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Pradeep Mishra

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University of Monastir

Journal of Computer Science IJCSIS

Dublin City University

Michael John Jacobson, Jr.

STMicroelectronics

Luca Breveglieri

Peking University

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Papers by Pradeep Mishra

Parallelizing Explicit Formula for Arithmetic in the Jacobian of Hyperelliptic Curves

Lecture Notes in Computer Science, 2003

One of the recent thrust areas in research on hyperelliptic curve cryptography has been to obtain... more One of the recent thrust areas in research on hyperelliptic curve cryptography has been to obtain explicit formulae for performing arithmetic in the Jacobian of such curves. We continue this line of research by obtaining parallel versions of such formulae. Our first contribution is to develop a general methodology for obtaining parallel algorithm of any explicit formula. Any parallel algorithm obtained using our methodology is provably optimal in the number of multiplication rounds. We next apply this methodology to Lange's explicit formula for arithmetic in genus 2 hyperelliptic curve-both for the affine coordinate and inversion free arithmetic versions. Since encapsulated add-and-double algorithm is an important countermeasure against side channel attacks, we develop parallel algorithms for encapsulated add-and-double for both of Lange's versions of explicit formula. For the case of inversion free arithmetic, we present parallel algorithms using 4, 8 and 12 multipliers. All parallel algorithms described in this paper are optimal in the number of parallel rounds. One of the conclusions from our work is the fact that the parallel version of inversion free arithmetic is more efficient than the parallel version of arithmetic using affine coordinates.

New Table Look-Up Methods for Faster Frobenius Map Based Scalar Multiplication Over GF(p n )

Lecture Notes in Computer Science, 2004

We describe a new scalar multiplication algorithm for elliptic and hyperelliptic curve cryptosyst... more We describe a new scalar multiplication algorithm for elliptic and hyperelliptic curve cryptosystems. The algorithm is obtained by combining Koblitz's idea of using Frobenius automorphism along with a very special kind of look-up table. In the case where the base point is unknown, we present an efficient algorithm to compute the look-up table online. Our algorithm applies to prime power fields GF (p n). One important subclass of such fields are Optimal Extension Fields (OEF's) which are believed to be ideal for efficient implementation of cryptographic primitives. Over prime power fields, our algorithm compares favourably to other known algorithms for scalar multiplication.

Fast Elliptic Curve Point Multiplication using Double-Base Chains

Among the various arithmetic operations required in implementing public key cryptographic algorit... more Among the various arithmetic operations required in implementing public key cryptographic algorithms, the elliptic curve point multiplication has probably received the maximum atten- tion from the research community in the last decade. Many methods for ecient and secure implementation of point multiplication have been proposed. The eciency of these methods mainly depends on the representation one uses for the scalar multiplier. In the current work we propose an ecient algorithm based on the so-called double-base number system. We introduce the new concept of double-base chains which, if manipulated with care, can significantly reduce the complexity of scalar multiplication on elliptic curves. Besides we have adopted some other measures to further reduce the operation count. Our algorithm compares favorably against classical and other similar approaches.

Application of Montgomery’s Trick to Scalar Multiplication for Elliptic and Hyperelliptic Curves Using a Fixed Base Point

Lecture Notes in Computer Science, 2004

We propose a scalar multiplication algorithm for elliptic and hyperelliptic curve cryptosystems, ... more We propose a scalar multiplication algorithm for elliptic and hyperelliptic curve cryptosystems, which uses affine arithmetic and is resistant against simple power attacks. Also, using a modification of known techniques the algorithm can be made immune against differential power attacks. The algorithm uses Montgomery's trick and a precomputed table consisting of multiples of the base point. Consequently, the algorithm is useful in a scenario where the base point is fixed, like Elgamal encryption or signature generation. Under such circumstances, for hyperelliptic curves, the algorithm compares favourably with other known algorithms over all fields. For elliptic curves, under similar circumstances, the algorithm performs better than other algorithms over prime fields. The increase in speed is due to a proper application of Montgomery's trick to efficiently perform the simultaneous inversion of several field elements.

Pipelined Computation of Scalar Multiplication in Elliptic Curve Cryptosystems

Lecture Notes in Computer Science, 2004

In the current work we propose a pipelining scheme for implementing Elliptic Curve Cryptosystems ... more In the current work we propose a pipelining scheme for implementing Elliptic Curve Cryptosystems (ECC). The scalar multiplication is the dominant operation in ECC. It is computed by a series of point additions and doublings. The pipelining scheme is based on a key observation: to start the subsequent operation one need not wait until the current one exits. The next operation can begin while a part of the current operation is still being processed. To our knowledge, this is the first attempt to compute the scalar multiplication in such a pipelined method. Also, the proposed scheme can be made resistant to side-channel attacks (SCA). Our scheme compares favourably to all SCA resistant sequential and parallel methods.

Efficient quintuple formulas for elliptic curves and efficient scalar multiplication using multibase number representation

Information Security, 2007

Abstract. In the current work we propose two efficient formulas for computing the 5-fold (5P) of ... more

Efficient and Secure Elliptic Curve Point Multiplication Using Double-Base Chains

Advances in Cryptology-ASIACRYPT …, 2005

Abstract. In this paper, we propose a efficient and secure point mul-tiplication algorithm, based... more

A General Methodology for Pipelining and Its Application to Point Multiplication in Curve Based Cryptography

Kishan C. Gupta, Pradeep K. Mishra, Pinakpani Pal. In Applied Cryptography and Network Security: ... more

Towards Minimizing Memory Requirement for Implementation of Hyperelliptic Curve Cryptosystems

Lecture Notes in Computer Science, 2007

Elliptic (ECC) and hyperelliptic curve cryptosystems (HECC) have emerged as cryptosystems for sma... more

A metric on the set of elliptic curves over

Applied Mathematics Letters, 2008

Elliptic curves over finite fields have found applications in many areas including cryptography. ... more Elliptic curves over finite fields have found applications in many areas including cryptography. In the current work we define a metric on the set of elliptic curves defined over a prime field F p , p > 3. Computing this metric requires us to solve an instance of a discrete log problem in F * p. This idea may have a possible application in devising some cryptographic primitives.

A General Methodology for Pipelining the Point Multiplication Operation in Curve Based Cryptography

Applied Cryptography and Network Security, 2006

Abstract. Pipelining is a well-known performance enhancing technique in computer science. Point m... more

A Graph Theoretic Analysis of Double Base Number Systems

Lecture Notes in Computer Science, 2007

Abstract. Double base number systems (DBNS) provide an elegant way to represent numbers. These re... more

A combinatorial interpretation of double base number system and some consequences

Advances in Mathematics of Communications, 2008

Page 1. Advances in Mathematics of Communications Web site: http://www.aimSciences. org Volume 2,... more

The double-base number system and its application to elliptic curve cryptography

Mathematics of Computation, 2008

We describe an algorithm for point multiplication on generic ellip- tic curves, based on a repres... more We describe an algorithm for point multiplication on generic ellip- tic curves, based on a representation of the scalar as a sum of mixed powers of 2 and 3. The sparseness of this so-called double-base number system, combined with some efficient point tripling formulae, lead to efficient point multiplication algorithms for curves defined over both prime and binary fields. Side-channel

Parallelizing Explicit Formula for Arithmetic in the Jacobian of Hyperelliptic Curves

Lecture Notes in Computer Science, 2003

One of the recent thrust areas in research on hyperelliptic curve cryptography has been to obtain... more One of the recent thrust areas in research on hyperelliptic curve cryptography has been to obtain explicit formulae for performing arithmetic in the Jacobian of such curves. We continue this line of research by obtaining parallel versions of such formulae. Our first contribution is to develop a general methodology for obtaining parallel algorithm of any explicit formula. Any parallel algorithm obtained using our methodology is provably optimal in the number of multiplication rounds. We next apply this methodology to Lange's explicit formula for arithmetic in genus 2 hyperelliptic curve-both for the affine coordinate and inversion free arithmetic versions. Since encapsulated add-and-double algorithm is an important countermeasure against side channel attacks, we develop parallel algorithms for encapsulated add-and-double for both of Lange's versions of explicit formula. For the case of inversion free arithmetic, we present parallel algorithms using 4, 8 and 12 multipliers. All parallel algorithms described in this paper are optimal in the number of parallel rounds. One of the conclusions from our work is the fact that the parallel version of inversion free arithmetic is more efficient than the parallel version of arithmetic using affine coordinates.

New Table Look-Up Methods for Faster Frobenius Map Based Scalar Multiplication Over GF(p n )

Lecture Notes in Computer Science, 2004

We describe a new scalar multiplication algorithm for elliptic and hyperelliptic curve cryptosyst... more We describe a new scalar multiplication algorithm for elliptic and hyperelliptic curve cryptosystems. The algorithm is obtained by combining Koblitz's idea of using Frobenius automorphism along with a very special kind of look-up table. In the case where the base point is unknown, we present an efficient algorithm to compute the look-up table online. Our algorithm applies to prime power fields GF (p n). One important subclass of such fields are Optimal Extension Fields (OEF's) which are believed to be ideal for efficient implementation of cryptographic primitives. Over prime power fields, our algorithm compares favourably to other known algorithms for scalar multiplication.

Fast Elliptic Curve Point Multiplication using Double-Base Chains

Among the various arithmetic operations required in implementing public key cryptographic algorit... more Among the various arithmetic operations required in implementing public key cryptographic algorithms, the elliptic curve point multiplication has probably received the maximum atten- tion from the research community in the last decade. Many methods for ecient and secure implementation of point multiplication have been proposed. The eciency of these methods mainly depends on the representation one uses for the scalar multiplier. In the current work we propose an ecient algorithm based on the so-called double-base number system. We introduce the new concept of double-base chains which, if manipulated with care, can significantly reduce the complexity of scalar multiplication on elliptic curves. Besides we have adopted some other measures to further reduce the operation count. Our algorithm compares favorably against classical and other similar approaches.

Application of Montgomery’s Trick to Scalar Multiplication for Elliptic and Hyperelliptic Curves Using a Fixed Base Point

Lecture Notes in Computer Science, 2004

We propose a scalar multiplication algorithm for elliptic and hyperelliptic curve cryptosystems, ... more We propose a scalar multiplication algorithm for elliptic and hyperelliptic curve cryptosystems, which uses affine arithmetic and is resistant against simple power attacks. Also, using a modification of known techniques the algorithm can be made immune against differential power attacks. The algorithm uses Montgomery's trick and a precomputed table consisting of multiples of the base point. Consequently, the algorithm is useful in a scenario where the base point is fixed, like Elgamal encryption or signature generation. Under such circumstances, for hyperelliptic curves, the algorithm compares favourably with other known algorithms over all fields. For elliptic curves, under similar circumstances, the algorithm performs better than other algorithms over prime fields. The increase in speed is due to a proper application of Montgomery's trick to efficiently perform the simultaneous inversion of several field elements.

Pipelined Computation of Scalar Multiplication in Elliptic Curve Cryptosystems

Lecture Notes in Computer Science, 2004

In the current work we propose a pipelining scheme for implementing Elliptic Curve Cryptosystems ... more In the current work we propose a pipelining scheme for implementing Elliptic Curve Cryptosystems (ECC). The scalar multiplication is the dominant operation in ECC. It is computed by a series of point additions and doublings. The pipelining scheme is based on a key observation: to start the subsequent operation one need not wait until the current one exits. The next operation can begin while a part of the current operation is still being processed. To our knowledge, this is the first attempt to compute the scalar multiplication in such a pipelined method. Also, the proposed scheme can be made resistant to side-channel attacks (SCA). Our scheme compares favourably to all SCA resistant sequential and parallel methods.

Efficient quintuple formulas for elliptic curves and efficient scalar multiplication using multibase number representation

Information Security, 2007

Abstract. In the current work we propose two efficient formulas for computing the 5-fold (5P) of ... more

Efficient and Secure Elliptic Curve Point Multiplication Using Double-Base Chains

Advances in Cryptology-ASIACRYPT …, 2005

Abstract. In this paper, we propose a efficient and secure point mul-tiplication algorithm, based... more

A General Methodology for Pipelining and Its Application to Point Multiplication in Curve Based Cryptography

Kishan C. Gupta, Pradeep K. Mishra, Pinakpani Pal. In Applied Cryptography and Network Security: ... more

Towards Minimizing Memory Requirement for Implementation of Hyperelliptic Curve Cryptosystems

Lecture Notes in Computer Science, 2007

Elliptic (ECC) and hyperelliptic curve cryptosystems (HECC) have emerged as cryptosystems for sma... more

A metric on the set of elliptic curves over

Applied Mathematics Letters, 2008

Elliptic curves over finite fields have found applications in many areas including cryptography. ... more Elliptic curves over finite fields have found applications in many areas including cryptography. In the current work we define a metric on the set of elliptic curves defined over a prime field F p , p > 3. Computing this metric requires us to solve an instance of a discrete log problem in F * p. This idea may have a possible application in devising some cryptographic primitives.

A General Methodology for Pipelining the Point Multiplication Operation in Curve Based Cryptography

Applied Cryptography and Network Security, 2006

Abstract. Pipelining is a well-known performance enhancing technique in computer science. Point m... more

A Graph Theoretic Analysis of Double Base Number Systems

Lecture Notes in Computer Science, 2007

Abstract. Double base number systems (DBNS) provide an elegant way to represent numbers. These re... more

A combinatorial interpretation of double base number system and some consequences

Advances in Mathematics of Communications, 2008

Page 1. Advances in Mathematics of Communications Web site: http://www.aimSciences. org Volume 2,... more

The double-base number system and its application to elliptic curve cryptography

Mathematics of Computation, 2008

We describe an algorithm for point multiplication on generic ellip- tic curves, based on a repres... more We describe an algorithm for point multiplication on generic ellip- tic curves, based on a representation of the scalar as a sum of mixed powers of 2 and 3. The sparseness of this so-called double-base number system, combined with some efficient point tripling formulae, lead to efficient point multiplication algorithms for curves defined over both prime and binary fields. Side-channel