

4.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.2 Proposed Di®erential Simple Power Analysis { DSPA . . . . . . . . . 27

4.2.1 The ¡Ârst attack on Montgomery ladder . . . . . . . . . . . . . 28

4.2.2 The second attack on Montgomery ladder . . . . . . . . . . . 29

4.2.3 Extended attacks to other algorithms . . . . . . . . . . . . . . 30

4.3 Analysis of the DSPA . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.3.1 Feasibility of the proposed attack . . . . . . . . . . . . . . . . 31

4.3.2 Attack scenario analysis . . . . . . . . . . . . . . . . . . . . . 32

4.4 Experimental Result . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

5 A More Powerful Attack against Ha-Moon's Countermeasure Based

on Randomized BSD 38

5.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

5.2 Review of Ha-Moon's DPA Countermeasure . . . . . . . . . . . . . . 39

5.3 Proposed Attack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

5.3.1 Attack model and notations . . . . . . . . . . . . . . . . . . . 40

5.3.2 Main idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

5.3.3 Description of the attack . . . . . . . . . . . . . . . . . . . . . 42

5.3.4 Key ¡Ânding step . . . . . . . . . . . . . . . . . . . . . . . . . 43

5.3.5 Attacking algorithm . . . . . . . . . . . . . . . . . . . . . . . 44

5.4 Experimental Result and Example . . . . . . . . . . . . . . . . . . . . 44

5.4.1 Experimental result . . . . . . . . . . . . . . . . . . . . . . . . 45

5.4.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.5 Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

6 Di®erential Doubling Attack on Montgomery Ladder 51

6.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

6.2 Fouque-Valette's Doubling Attack on SMA Algorithm . . . . . . . . . 52

6.2.1 Attack model . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

6.2.2 Description of the doubling attack . . . . . . . . . . . . . . . . 52

6.3 Proposed Di®erential Doubling Attack { DDA . . . . . . . . . . . . . 54

6.3.1 Description of the di®erential doubling attack . . . . . . . . . 54

6.3.2 Attacking algorithm . . . . . . . . . . . . . . . . . . . . . . . 56

6.4 Experimental Result . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

7 Conclusions 61

7.1 Brief Review of Main Contributions . . . . . . . . . . . . . . . . . . . 61

7.2 Further Research Topics and Directions . . . . . . . . . . . . . . . . . 62Reference [1] C. AumÄuller, P. Bier, W. Fischer, P. Hofreiter, and J.-P. Seifert, Fault At-

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