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The Comparative Performance Analysis of Lattice Based Ntru Cryptosystem with Other Asymmetrical Cryptosystems

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The Comparative Performance Analysis of Lattice Based Ntru Cryptosystem with Other Asymmetrical Cryptosystems View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by DSpace@IZTECH Institutional Repository THE COMPARATIVE PERFORMANCE ANALYSIS OF LATTICE BASED NTRU CRYPTOSYSTEM WITH OTHER ASYMMETRICAL CRYPTOSYSTEMS A Thesis Submitted to the Graduate School of Engineering and Science of İzmir Institute of Technology in Partial Fullfillment of the Requirements for the Degree of MASTER OF SCIENCE in Computer Software by Ali MERS İN September 2007 İZM İR We approve the thesis of Ali MERS İN Date of Signature ............................................................................ 10 September 2007 Assoc. Prof. Dr. Ahmet KOLTUKSUZ Supervisor Department of Computer Engineering İzmir Institute of Technology ............................................................................ 10 September 2007 Prof. Dr. Şaban EREN Department of Computer Engineering Maltepe University ............................................................................ 10 September 2007 Dr. Serap ATAY Department of Computer Engineering İzmir Institute of Technology ............................................................................ 10 September 2007 Prof. Dr. Sıtkı AYTAÇ Head of Department Department of Computer Engineering İzmir Institute of Technology .................................................... Prof. Dr. M. Barı ş ÖZERDEM Head of the Graduate School ACKNOWLEDGEMENT Foremost, I would like to thank my advisor, Assoc. Prof. Dr. Ahmet Koltuksuz, for his guidance, patience, and encouragement. Furthermore, I would like to thank Dr. Serap Atay for lending her source code to measure ECC timings; Hüseyin Hı şıl for our CRYMPIX education and his contributions for my cryptographic knowledge; my room mate Selma Tekir for her support and patience. I also would like to thank my room mate Mutlu Beyazıt for his huge support; if he had not helped, this thesis would not be completed. I would like to thank my parents for their support throughout my education as well as in my graduate study. Finally, I would like to thank my dear girl friend, Gönül Karahan, who always stands by me. She is the one who ensured that I completed my graduate study. ABSTRACT THE COMPARATIVE PERFORMANCE ANALYSIS OF LATTICE BASED NTRU CRYPTOSYSTEM WITH OTHER ASYMMETRICAL CRYPTOSYSTEMS Current popular asymmetrical cryptosystems are based on hardness of number theoretic problems. In the future, these problems may become practically solvable with the improvements of processing power, the development of quantum computation and distributed computation. So the need for new cryptosystems which are not based on these problems has risen. Researches on hardness of lattice problems have brought a new candidate for asymmetrical cryptography: Lattice Based Cryptography. NTRU is one of these cryptosystems. For practical use, the actual performance results of NTRU with respect to current asymmetrical cryptosystems should be known. This thesis is developed based on this purpose. iv ÖZET KAFES TABANLI NTRU KR İPTOS İSTEM İNİN D İĞ ER AS İMETR İK KR İPTOS İSTEMLERLE KAR ŞILA ŞTIRMALI PERFORMANS ANAL İZİ Günümüzde aktif olarak kullanılan asimetrik kriptosistemlerde güvenlik sayı teorisi problemlerinin zorluklarına dayanmaktadırlar. İş lemci gücünün gün geçtikçe artması, kuantum ve da ğıtık hesaplamanın geli şmesiyle, sayı teorisi problemlerinin zorlu ğunu temel alan kriptosistemlerin güvenilirlikleri gelecekte tehlike altına girebilecektir. Bu nedenle asimetrik kriptografide kullanılabilecek, ba şka matematiksel problemleri kullanan, yeni kriptosistemler üretilmeli ve bir alternatif olarak sunulmalıdır. Bu amaçla, kafes problemlerinin zorlukları üzerine yapılan çalı şmalar, “Kafes Tabanlı Kriptografi”yi ortaya çıkarmı ş ve kafes tabanlı NTRU kriptosistemi geli ştirilmi ştir. NTRU kriptosisteminin günümüz kriptosistemlerinin yerini alabilmesi için; NTRU kriptosisteminin bu kriptosistemlerle kar şıla ştırmalı performans analizi yapılmalıdır ve bu gereksinim tezimin hedefini olu şturur. v TABLE OF CONTENTS LIST OF FIGURES .......................................................................................................viii LIST OF TABLES...........................................................................................................ix CHAPTER 1 INTRODUCTION...................................................................................... 1 CHAPTER 2 MATHEMATICAL BACKGROUND....................................................... 3 2.1. Lattice..................................................................................................... 3 2.1.1. Lattice ............................................................................................... 3 2.1.2. Modular Lattice ................................................................................ 4 2.1.3. NTRU Lattice ................................................................................... 5 2.2. Quotient Polynomial Ring...................................................................... 5 2.2.1. Addition on Quotient Polynomial Ring............................................ 5 2.2.2. Multiplication on Quotient Polynomial Ring ................................... 6 2.2.3. Modular Lattices and Quotient Polynomial Ring............................. 6 2.3. Hard Lattice Problems............................................................................ 7 2.3.1. The Shortest Vector Problem ........................................................... 7 2.3.2. The Closest Vector Problem............................................................. 7 2.3.3. The Smallest Basis Problem............................................................. 8 2.3.4. Notes on Lattice Problems................................................................ 8 CHAPTER 3 ASYMMETRICAL CRYPTOGRAPHY AND THE HISTORY OF LATTICE BASED CRYPTOGRAPHY.................................................... 9 3.1. Asymmetrical Cryptosystems................................................................. 9 3.1.1. Introduction to Asymmetrical Cryptography ................................... 9 3.1.2. Some Asymmetrical Cryptosystems............................................... 10 3.2. A Historical Perspective....................................................................... 12 CHAPTER 4 NTRU CRYPTOSYSTEM....................................................................... 14 4.1. Domain Parameters and Some Definitions........................................... 14 4.2. Key Generation..................................................................................... 16 vi 4.3. Encryption ............................................................................................ 17 4.4. Decryption ............................................................................................ 17 4.5. Conditions............................................................................................. 18 4.6. Digital Envelope................................................................................... 19 CHAPTER 5 IMPLEMENTATION OF NTRU ............................................................ 21 5.1. Instantiation of Cryptosystem............................................................... 21 5.1.1. Choosing Parameters ...................................................................... 21 5.1.2. Choosing n...................................................................................... 22 5.1.3. Choosing f, g, r and m..................................................................... 22 5.1.4. Choosing p and q ............................................................................ 23 5.1.5. Alternatives..................................................................................... 23 5.1.6. NAEP Encryption Scheme ............................................................. 24 5.1.6.1. Encryption ................................................................................ 24 5.1.6.2. Decryption................................................................................ 25 5.2. Implementation Details ........................................................................ 25 5.2.1. Programming Language ................................................................. 26 5.2.2. Representation of Polynomials....................................................... 26 5.2.3. Memory Management..................................................................... 26 5.2.4. Details of Some Instructions........................................................... 27 CHAPTER 6 PERFORMANCE ANALYSIS OF NTRU.............................................. 29 6.1. Parameters of NTRU Used for Comparison......................................... 29 6.2. Performance Comparison of NTRU with ECC and RSA .................... 30 6.2.1. Comparison of Key Sizes ............................................................... 30 6.2.2. Comparison of Key Generation, Encryption and Decryption Performance ................................................................................... 31 CHAPTER 7 CONCLUSION ........................................................................................ 44 REFERENCES ............................................................................................................... 46 vii LIST OF FIGURES Figure Page Figure 5.1. Add Operation. ............................................................................................. 27 Figure 5.2. Convolution Product Operation.................................................................... 28 Figure 6.1. Key Size
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