quadratic residues and applications in cryptographynica.anca/teza doctorat... · 2020. 2. 4. ·...

Alexandru Ioan Cuza University of Iaşi, RomâniaDepartment of Computer Science

Quadratic Residues and Applications inCryptography

by

Anca-Maria Nica

supervisor

Prof. Dr. Cătălin Dima

2020

.

Doctoral committee:

Conf.Dr. Adrian Iftene - committee chairmanAlexandru Ioan Cuza University of IaşiProf.Dr. Cătălin Dima - doctoral supervisorAlexandru Ioan Cuza University of Iaşi /“Paris Est Creteil - Val de Marne”Prof.Dr. Constantin Popescu - reviewerUniversity of OradeaProf.Dr. Ferucio Laurenţiu Ţiplea - reviewerAlexandru Ioan Cuza University of IaşiConf.Dr. Octavian Catrina - reviewerUniversity Politehnica of BucharestConf.Dr. Mihai Dumitru Prunescu - reviewerUniversity of Bucharest

Acknowledgements

I became more and more concerned about the meaning of life, whose essence can

be summarized in one word: giving. But you cannot give what you do not have, so

growing is another leading word in my life. I would like to have a positive impact on

others’ lives, and I am doing this profoundly inspired by the influence I got, in turn,

from the most important people in my life.

I’m looking around me and I can not feel anything else than gratefulness. I am

grateful for the models I have, because life teaches me a lot by their examples. I

am surrounded by special people, beginning with my mentor, Fr. Teodosie, who is a

true father to me. He is sustaining me in all situations, he is a live model of being

a Christian for me, an example of empathizing and communicating with people. I

learned from him that you have to be very patient with people, as he is with me all

the time. He taught me, by his life, that the strongest way of teaching others is by

your own example. He taught me that before night you are the leader who establishes

the timetable for the next day. Then, in the morning, you have to be a committed

employee and not to negotiate the things you have already planned to do. He also

showed me how one can make a masterpiece from each day and praise God for all.

I would like to thank my supervisors Prof. Dr. Ferucio Laurenţiu Ţiplea and Prof.

Dr. Cătălin Dima for all their help and support.

Professor Ţiplea taught me that you can always be kind with others, no matter

how they act or speak to you. I realized through his example that you always have

to see value in people, you have to focus on their strengths, you have to appreciate,

respect, and believe in them and also that you have to add value to people all the

time - as John Maxwell said - these are the seeds for success. He gently guided me

all these six years, and still does in a very efficient and thoughtful way.

From Lect. Dr. Sorin Iftene I have learned that whenever you have the opportunity

to encourage people, it is a great idea to do so. He also taught me by his example

how to always be thoughtful and attentive to others’ needs.

From FCS I have learned how to act with yourself and the fact that you can be

as strict as you wish with yourself but very lenient with others. I am also grateful to

FCS for its constant support and mentoring and for offering the perfect environment

for writing this thesis.

iii

They are like a lighthouse showing the direction. I look forward to giveback, to

reward the trust that they invested in me and without which I would not have gotten

here.

Even if words are too poor, I would like to thank them all, along with other great

people that surrounded me throughout the process, for their contribution.

I express here my profound gratitude to God, to the Holy Theotokos, to all Saints

and to my guardian angel who took care of me all the time.

This thesis does not represent the ending but rather the beginning of a new period

of research in this area. In the last five years of study I had the chance to attend

many (inter)national conferences and winter/summer schools on related topics that

opened up new horizons in my research and also spurred me to improve my English

enough to be able to teach in English. I am ever so grateful to our faculty and to

all those who have facilitated such opportunities. One of them is Lect. Dr. Emanuel

Onica who helped me to attend a lot of interesting and useful scientific events by his

projects.

Last but not least, I want to thank my parents and my friends who understood

me patiently and sustained me along the way. Words are never enough to express my

gratitude. Thank you! God bless you all!

iv

.

To Fr. Teodosie,

v

Contents

Preface 5

Thesis overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Thesis contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

List of publications 11

1 Introduction to cryptography and quadratic residues 13

1.1 Some history . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.2 Principles, goals and security in modern cryptography . . . . . . . . . 16

1.3 Quadratic residues in mathematics . . . . . . . . . . . . . . . . . . . 22

1.4 Quadratic residues in cryptology . . . . . . . . . . . . . . . . . . . . . 25

1.5 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2 Prerequisites 31

2.1 Congruence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.2 Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.3 Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.4 Quadratic residues . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.4.1 Legendre and Jacobi symbols . . . . . . . . . . . . . . . . . . 37

2.4.2 Computing square roots . . . . . . . . . . . . . . . . . . . . . 40

3 On the distribution of quadratic residues 45

3.1 Counting quadratic residues and non-residues in the set a+X . . . . 49

3.1.1 The case of prime moduli . . . . . . . . . . . . . . . . . . . . 50

3.1.2 The case of RSA moduli . . . . . . . . . . . . . . . . . . . . . 56

3.2 Computing probabilities on sets Y(a+X) . . . . . . . . . . . . . . . 68

vii

3.3 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4 Applications of QR to IBE 71

4.1 Cocks’ IBE scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.1.1 Cocks’ IBE ciphertexts . . . . . . . . . . . . . . . . . . . . . . 73

4.1.2 Galbraith’s test . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.1.3 Anonymous Cocks’ schemes . . . . . . . . . . . . . . . . . . . 81

4.1.4 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . 87

4.2 Boneh-Gentry-Hamburg’s IBE scheme . . . . . . . . . . . . . . . . . 87

4.2.1 Associated polynomials . . . . . . . . . . . . . . . . . . . . . . 89

4.2.2 The BGH scheme and its security . . . . . . . . . . . . . . . . 89

4.2.3 A new security analysis for BasicIBE scheme . . . . . . . . . 95


4.3 QR-based IBE schemes that fail security . . . . . . . . . . . . . . . . 98

4.3.1 Jhanwar-Barua scheme . . . . . . . . . . . . . . . . . . . . . . 98

4.3.2 Other insecure IBE schemes based on QR . . . . . . . . . . . 103


4.4 Continuous mutual authentication . . . . . . . . . . . . . . . . . . . . 105

4.4.1 Real privacy management . . . . . . . . . . . . . . . . . . . . 106

4.4.2 RPM description . . . . . . . . . . . . . . . . . . . . . . . . . 111

4.4.3 Continuous mutual authentication and data security . . . . . . 116


4.5 Pseudo-random generators . . . . . . . . . . . . . . . . . . . . . . . . 119

4.5.1 Pseudo-randomness from QR . . . . . . . . . . . . . . . . . . 120


5 From identity-based to attribute-based encryption 123

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

5.2 ABE and the backtracking attack . . . . . . . . . . . . . . . . . . . . 126

5.3 KP-ABE for Boolean circuits using secret sharing and bilinear maps . 131

5.3.1 The secure KP-ABE Scheme 1 . . . . . . . . . . . . . . . . . . 131


5.4 KP-ABE for Boolean circuits using secret sharing and multilinear maps 143

viii

5.4.1 The secure KP-ABE Scheme 2 . . . . . . . . . . . . . . . . . . 144


6 Conclusion and open problems 157

Bibliography 163

ix

Preface

About this thesis

The most inspiring aspect in doing this thesis was “the improvement”, not only re-

garding some schemes and boundaries in security proofs, but reaching the next level in

the research process, growth and comprehending. This is what guarantees the future

results and gives beauty to the process.

We started five years ago from some problems which are of great interest in cryp-

tography. Searching an efficient variant of Cocks’ IBE scheme was one of them. Then,

starting from it, we investigated the set of integers which are obtained by adding a

quadratic residue to an integer in Z∗n, i.e. the set a+QRn, as we will deeply discuss

in Chapter 3 of this thesis. Another starting point in this research was the proof of

Galbraith’s test, addressed in detail in Section 4.1.2, the anonymization and the se-

curity of Cocks’ IBE scheme, together with applications of this scheme and attribute

based encryption, which is considerable useful in cloud computing, access control in

cloud and other fields. These are the main subjects which we describe in this thesis.

Thesis overview

Chapter 1: Introduction to cryptography and quadratic residues

In the first chapter, after a short review of the thesis, we present some phases in the

history of cryptology - one of the areas regarding information hiding (see Figure 1.1 on

page 16). We emphasize the niche of Public Key Cryptography (PKE) and specially

Identity-based Encryption (IBE), until we get to IBE based on quadratic residues

(QR). This is one of the areas where we applied some of our mathematical results in

5

6 Preface

Chapter 3.

In Section 1.2 we state two main principles of cryptology followed immediately

by the objectives of cryptography together with the security goals that have to be

satisfied according to the security model which a cryptosystem reaches. In Figure 1.2

on page 21 we can see the relation between these security models. The security level

of a cryptographic scheme is usually proved using security games, as the one presented

in the end of Section 1.2.

In the following two sections we point out some of the areas where quadratic

residues are of great interest, focusing mainly on mathematical aspects, Section 1.3,

and cryptographic aspects, Section 1.4. In the last section of Chapter 1 we shortly

present the literature review regarding mainly four key aspects around which our

study shall be structured. The first one is related to the mathematical results in

Chapter 3, i.e. the distribution of QR and the Jacobi patterns, aiming to get the

exact cardinality of sets like QRn(a+QRn) - the set of QR in the set a+QRn. These

results bring the second key point which consists of QR-based IBE schemes, including

the anonymous variants. This subject is addressed in Chapter 4. The third aspect of

our study is the application of such schemes in Real Privacy Management (RPM) in

order to provide Continuous Mutual Authentication (CMA). In the end of the section,

the state of the art regarding ABE is presented, focusing on KP-ABE.

Chapter 2: Prerequisites

This chapter introduces some notations, definitions, and basic results from number

theory, probabilities, and complexity which we are going to use along the thesis. A

special place here is taken by quadratic residues, the Legendre and Jacobi symbols

together with some square root extraction algorithms.

Chapter 3: On the distribution of quadratic residues

This chapter begins with the motivation of the study we did regarding the distribution

of quadratic residues1. Unfortunately, the Cocks’ IBE scheme was proved not to be

anonymous by Galbraith’s test. This test was briefly presented in two papers [41, 14]

1These results are attained in a joint work with F.L. Ţiplea, S. Iftene, and G. Teşeleanu and werepublished in [280, 78]

Preface 7

but we felt that a more rigorous proof of this test and explicit computations would’ve

been useful. Our research has lead to important results with exact formulas for the

cardinality of a multitude of sets with different Jacobi patterns. In Section 3.2 few

examples of calculating probabilities using these distributions were presented. These

probabilities are of great interest not only for encryption schemes, but also in diverse

issues like security of cryptosystems or pseudo-random generators.

Chapter 4: Applications of quadratic residues to identity-

based encryption

This chapter presents some applications of our results from Chapter 3. First we briefly

recall Cocks’ scheme then we deeply analyze its cryptotexts in order to prepare the

foundation for the proof of Galbraith test. Then we present an anonymous variant of

this scheme, proposed by G.A. Schipor in [251], followed by a much simple description

of Joye’s anonymous variant of Cocks’ IBE scheme, which was detailed in [215]. In

Section 4.2 some QR-based IBE schemes are described starting with BGH [44], which

is an IND-ID-CPA secure scheme, with improved ciphertext expansion, compared to

Cocks’, but less time efficient. We obtained in [253] a better upper bound for the

BGH scheme which is described in Section 4.2.3. In Section 4.3 we will see other

attempts of improving time efficiency of BGH, but, unfortunately, they are insecure,

as Schipor proved in [250]. These results were clearly presented in [279].

Section 4.4 describes a technique for continuous mutual authentication, namely

RPM, with its four configurations, while in Section 4.4.3 we showed how, using Cocks’

scheme in one of the configurations, results an improved variant of RPM.

Chapter 5: From identity-based to attribute-based encryption

Chapter 5 is very important by the fact that it presents a generalization of IBE with

applications in a huge variety of niches as cloud computing and IoT. It begins with

a brief introduction on ABE, utility, types of ABE and the state of the art, followed,

in Section 5.2, by some definitions and notations regarding ABE, together with the

general structure and correctness of an ABE scheme, the backtracking attack and some

deeper details on KP-ABE schemes - the core topic of the chapter. In Sections 5.3

8 Preface

and 5.4 two efficient KP-ABE schemes are presented, accompanied by their security

proofs, implementation issues, applications, complexity and comparisons.

In Chapters 3 to 5, there are some sections called Concluding remarks. They sum

up the key-ideas and results discussed in the sections above them and emphasize the

contribution on those areas.

Chapter 6: Conclusion and open problems

In this last chapter we draw conclusions and present some open problems regarding

the results obtained in the thesis and further work.

Thesis contributions

After the introduction and preliminaries in Chapters 1 and 2 the next chapters expose

our work as follows. Chapter 3 presents some results we developed regarding sets such

as QNRm(a + QRm), the set of integers of the form a + QRm which are quadratic

non-residues modulo m. These sets are very useful for cryptography due to the fact

that cryptographic schemes can be created using them [71, 44, 123].

In order to develop new results we analyzed the state of the research. Thus, a

useful timeline expressing the state of the art regarding the distribution of residues is

presented in Figures 3.1 and 3.2 on pages 47, 48. So, the reader can create his own

view about the importance and the great interest on this topic.

Perron’s work on the distribution of quadratic residues and non-residues in sets

like a + QRm focuses on prime moduli [223]. We extended these results to the case

where the modulus is an RSA integer. We also generalized the case a + QRm and

studied sets of the form a+X, where X can be one of the sets Zm, Z∗m, QRm, QNRm,

and the modulus can be either a prime or an RSA integer. In the last case, when m

is of the form p · q, for some distinct primes p and q, X may also be one of the sets J±mand J∓m. For all these sets a+X we presented not only their cardinals, but we counted

the number of elements for all Jacobi patterns on these sets. Section 3.2 shows how

to compute probabilities on these sets, for example, the probability that x is in J−n

when it is extracted uniformly at random from the set a+ Z∗n, see Corollary 3.2.1.

In Chapter 4 some applications of the results in Chapter 3 were detailed, together

Preface 9

with an interesting combination between a continuous mutual authentication protocol

and Cocks’ IBE scheme.

In Section 4.1 we deeply analyzed Cock’s IBE scheme and its cryptotexts structure

in order to be able to compute the exact probability that a given cryptotext was

encrypted for a given identity, see Section 4.1.2. Thus, in Section 4.1.1, first we studied

the way that the messages are encrypted, and how the sets of cryptotexts outputted

by this scheme look like. Thus, the computations in Section 4.1.2 were done using

the results achieved in Chapter 3 and the cardinalities in Section 4.1.1. Then we have

shown in section 4.1.3 how efficient anonymized Cocks’ cryptotexts can be obtained

from non-anonymous ones as an independent process. One such secure anonymous

scheme is due to G.A. Schipor [251]. Right after this scheme, in Section 4.1.3, we

showed how easily the anonymization variant of Cocks’ IBE scheme due to Joye [158]

can be described, without using cyclotomic polynomials and algebraic toruses, as it

was presented in [215].

Cocks’ IBE scheme, notwithstanding its simplicity and elegance, outputs quite

large cryptotexts, 2logn bits per bit of plaintext. Section 4.2 describes a solution

proposed in 2007 by Boneh et al., the BasicIBE (shortened here into BGH) which

improves the length of the cryptotexts at the cost of increasing the time complexity

to quartic in the security parameter. This scheme is proven to be IND-ID-CPA secure

under the QR assumption for the RSA generator in the random oracle model (ROM),

as we can see in Section 4.2.2. A better upper bound for BGH scheme has been

obtained in [253] and it is detailed in Section 4.2.3.

Starting from [44] Jhanwar and Barua tried to make the encryption/decryption

processes faster, as it is presented in Section 4.3.1 (their scheme will be called here JB

for short). The bottleneck of the scheme proposed by Boneh et al. was the algorithm

for solving Equation (4.2).

In [156], the same two researchers, Jhanwar and Barua, found a very useful prob-

abilistic algorithm for finding solutions to Equation (4.2) on page 88 instead of the

deterministic one of Boneh et al. Unfortunately, the scheme proposed by them is no

longer a secure variant of Cocks’ scheme due to the method of combining the solutions

of two congruential equations in order to get a third solution to another equation. As

A. Schipor showed, the variants of the schemes presented by Elashry, Mu, and Susilo

10 Preface

in [105] and [103] suffer from the same security weakness. Thus, for the moment,

the QR-based IBE schemes which remain secure are Cocks’ scheme, BGH and their

anonymous variants, as it is detailed in [279]. For a comparison between Cocks’ and

BGH cryptosystems see Table 4.1 on page 105.

An important contribution of the thesis relies to continuous mutual authentication.

When two parts wish to communicate securely they (both) will want to be sure, at

each moment during the process, that on the other end of the “line” is the person

that they aspect to be and not a third party, not an eavesdropper. In order to

achieve this, continuous (mutual) authentication is needed. But what if, at a certain

point, an intruder will decode their communication? Is there any possibility that the

communication become secure again during the same process, without interrupting

it and start it over? This property was first defined by Elashry et al. in [104], who

called it resiliency. We found a way to achieve this property using Cocks’ IBE scheme,

which perfectly fits to RPM configurations, see Section 4.4.

In the end of Chapter 4 we will see how pseudorandom generators can be created

using quadratic residues, which is another important application of QR in cryptogra-

phy.

Thus, in Chapter 5 we outlined the latest ideas developed in the area of KP-

ABE schemes based on bilinear maps and secret sharing. We conclude that, for

safety, leveled multi-linear maps should be avoided. However, the current solutions

for Boolean circuits in general which use bilinear maps are not efficient. So, finding a

balanced variant for this kind of circuits remains an open problem.

List of publications

1. F. L. Ţiplea, S. Iftene, G. Teşeleanu, and A.-M. Nica. On the distribution of

quadratic residues and non-residues modulo composite integers and applications

to cryptography. Applied Mathematics and Computation, vol. 372, May

2020 (Journal impact factor: 3.092), available on-line,

doi.org/10.1016/j.amc.2019.124993.

2. A.-M. Nica, Continuous mutual authentication and data security. Interna-

tional Journal of Computer Science and Information Security (IJCSIS),

vol. 17, February 2019 (Journal impact factor: 0.702).

3. A.-M. Nica and F. L. Ţiplea. On anonymization of Cocks identity-based en-

cryption scheme (extended version of the conference paper). In Computer

Science Journal of Moldova, vol.27, no.3(81), pp.283-298, 2019 http:

//www.math.md/publications/csjm/issues/v27-n3/13001/

(Journal indexed in Web of Science).

4. A.-M. Nica and F. L. Ţiplea. On anonymization of Cocks identity-based en-

cryption scheme. In Proceedings of the 5th Conference on Mathematical

Foundations of Informatics, MFOI 2019, Iasi, Romania, July 3-6, 2019, Ed-

itura Universităţii “Alexandru Ioan Cuza”, Iasi, pages 75-85, 2019.

5. G. Teşeleanu, F. L. Ţiplea, S. Iftene, and A.-M. Nica. Boneh-Gentry-Hamburg’s

identity-based encryption schemes revisited. In Proceedings of the 5th Con-

ference on Mathematical Foundations of Informatics, MFOI2019, July

3-6, 2019, Iasi, Romania, pages 45 – 58, 2019.

6. F. L. Ţiplea, C. C. Drăgan, and A.-M. Nica, Key-policy attribute-based en-

cryption from bilinear maps, in Innovative Security Solutions for Information

11

http://www.math.md/publications/csjm/issues/v27-n3/13001/http://www.math.md/publications/csjm/issues/v27-n3/13001/

Technology and Communications - 10th International Conference, SecITC 2017,

Bucharest, Romania, June 8-9, 2017, Revised Selected Papers, Lecture Notes

in Computer Science 10543, pp. 28–42, 2017.

7. F. L. Ţiplea, S. Iftene, G. Teşeleanu, and A.-M. Nica, Security of identity-based

encryption schemes from quadratic residues, in Innovative Security Solutions for

Information Technology and Communications - 9th International Conference,

SecITC 2016, Bucharest, Romania, June 9-10, 2016, Revised Selected Papers,

Lecture Notes in Computer Science 10006, pp. 63–77, 2016.

8. G. Teşeleanu, F. L. Ţiplea, S. Iftene, and A.-M. Nica. Boneh-Gentry-Hamburg’s

identity-based encryption schemes revisited, IET Information Security (un-

der review)

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