1 Montgomery Modular Multiplication in Hard- ware

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FEI KEMT 4 Elliptic Curve Method in Hardware 55 4.1 Parameterisation of the ECM Algorithm . . . . . . . . . . . . . . . . 56 4.1.1 Phase 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 4.1.2 Phase 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.2 Design of the ECM Unit . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.2.1 Control Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.2.2 Memory Management . . . . . . . . . . . . . . . . . . . . . . . 59 4.2.3 Choice of the Arithmetic Algorithms . . . . . . . . . . . . . . 60 4.2.4 Parallelization of the Algorithm . . . . . . . . . . . . . . . . . 64 4.3 Implementation of the ECM Unit . . . . . . . . . . . . . . . . . . . . 65 4.3.1 Hardware Platform . . . . . . . . . . . . . . . . . . . . . . . . 65 4.3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 4.3.3 ECM-Based Acceleration of GNFS: a Case Study . . . . . . . 67 4.4 Conclusions and Future Steps . . . . . . . . . . . . . . . . . . . . . . 69 5 True Random Number Generator - preliminaries 71 5.1 Randomness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 5.1.1 Definitions of Randomness . . . . . . . . . . . . . . . . . . . . 72 5.1.2 Random Number Generator . . . . . . . . . . . . . . . . . . . 73 5.1.3 Applications of Random Numbers . . . . . . . . . . . . . . . . 75 5.2 TRNG Implementations in Digital Systems . . . . . . . . . . . . . . . 76 5.2.1 Sources of Randomness . . . . . . . . . . . . . . . . . . . . . . 77 5.2.2 Survey of Designs Based on Jitter . . . . . . . . . . . . . . . . 82 5.3 PLL-Based TRNG on FPGA . . . . . . . . . . . . . . . . . . . . . . 85 5.3.1 Randomness Extraction Method . . . . . . . . . . . . . . . . . 85 5.3.2 Coherent Sampling . . . . . . . . . . . . . . . . . . . . . . . . 88 5.4 Testing of TRNGs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 5.5 Attacks against TRNG . . . . . . . . . . . . . . . . . . . . . . . . . . 91 5.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 6 True Random Number Generator 94 6.1 Clock Synthesis in FPGAs . . . . . . . . . . . . . . . . . . . . . . . . 94 6.1.1 PLL as Source of Randomness . . . . . . . . . . . . . . . . . . 96 6.2 PLL-Based TRNG on FPGA . . . . . . . . . . . . . . . . . . . . . . 101 6.2.1 PLL Configurations . . . . . . . . . . . . . . . . . . . . . . . . 101 6.2.2 Analysis of TRNG in Altera Stratix FPGAs . . . . . . . . . . 103 x
FEI KEMT 6.2.3 Analysis of TRNG in Actel FPGAs . . . . . . . . . . . . . . . 105 6.2.4 Stochastic Model of PLL-TRNG . . . . . . . . . . . . . . . . . 109 6.3 Active Non-Invasive Attack on TRNG . . . . . . . . . . . . . . . . . 114 6.3.1 Attack description . . . . . . . . . . . . . . . . . . . . . . . . 114 6.3.2 Measurements results . . . . . . . . . . . . . . . . . . . . . . . 115 6.4 Conclusions and Further Research . . . . . . . . . . . . . . . . . . . . 120 7 Research Contribution 124 Bibliography 127 xi
Page 1 and 2: Technical University of Koˇsice Fa
Page 3 and 4: Metadata Sheet Author: Martin ˇ Si
Page 5 and 6: FEI KEMT ciel’ovej platformy a vl
Page 7 and 8: Acknowledgement There are several p
Page 9: Contents Introduction 1 1 Montgomer
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
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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
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
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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
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FEI KEMT [68] what can be expressed
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FEI KEMT and a harvesting mechanism
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FEI KEMT control also all the syste
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FEI KEMT we can mention a generator
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FEI KEMT noise to binary signal a c
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FEI KEMT over time. For this reason
Page 103 and 104:
FEI KEMT The Bucci and Luzzi Testab
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FEI KEMT CLI PLL PLL 1 2 CLJ CLK D
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FEI KEMT For R it holds that the in
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FEI KEMT extraction. In other words
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FEI KEMT and effort needed for repr
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FEI KEMT 6 True Random Number Gener
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FEI KEMT clock input F IN F FB Phas
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FEI KEMT Table 6 - 2 Parameters of
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FEI KEMT Since the size of the jitt
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FEI KEMT Table 6 - 3 Parameters set
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FEI KEMT where N1 is the number of
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FEI KEMT oscillator with frequency
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FEI KEMT Table 6 - 7 Area occupatio
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FEI KEMT 0,75 0,5 0,25 1 0 1 30 59
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FEI KEMT Stochastic Model The clock
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FEI KEMT Table 6 - 8 Mean values me
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FEI KEMT Figure 6 - 8 Sampled wavef
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FEI KEMT Table 6 - 9 Results of sta
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FEI KEMT Figure 6 - 11 Amount of sa
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FEI KEMT Figure 6 - 13 Amount of sa
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FEI KEMT 7 Research Contribution Wi
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FEI KEMT Curriculum vitae Professio
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FEI KEMT [16] Altera Corporation. S
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FEI KEMT [37] Bundesamt für Sicher
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FEI KEMT [54] Federal Information P
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FEI KEMT [71] Gura, N., Chang, S.,
Page 155 and 156:
FEI KEMT of Commerce, month = aug,
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FEI KEMT and C. Paar, Eds., no. 216
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FEI KEMT [128] Zimmermann, P. ECMNE
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