1 Montgomery Modular Multiplication in Hard- ware

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Technical University of Koˇsice Faculty of Electrical Engineering and Informatics Department of Electronics and Multimedia Communications Analysis and Implementation of Selected Blocks for Public-Key Cryptosystems in FPGAs Montgomery Modular Multiplier and True Random Number Generator Doctoral Thesis Discipline: 26-13-9 Electronics Department: Department of Electronics and Multime- dia Communications (FEI) Supervisor: doc. Ing. Miloˇs Drutarovský, PhD. Consultant: prof. Ing. Viktor Fischer, PhD. Koˇsice 2010 Martin ˇSimka
Metadata Sheet Author: Martin ˇ Simka Thesis title: Analysis and Implementation of Selected Blocks for Public- Key Cryptosystems in FPGAs Subtitle: Montgomery Modular Multiplier and True Random Num- ber Generator Language: English Type of Thesis: Doctoral Thesis Number of Pages: 126 Degree: PhD. University: Technical University of Koˇsice Faculty: Faculty of Electrical Engineering and Informatics (FEI) Department: Department of Electronics and Multimedia Communica- tions (KEMT) Discipline: 26-13-9 Electronics Town: Koˇsice, Slovakia Supervisor: doc. Ing. Miloˇs Drutarovský, PhD. Consultant(s) : prof. Ing. Viktor Fischer, PhD. Date of Submission: 2. 8. 2010 Date of Defence: 9. 2010 Keywords: modular multiplication, elliptic curve method, factorisation, random number generator Category Conspectus: Technika, technológia, inˇzinierstvo; Elektronika Thesis Citation: Martin ˇ Simka: Analysis and Implementation of Selected Blocks for Public-Key Cryptosystems in FPGAs. Koˇsice: Technical University of Koˇsice, Faculty of Electrical Engi- neering and Informatics. 2010. 126 pages Title SK: Analýza a implementácia vybraných blokov pre kryp- tografické systémy s verejným kl’účom Subtitle SK: Montgomeryho modulárna násobička a generátor skutočne náhodných čísel Keywords SK: modulárne násobenie, metóda eliptických kriviek, fak- torizácia, generátor náhodných čísel
Page 1: Technical University of Koˇsice Fa
Page 5 and 6: FEI KEMT ciel’ovej platformy a vl
Page 7 and 8: Acknowledgement There are several p
Page 9 and 10: Contents Introduction 1 1 Montgomer
Page 11 and 12: FEI KEMT 6.2.3 Analysis of TRNG in
Page 13 and 14: FEI KEMT 6 - 2 Block diagram of dig
Page 15 and 16: List of Algorithms 1 - 1 Montgomery
Page 17 and 18: FEI KEMT π(p) prime counting funct
Page 19 and 20: FEI KEMT MWR2MM Multiple Word Radix
Page 21 and 22: FEI KEMT and Elliptic Curve Cryptog
Page 23 and 24: FEI KEMT The hardware implementatio
Page 25 and 26: FEI KEMT data inputs clock Look-up
Page 27 and 28: FEI KEMT A (X) request for B’s pr
Page 29 and 30: FEI KEMT Algorithm 1 - 1 Montgomery
Page 31 and 32: FEI KEMT In the Algorithm 1 - 1 the
Page 33 and 34: FEI KEMT by the radix b changes to
Page 35 and 36: FEI KEMT the ECC instead of the RSA
Page 37 and 38: FEI KEMT Algorithm 1 - 5 Key genera
Page 39 and 40: FEI KEMT 2 Montgomery Modular Multi
Page 41 and 42: FEI KEMT not significant. On the ot
Page 43 and 44: FEI KEMT Algorithm 2 - 2 The multip
Page 45 and 46: FEI KEMT Beside the internal struct
Page 47 and 48: FEI KEMT q x i S (j) 2 w-1 S (j) 1
Page 49 and 50: FEI KEMT x i x i-1 xi-n+1 Y (j) M (
Page 51 and 52: FEI KEMT especially if the access t
Page 53 and 54:
FEI KEMT 2.2.3 Interface to Control
Page 55 and 56:
FEI KEMT Clock Signal Distribution
Page 57 and 58:
FEI KEMT 2.3.2 Montgomery Multiplic
Page 59 and 60:
FEI KEMT 1. Fully software solution
Page 61 and 62:
FEI KEMT 2.3.4 Implementation Resul
Page 63 and 64:
FEI KEMT 3 Elliptic Curve Method in
Page 65 and 66:
FEI KEMT improving the area-time pr
Page 67 and 68:
FEI KEMT 3.3.1 Pollard’s (p − 1
Page 69 and 70:
FEI KEMT If the order of P ∈ E(Fq
Page 71 and 72:
FEI KEMT handicap of the Montgomery
Page 73 and 74:
FEI KEMT Two major improvements hav
Page 75 and 76:
FEI KEMT and case study with GNFS b
Page 77 and 78:
FEI KEMT Table 4 - 1 Computational
Page 79 and 80:
FEI KEMT from or writing to a singl
Page 81 and 82:
FEI KEMT Algorithm 4 - 1 Modified M
Page 83 and 84:
FEI KEMT Algorithm 4 - 2 Modular ad
Page 85 and 86:
FEI KEMT microprocessor, e.g. Alter
Page 87 and 88:
FEI KEMT timings of ECM implementat
Page 89 and 90:
FEI KEMT hardware was implemented o
Page 91 and 92:
FEI KEMT [68] what can be expressed
Page 93 and 94:
FEI KEMT and a harvesting mechanism
Page 95 and 96:
FEI KEMT control also all the syste
Page 97 and 98:
FEI KEMT we can mention a generator
Page 99 and 100:
FEI KEMT noise to binary signal a c
Page 101 and 102:
FEI KEMT over time. For this reason
Page 103 and 104:
FEI KEMT The Bucci and Luzzi Testab
Page 105 and 106:
FEI KEMT CLI PLL PLL 1 2 CLJ CLK D
Page 107 and 108:
FEI KEMT For R it holds that the in
Page 109 and 110:
FEI KEMT extraction. In other words
Page 111 and 112:
FEI KEMT and effort needed for repr
Page 113 and 114:
FEI KEMT 6 True Random Number Gener
Page 115 and 116:
FEI KEMT clock input F IN F FB Phas
Page 117 and 118:
FEI KEMT Table 6 - 2 Parameters of
Page 119 and 120:
FEI KEMT Since the size of the jitt
Page 121 and 122:
FEI KEMT Table 6 - 3 Parameters set
Page 123 and 124:
FEI KEMT where N1 is the number of
Page 125 and 126:
FEI KEMT oscillator with frequency
Page 127 and 128:
FEI KEMT Table 6 - 7 Area occupatio
Page 129 and 130:
FEI KEMT 0,75 0,5 0,25 1 0 1 30 59
Page 131 and 132:
FEI KEMT Stochastic Model The clock
Page 133 and 134:
FEI KEMT Table 6 - 8 Mean values me
Page 135 and 136:
FEI KEMT Figure 6 - 8 Sampled wavef
Page 137 and 138:
FEI KEMT Table 6 - 9 Results of sta
Page 139 and 140:
FEI KEMT Figure 6 - 11 Amount of sa
Page 141 and 142:
FEI KEMT Figure 6 - 13 Amount of sa
Page 143 and 144:
FEI KEMT 7 Research Contribution Wi
Page 145 and 146:
FEI KEMT Curriculum vitae Professio
Page 147 and 148:
FEI KEMT [16] Altera Corporation. S
Page 149 and 150:
FEI KEMT [37] Bundesamt für Sicher
Page 151 and 152:
FEI KEMT [54] Federal Information P
Page 153 and 154:
FEI KEMT [71] Gura, N., Chang, S.,
Page 155 and 156:
FEI KEMT of Commerce, month = aug,
Page 157 and 158:
FEI KEMT and C. Paar, Eds., no. 216
Page 159:
FEI KEMT [128] Zimmermann, P. ECMNE
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