1 Montgomery Modular Multiplication in Hard- ware

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FEI KEMT in any given range of values. Other definition comes from Schneier [101], who says that random is a sequence that has the same statistical properties as random bits, is unpredictable and cannot be reliably reproduced. Kolmogorov defines a string of bits as being random if and only if it is shorter than any computer program that can produce that string. From all three definitions we can extract a common re- quirement (necessary but insufficient) for having the numbers in a random sequence uncorrelated 6 . Unpredictable sequence is the one for which the knowledge of all generated values in the past does not increase probability to guess the subsequent value, or in other words knowing one of the numbers in the sequence must not help predicting the other ones. The same fact we can illustrate by another of unpredictability definitions which defines it as a status that there is no polynomial algorithm, by which knowing l bits of the generated sequence S one is able to predict (l+1)-th bit with probability bigger than 0.5 [86]. No correlation also causes that the generated random sequence cannot be produced by other computer program than the one that prints the whole random sequence as it is. Under truly random sequence of bits we understand an uncorrelated sequence that cannot be reproduced or predicted, has equal probability of all possible outputs (equiprobability) and its generation is based on a random process. A sequence that keeps the statistical properties of random sequence, but its members are correlated or the sequence can be reproduced is called pseudo-random. The pseudo-random sequence looks random, but its origin is not in a random process and the sequence generation can be reproduced and described as an algorithm. One of the issues discussed in the thesis is the ability to distinguish between the truly random and pseudo-random sequence by exploration of the generation process in generator. 5.1.2 Random Number Generator A RNG is an electronic device or software routine designed to yield a sequence of random numbers. A pseudo-random number generator (PRNG) is based on an algebraic function that expands the initial random value (a seed) into a random-like looking sequence. A true-random number generator (TRNG) includes a physical source of randomness 6 However, simple (linear) and known correlation relations between the members of sequence do not exclude such source. In these cases a corrector that removes the correlated samples may be applied. More dangerous are masked correlations of higher order that are difficult in detection. 73
FEI KEMT and a harvesting mechanism which extracts the randomness and generates truly random values. Security level of PRNG depends on complexity of the generating function, the period length of the generated sequence, and the amount of entropy in the seed. As a result, the pseudo-random sequences may achieve a high level of unpredictability in case of sufficient complexity of the generating function. However, the pseudo- random sequence has always a finite period and remains reproducible as far as initial conditions are sustained. The PRNG is the only choice for software implementations and thanks to deterministic components it attracts also the designers of electronic digital systems. Note that also pseudo-random sequence can be unpredictable when produced by cryptographically secure PRNG e.g. based on hash (one-way) functions, stream ci- phers or Blum Blum Shub principle [28]. The PRNG requires a random seed (from a TRNG or other reliable source of entropy, if available) to obtain the starting level of entropy. As the system is deterministic, for identical seeds the PRNG generates identical output pseudo-random sequences, too. No more entropy is added during exploitation of the seed, therefore the seed’s entropy designates the unpredictability of the generated sequence. The term generator is not completely correct in case of TRNG as the randomness is not generated but rather extracted from a source of randomness (see Figure 5 – 1). In TRNG the occurrence of random events is sampled by an extractor and transformed into a sequence of numerical values usually expressed as a binary stream. Source of randomness A/D conversion analogue part digital part noise signal Postprocessing digitised noise signal internal random sequence Output buffer external interface random number sequence Figure 5 – 1 Schematic diagram of a TRNG with designation of internal signals and interfaces The Figure 5 – 1 represents a typical design of TRNG based on a physical phe- nomenon. Using a proper harvest mechanism the analogue signal is converted into its digitised form. According to statistical properties of the signal it may be required to apply a post-processing in order to produce an internal random sequence. The generated sequence can be further accumulated in output buffer before leaving the 74
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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
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: FEI KEMT [68] what can be expressed
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
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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.,
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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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1 Montgomery Modular Multiplication in Hard- ware

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