Analyzing and comparing Montgomery multiplication algorithms

Koç, Çetin Kaya; Acar, Tolga; Kaliski, Burt

doi:10.1109/40.502403

Cited by 408 publications

(38 citation statements)

References 6 publications

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“…Software implementations of Montgomery multiplication usually utilize the straightforward algorithm described in [1] or the multiple-precision algorithms discussed in [2]. Algorithm 1 shows the straightforward Montgomery multiplication algorithm for two k-bit inputsā andb.…”

Section: A Software-based Montgomery Multiplicationmentioning

confidence: 99%

A Karatsuba-Based Montgomery Multiplier

Chow

Eguro

Luk

et al. 2010

2010 International Conference on Field Programmable Logic and Applications

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Modular multiplication of long integers is an important building block for cryptographic algorithms. Although several FPGA accelerators have been proposed for large modular multiplication, previous systems have been based on O(N 2 ) algorithms. In this paper, we present a Montgomery multiplier that incorporates the more efficient Karatsuba algorithm which is O(N (log 3/ log 2) ). This system is parameterizable to different bitwidths and makes excellent use of both embedded multipliers and fine-grained logic. The design has significantly lower LUT-delay product and multiplier-delay product compared with previous designs. Initial testing on a Virtex-6 FPGA showed that it is 60-190 times faster than an optimized multi-threaded software implementation running on an Intel Xeon 2.5 GHz CPU. The proposed multiplier system is also estimated to be 95-189 times more energy efficient than the software-based implementation. This high performance and energy efficiency makes it suitable for server-side applications running in a datacenter environment.

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Section: A Software-based Montgomery Multiplicationmentioning

confidence: 99%

A Karatsuba-Based Montgomery Multiplier

Chow

Eguro

Luk

et al. 2010

2010 International Conference on Field Programmable Logic and Applications

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“…They all improve efficiency by interleaving partial products generation and modular reduction steps to reduce the width of intermediate data and to gain some speedup. One famous variant is the Coarsely Integrated Operand Scanning (CIOS) method presented by Koc et al in [5]. However, besides these improvements, Montgomery multiplication still suffers from strong dependencies inside the main loop of partial products accumulation and modular reduction.…”

Section: Background On F P Finite Field Multipliersmentioning

confidence: 99%

“…1. Each one corresponds to one step of the iteration described Algorithm 1 CIOS algorithm (from [5]). Require: R = 2 n such as 4×P < R, P = −P −1 (mod w) and A, B two integers such that 0 ≤ A, B < 2 × P Ensure:…”

Section: Hyper-threaded Multipliermentioning

confidence: 99%

Hyper-threaded multiplier for HECC

Gallin

Tisserand

2017

2017 51st Asilomar Conference on Signals, Systems, and Computers

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Abstract-Modular multiplication is the most costly and common operation in hyper-elliptic curve cryptography. Over prime fields, it uses dependent partial products and reduction steps. These dependencies make FPGA implementations with fully pipelined DSP blocks difficult to optimize. We propose a new multiplier architecture with hyper-threaded capabilities. Several independent multiplications are handled in parallel for efficiently filling the pipeline and overlapping internal latencies by independent computations. It increases the silicon efficiency and leads to a better area / computation time trade-off than current state of the art. We use this hyper-threaded multiplier into small accelerators for hyper-elliptic curve cryptography in embedded systems.

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“…10 9 M might try to parallelize the multiplication of large numbers by splitting the multiplicands into smaller "words" and involving other processors in the multiplication of these words. Further details about this process can be found in [36]. However, this attack incurs a significant communication overhead that prevents an M from gaining any substantial speedup; given a large number of squaring rounds, the RTT between the cooperating processors needs to be in the order of few nanoseconds to achieve even a modest speedup.…”

Section: Low-cost Variable-exponent Modular Exponentiation Puzzlementioning

confidence: 99%

Low-Cost Client Puzzles Based on Modular Exponentiation

Karame

Čapkun

2010

Computer Security – ESORICS 2010

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Abstract. Client puzzles have been proposed as a useful mechanism for mitigating Denial of Service attacks on network protocols. While several puzzles have been proposed in recent years, most existing nonparallelizable puzzles are based on modular exponentiations. The main drawback of these puzzles is in the high cost that they incur on the puzzle generator (the verifier). In this paper, we propose cryptographic puzzles based on modular exponentiation that reduce this overhead. Our constructions are based on a reasonable intractability assumption in RSA: essentially the difficulty of computing a small private exponent when the public key is larger by several orders of magnitude than the semi-prime modulus. We also discuss puzzle constructions based on CRT-RSA [11]. Given a semi-prime modulus N , the costs incurred on the verifier in our puzzle are decreased by a factor of |N | k when compared to existing modular exponentiation puzzles, where k is a security parameter. We further show how our puzzle can be integrated in a number of protocols, including those used for the remote verification of computing performance of devices and for the protection against Denial of Service attacks. We validate the performance of our puzzle on PlanetLab nodes.

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Analyzing and comparing Montgomery multiplication algorithms

Cited by 408 publications

References 6 publications

A Karatsuba-Based Montgomery Multiplier

A Karatsuba-Based Montgomery Multiplier

Hyper-threaded multiplier for HECC

Low-Cost Client Puzzles Based on Modular Exponentiation

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