Fast Implementation of Elliptic Curve Arithmetic in GF(p n )

Lim, Chae Hoon; Hwang, Hyo Sun

doi:10.1007/978-3-540-46588-1_27

Cited by 54 publications

(50 citation statements)

References 25 publications

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“…In general, if m is even, the exact number of p i -th power is: Table 2 shows the new exact analyses of BP algorithm and it is compared with IM algorithm. Note that we corrected minor counting errors that were shown in previous works [1,10]. We also show analyses for the 'special case' when the product of two subfield elements can be accumulated without overflow, and when α is small, that is our case.…”

Section: Table 1 Example Of Computingmentioning

confidence: 75%

“…Since r − 1 is in a special form we can compute A(x) r−1 efficiently by addition chain and p i -th power. The efficient method was already shown in [1], but we want to show that the analysis shown in [1] and [10] is incorrect.…”

Section: A(x)mentioning

confidence: 98%

“…It takes 670 cycles in our implementation. Among the various finite field inversion algorithms we consider IM (Inversion with Multiplication) [10] and BP [1] algorithms since only they require just one subfield inversion. Here we review the IM algorithm and the BP algorithm.…”

Section: Field Inversionmentioning

confidence: 99%

“…Since Koblitz [8] and Miller [11] first introduced elliptic curve cryptography (ECC), many works [7,10,2] have shown that ECC can be very efficiently embedded into restricted hardware such as smart cards. During the past few years, most people believed that elliptic curve defined over GF (2 m ) was the only useful one for hardware implementation since it can be implemented with only simple bit operations.…”

Section: Introductionmentioning

confidence: 99%

“…Not only is it easy to implement but also more efficient, since there are no carry propagation and the inversion in GF (p m ) is far more efficient than that in GF (p). Many works [10,2,1] have shown that GF (p m ) is very suitable choice for computer software implementation.…”

Section: Introductionmentioning

confidence: 99%

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Fast Implementation of Elliptic Curve Defined over GF(pm) on CalmRISC with MAC2424 Coprocessor

Chung

Sim

Lee

2000

Cryptographic Hardware and Embedded Systems — CHES 2000

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Abstract. In this paper, we propose fast finite field and elliptic curve (EC) algorithms useful for embedding cryptographic functions on high performance device such that most instructions take just one cycle. In such case, the integer multiplications and additions have the same computational cost so that the computational cost analyses that were previously done in traditional manner may be invalid and in some cases the new algorithms should be introduced for fast computation. In our implementation, column major method for field multiplication and BP inversion algorithm are used for fast field arithmetic, and mixed coordinates method is used for efficient EC exponentiation. We give here analyses on various algorithms that are useful for implementing EC exponentiation on CalmRISC microcontroller with MAC2424 coprocessor, as well as new exact analyses on BP (Bailey-Paar) inversion algorithm and EC exponentiation. Using techniques shown in this paper, we implemented EC exponentiation for various coordinate systems and the best result took 122ms, assuming 50ns clock cycle.

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Section: Table 1 Example Of Computingmentioning

confidence: 75%

Section: A(x)mentioning

confidence: 98%

Section: Field Inversionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Fast Implementation of Elliptic Curve Defined over GF(pm) on CalmRISC with MAC2424 Coprocessor

Chung

Sim

Lee

2000

Cryptographic Hardware and Embedded Systems — CHES 2000

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Accelerated precomputation points–based scalar reduction on elliptic curve cryptography for wireless sensor networks

Faye

Guyennet

Shou

et al. 2017

Int J Communication

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Sensor devices are limited resource power and energy, thus providing security services for sensor networks is very difficult. Elliptic curve cryptography (ECC) is one of the most famous asymmetric cryptographic schemes, which offers the same level of security with much shorter keys compared to the other widely used asymmetric cryptographic algorithm, RSA (Rivest, Shamir, and Adleman). In ECC, the main and most-heavily used operation is the scalar multiplication kP, where the scalar value k is a private integer and must be secured. In this work, we present a new approach to accelerate the main scalar multiplication on ECC over prime fields for sensor networks. This approach uses an equivalent representation of points and can act as a support for existing schemes in a selected interval. The simulation results showed that the proposed technique increases the efficiency of the computation time. For example, on this scalar multiplication, we obtain a gain of 4 bits in 161 bits for 6.25% of the scalars. This gain can sometimes reach 100% in some cases. After this significant reduction of the scalar k, we present a fast precomputation algorithm in a distributed scalar multiplication on kP to avoid storage of precomputation points, which requires extra memory. KEYWORDS elliptic curve cryptography, fast scalar multiplication, precomputation point, wireless sensor networks Int J Commun Syst. 2017;30:e3327.wileyonlinelibrary.com/journal/dac

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Speeding Up Elliptic Scalar Multiplication with Precomputation

Lim¹,

Hwang²

2000

Lecture Notes in Computer Science

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Fast Implementation of Elliptic Curve Arithmetic in GF(p n )

Cited by 54 publications

References 25 publications

Fast Implementation of Elliptic Curve Defined over GF(pm) on CalmRISC with MAC2424 Coprocessor

Fast Implementation of Elliptic Curve Defined over GF(pm) on CalmRISC with MAC2424 Coprocessor

Accelerated precomputation points–based scalar reduction on elliptic curve cryptography for wireless sensor networks

Speeding Up Elliptic Scalar Multiplication with Precomputation

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