1 Montgomery Modular Multiplication in Hard- ware

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FEI KEMT Figure 6 – 14 Comparison of probability histograms for the jitter measured by temperature 20 ◦ C in TRNG with configuration A and B. Data measured were around the rising edge of the sampled clock waveform. Figure 6 – 15 Difference in number of sampled ones for critical samples by boundary temperatures −40 ◦ C and +30 ◦ C in TRNG with configuration A and B around the rising edge of the sampled clock waveform. 123
FEI KEMT 7 Research Contribution With this thesis we contributed to the field of hardware implementation of public-key cryptographic system elements. We discussed the aspects of algorithm adaptations and system architectures for modular multiplier and cryptanalytic hardware. Ran- domness extraction method based on clock circuitry was evaluated and new findings were presented. The research contribution were achieved in the following topics: • Optimised Montgomery modular multiplier implementation in hardware. • The elliptic curve method implementation in hardware. • Evaluation of random number generator based on clock circuitry in FPGAs. Optimised Montgomery modular multiplier implementation in hardware Two most popular public-key cryptographic algorithms – the RSA and ECC use extensively modular operations with large numbers. The MM can be a very slow operation when performed on general-purpose computers, therefore can be accelerated by an effective hardware implementation. We analysed algorithms for Montgomery MM and architectures for their effective implementation suitable for reconfigurable hardware structures. Our attention was paid to keep the scalability and parametrisation of multiplier unit also in the other parts of the system and find an optimal model for division of computational load between the software and hardware part of the system. The results of area oc- cupation and timing analysis were presented after application of hardware-software co-design. The elliptic curve method implementation in hardware The security of the most applied public-key cryptographic algorithm – RSA depends on hardness of factoring large numbers. In the currently best known method for factoring large integers – the GNFS one important step is the factorisation of mid-sized integers for which an ECM is an efficient algorithm. The ECM algorithm is a classical example of algorithm that can be significantly accelerated thanks to special-purpose hardware. We provided a detailed description of efficient ECM architecture, especially suited for hardware implementation. The modular multiplier obtained as a result of our research described in the previous point presents a core element of the ECM unit and allows fast prototyping. For 124
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Technical University of Koˇsice Fa
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Metadata Sheet Author: Martin ˇ Si
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FEI KEMT ciel’ovej platformy a vl
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Acknowledgement There are several p
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Contents Introduction 1 1 Montgomer
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FEI KEMT 6.2.3 Analysis of TRNG in
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FEI KEMT 6 - 2 Block diagram of dig
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List of Algorithms 1 - 1 Montgomery
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FEI KEMT π(p) prime counting funct
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FEI KEMT MWR2MM Multiple Word Radix
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FEI KEMT and Elliptic Curve Cryptog
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FEI KEMT The hardware implementatio
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FEI KEMT data inputs clock Look-up
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FEI KEMT A (X) request for B’s pr
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FEI KEMT Algorithm 1 - 1 Montgomery
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FEI KEMT In the Algorithm 1 - 1 the
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FEI KEMT by the radix b changes to
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FEI KEMT the ECC instead of the RSA
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FEI KEMT Algorithm 1 - 5 Key genera
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FEI KEMT 2 Montgomery Modular Multi
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FEI KEMT not significant. On the ot
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FEI KEMT Algorithm 2 - 2 The multip
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FEI KEMT Beside the internal struct
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FEI KEMT q x i S (j) 2 w-1 S (j) 1
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FEI KEMT x i x i-1 xi-n+1 Y (j) M (
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FEI KEMT especially if the access t
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FEI KEMT 2.2.3 Interface to Control
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FEI KEMT Clock Signal Distribution
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FEI KEMT 2.3.2 Montgomery Multiplic
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FEI KEMT 1. Fully software solution
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FEI KEMT 2.3.4 Implementation Resul
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FEI KEMT 3 Elliptic Curve Method in
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FEI KEMT improving the area-time pr
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FEI KEMT 3.3.1 Pollard’s (p − 1
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FEI KEMT If the order of P ∈ E(Fq
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FEI KEMT handicap of the Montgomery
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FEI KEMT Two major improvements hav
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FEI KEMT and case study with GNFS b
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FEI KEMT Table 4 - 1 Computational
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FEI KEMT from or writing to a singl
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FEI KEMT Algorithm 4 - 1 Modified M
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FEI KEMT Algorithm 4 - 2 Modular ad
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FEI KEMT microprocessor, e.g. Alter
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FEI KEMT timings of ECM implementat
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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: FEI KEMT Figure 6 - 13 Amount of sa
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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