1 Montgomery Modular Multiplication in Hard- ware

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FEI KEMT do not assure the random base of generated sequence. In addition the working environment may also have a significant impact on the parameters of output bits. Requirements on RNGs applied in cryptography cover security parameters of the design, unpredictability of the generated sequence and specific statistical properties of the output sequence. The generator chosen for our research - the PLL-TRNG proposed in [60] will be further tested and analysed in order to provide better tools for choosing its parameters and understand its behaviour in changing environment. Described theoretical background on testing and attacks of RNGs has been applied and the results are given in the following chapter. 93
FEI KEMT 6 True Random Number Generator The chapter is dedicated to analysis of jitter-based random generator under various aspects. Our work is based on the TRNG design proposed by Viktor Fischer and Miloˇs Drutarovsk´y published in 2002 [60]. We enhance the already published results summarised in the previous chapter. Our focus is put on analysis of the generator in changing working conditions and configurations settings. Results of the research were published in the following list of papers [47, 48, 61, 62,114–116,119]. The main achievements of our research were done in the following areas: • Analysis of PLL circuitry as a source of randomness – implementation issues, possible PLL configurations, verification of vendor parameters, • Analysis of TRNG implementation in different FPGAs – achievability study, design consideration, practical results, • Stochastic model of PLL-TRNG – proposal and practical verification, • Temperature influence on PLL-TRNG – practical attack on TRNG with results and suggestions for design. The chapter is structured as follows. In the Section 6.1 we describe two ways of clock synthesis in modern FPGAs and summarise the parameters of the clock circuitry verified by practical measurements of the PLL parameters. The Section 6.2 provides an analysis of PLL configurations, practical results from Altera and Actel FPGA implementations of the generator and a stochastic model of the generator. In Section 6.3 we describe a non-invasive attack on the generator together with practical outcomes. In the last part (Section 6.4) we discuss the obtained results and provide ideas on the further research. 6.1 Clock Synthesis in FPGAs In present-day integrated digital systems, there is a need for numerous clock signals with various frequencies. The synthesis of the clocks in separated circuits is not effective and the frequencies are too high to be generated by an external crys- tal. FPGA vendors offer for this purpose a clock circuitry embedded on the FPGA chip. Beside synthesis of clock signals with required frequencies it provides addi- tional functions making possible a processing of signals with very high frequencies. 94
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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
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: FEI KEMT and effort needed for repr
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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