1 Montgomery Modular Multiplication in Hard- ware

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FEI KEMT 5.4 Testing of TRNGs Randomness of the generated numbers cannot be proven only by passing generally used statistical tests. Instead of that each RNG implementation has to be evaluated individually as an unique system. However, if the prototype in the lab generates acceptable random numbers this may not be true for each piece of TRNG of the same type during the whole operation time and therefore a continual testing of the generated output is required. It is well-known that most of the attacks are directed towards the implementa- tions of the cryptographic algorithms and not to the algorithms themselves. This means that special attention should be paid to avoid all weaknesses helping an at- tacker in breaking of a system. The topic of tests is highly accurate in case of attacks. The generators as sources of secrets, on which the security of the whole cryptosystems is based, are popular target of attacks and attempts to obscure the generated output. The topic of attacks is also included in the chapter. Changing the working conditions may have a degrading influence on the parameters of generated sequence. In [74] an approach for the evaluation of physical random number generators is given which takes the construction of the TRNG into account. The document presents a theory how the TRNGs used in cryptographic systems should be evaluated. For the TRNGs testing we have to accept the following facts [100]: • A final set of statistical tests may detect defects of a random source, but these tests cannot verify the randomness of the source. • Good statistical properties of the random numbers are clearly not sufficient for sensitive cryptographic applications as the generation of the keys, signature key pars or signature parameters. • The key criterion is not the statistical behavior of the numbers but their entropy. • For good TRNG it has to be given that the increase of entropy per generated number is sufficiently large. In [74], there is proposed a set of tests that should be passed, including the Coron’s test of entropy increase. In addition to the proof that the generated numbers have desired properties, it is needed to provide an explanation of randomness 89
FEI KEMT extraction. In other words, the principle of random numbers generation has to be described for better understanding and for better analysis of possible attacks on the TRNG. Startup Test, Online Test, TOT Tests If RNG prototype in a lab generates acceptable random numbers this may not be true for each TRNG of the same type during the whole operation time. The reason for this could be found in tolerances of components of the noise source, ageing effects, or outside attacks. In the worst case the TRNG breaks down totally and the output numbers are constant from that moment on. Therefore, the developer of the TRNG should implement also tests that will detect similar cases of the randomness degradation of the output bits. We distinguish between 3 types of tests [74]: 1. startup test is used to verify the principle functionality of the noise source when the TRNG has been started. 2. online test should detect if the quality of the random numbers is not sufficient for this particular TRNG or deteriorates in the course of the time. 3. tot test (’tot’ stands for ’total failure of the noise source’) should detect a total breakdown of the noise source. Implementation of the tests For implementation of the tests one has to consider the limitations that are given by the platform on which the TRNG is implemented. Not rarely the implementation target are smart cards, or field programmable gate arrays (FPGAs) with limited memory space. Therefore the chosen tests should require only small additional logic resources. Moreover the tests should be selected according to the features of the TRNG and the basic principle of the random source. It is possible to create also new tests that are more suitable for the particular TRNG and detect better the possible defects. Due to the limited memory resources of target platforms it is impossible to test the statistical properties on very long sequences (up to Mbits of data) as some tests (e.g. [97]) require. The goal is to find tests that are able continually evaluate the quality of the random source without the need of storing the output bits. Require- ments which appropriate online tests should fulfil are formulated in [100]. Two another requirements are given on the tests. On one side we expect detection of even small deviation from ideal random source, but on the other side often random 90
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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
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: FEI KEMT For R it holds that the in
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
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FEI KEMT [128] Zimmermann, P. ECMNE
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1 Montgomery Modular Multiplication in Hard- ware

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