1 Montgomery Modular Multiplication in Hard- ware

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FEI KEMT in shorter words taking into account the physical limitations of the structures in se- lected hardware platform. Optimal solution in case when the operands length may change would provide a design for which the length of operands determines only the computational time for an operation but not the overall performance of the unit that is constant for arbitrary length. Montgomery Methods The MMM provides a very efficient way for computing the modular exponentiation. Input operands for the baseline algebraic operations of the RSA algorithm described by Equations 1.5-1.8 have very long length due to security reasons. Nowadays, the key length for the RSA is switched from 1024 to 2048 bits as the factorisation effort brings better results, closer to the bottom standard value. Having a need to use operands with doubled precision it is even more desirable to find algorithms that minimise the number of the algebraic operations together with their complexity. The Montgomery reduction allows efficient implementation of the MM without using the classical modular reduction step that is even more expensive operation in comparison to the multiplication. Therefore it pays off to minimise the number of required reductions or to use algorithms avoiding the division. In Montgomery exponentiation algorithm (Algorithm 1 – 1 [86]) the modular exponentiation unrolls into series of the MMM. Thanks to the transformation to a Montgomery domain and application of the MMM, it is possible to avoid the un- wanted modular reduction during computations. We continue with description of the MMM and conversion operations applied in the Algorithm 1 – 1. Given two integers X and Y (X, Y < M < R), and the prime k-bit modulus M, the MMM algorithm computes S = MMM(X, Y ) = (XY R −1 ) mod M , (1.9) where R −1 is the inverse of R = b k and b denotes a base or radix. The M-residue X, of an integer X < M is defined as [41]: X = XR mod M (1.10) For conversion to the Montgomery domain we can use the MMM function as follows: MMM(X, R 2 ) = XR 2 R −1 mod M (1.11) = XR mod M = X 9
FEI KEMT Algorithm 1 – 1 Montgomery exponentiation algorithm [86], the definition of M ′ requires that gcd(M, R) = 1, b denotes base or radix. Require: M = (mk−1 . . . m0)b, R = b k , M ′ = −M −1 mod b, E = (et . . . e0)2 with et = 1, and an integer X, 1 ≤ X < M. The values R 2 mod M and R mod M may be also provided as precomputed inputs. Ensure: A = X E mod M. 1: X ⇐ MMM(X, R 2 mod M) 2: A ⇐ R mod M 3: for i = t down to 0 do 4: A ⇐ MMM(A, A) 5: if ei = 1 then 6: A ⇐ MMM(A, X) 7: end if 8: end for 9: A ⇐ MMM(A, 1) 10: return A Therefore the first operation in the Algorithm 1 – 1 (Step 1) maps the input value X to its M-residue X. Now we show how to re-map the value X to its ordinary form of integer X what is done in the last operation of the exponentiation (Algorithm 1 – 1, Step 9). It can be seen that the Montgomery product of two M-residues X, Y is itself the M-residue S: S = MMM(A, B) (1.12) = XY R −1 mod M = XRY RR −1 mod M = XY R mod M = SR mod M 10
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 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: FEI KEMT A (X) request for B’s pr
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
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FEI KEMT from or writing to a singl
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FEI KEMT Algorithm 4 - 1 Modified M
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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
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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.,
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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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