1 Montgomery Modular Multiplication in Hard- ware

More documents

Recommendations

Info

FEI KEMT 1 Montgomery Modular Multiplication in Hard- ware - preliminaries Many popular public-key cryptographic algorithms and protocols, such as RSA, ElGamal, elliptic curve cryptography (ECC), Diffie-Hellman, etc. [86] extensively use modular operations with large numbers. Typical size of operands in ECC and RSA is 160-300 bits and 1000-2000 bits, respectively. We start the chapter with discussion on optimal choice of the computation method and way of its implementation according to chosen implementation platform (the Section 1.1). In Section 1.2 we bring a summary on RSA algorithm together with a short analysis of available algorithms for modular multiplication. We mention the aspects of hardware implementation and review the available pa- pers in this area. Finally, the further implemented algorithm and its modification are introduced. The Section 1.3 we start with definition of elliptic curves (EC) and continue with their application in cryptography. The last section summarises the most important features of the presented public-key algorithms and identifies the most important part of the system for effective implementation. 1.1 Implementation Platforms By having all parts of cryptosystem (encryption, authentication, key storage, gen- eration of random numbers . . . ) implemented on the same platform one is able to achieve highly compact and therefore potentially secure implementation. The more signals are available for an adversary for observation, the more information about processed data can be obtained. While in the past the development of hardware and software platforms was done separately, beside the initial requirements and definitions of data formats and interfaces, nowadays with so called hardware-software co-design one tries to find optimum in effective utilisation of resources. In such case some of operations are implemented as a hardware structure and the others as a software function. With reconfigurable devices and embedded soft-core processors the situation is very suitable for such an approach. However, development of mixed systems is not a trivial task for designers, especially on the level when decision on tasks division is done. Systems making possible to simulate and evaluate system performance by proposed software-hardware architecture before a real (and expensive) implementation are only on the early stage of development (check e.g. GEZEL language and design environment [99]). 3
FEI KEMT The hardware implementation platforms offer higher level of security thanks to possibility to separate physically a sensitive data and in dependency on operations also higher performance as similar software implementations. As a hardware platform can be considered: • ASIC (Application-Specific Integrated Circuit), • FPGA (Field-Programmable Gate Array) or • RFID (Radio Frequency Identification) chip. There are different approaches by implementation of cryptosystems. Implementa- tion can provide some supporting functions for general-purpose processor, cover all crypto-related operations in a standard system or even represent complete system able to substitute the original non-secured system. In dependency on the application the implementation can be done in the form of a smart card, IP (Intellectual Property) core, co-processor, PCI card, router etc. With enlarging area of chips it is possible to implement a CPU, memory blocks, peripherals, interfaces and co-processor on a single chip providing a such called system-on-a-chip (SOC). Especially in cryptography there is a requirement for implementation systems as SOC which hides the internal signals from possible abuse by the adversary. SOC raises another requirement, namely to find a way for implementation of all parts of SOC on the same chip, same platform, if possible by sharing the same resources. Applications have various requirements for area, speed, energy, or power con- sumption. Additionally, in case of cryptosystems we define also level of security taking into account the vulnerability against eavesdropping and side-channel at- tacks, or ability of the system to detect an attack and thereafter delete the sensitive data in a way making impossible to restore them by an adversary (tamper resis- tance). Definitions of conditions required to certify cryptographic implementations on a certain level of security and areas where such systems can be used are set in standards of well-known standardisation organisations [57]. Reconfigurable Devices Reconfigurable device is an hardware architecture with both a functionality of processing elements and an interconnection between them can be modified after fabrication time. The most known reconfigurable hardware components are FPGAs. 4
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 and 10: Contents Introduction 1 1 Montgomer
Page 11 and 12: FEI KEMT 6.2.3 Analysis of TRNG in
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: FEI KEMT and Elliptic Curve Cryptog
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
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 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
Page 159:
FEI KEMT [128] Zimmermann, P. ECMNE
show all

1 Montgomery Modular Multiplication in Hard- ware

Create successful ePaper yourself

Delete template?

Save as template?