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45
An Extension of Kedlaya's Algorithm to Hyperelliptic Curves in Characteristic 2
, 2002
"... We present an algorithm for computing the zeta function of an arbitrary hyperelliptic curve over a finite field Fq of characteristic 2, thereby extending the algorithm of Kedlaya for odd characteristic. For a genus g hyperelliptic curve defined over F2 n , the averagecase time complexity is O(g ) a ..."
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Cited by 16 (5 self)
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We present an algorithm for computing the zeta function of an arbitrary hyperelliptic curve over a finite field Fq of characteristic 2, thereby extending the algorithm of Kedlaya for odd characteristic. For a genus g hyperelliptic curve defined over F2 n , the averagecase time complexity is O(g ) and the averagecase space complexity is O(g ), whereas the worstcase time and space complexities are O(g ) and ) respectively.
High Performance Arithmetic for Hyperelliptic Curve Cryptosystems of Genus Two
, 2003
"... Nowadays, there exists a manifold variety of cryptographic applications: from low level embedded crypto implementations up to high end cryptographic engines for servers. The latter require a exible implementation of a variety of cryptographic primitives in order to be capable of communicating wi ..."
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Cited by 14 (7 self)
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Nowadays, there exists a manifold variety of cryptographic applications: from low level embedded crypto implementations up to high end cryptographic engines for servers. The latter require a exible implementation of a variety of cryptographic primitives in order to be capable of communicating with several clients. On the other hand, on the client it only requires an implementation of one speci c algorithm with xed parameters such as a xed eld size or xed curve parameters if using ECC/ HECC. In particular for embedded environments like PDAs or mobile communication devices, xing these parameters can be crucial regarding speed and power consumption. In this contribution, we propose a highly ecient algorithm for a hyperelliptic curve cryptosystem of genus two, well suited for these constraint devices.
POINT COUNTING IN FAMILIES OF HYPERELLIPTIC CURVES IN CHARACTERISTIC 2
"... Let ĒΓ be a family of hyperelliptic curves over F2 alg cl with general Weierstrass equation given over a very small field F. We describe in this paper an algorithm for computing the zeta function of Ē¯γ, with ¯γ in a degree n extension field of F, which has as time complexity Õ(n3) bit operations ..."
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Cited by 12 (5 self)
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Let ĒΓ be a family of hyperelliptic curves over F2 alg cl with general Weierstrass equation given over a very small field F. We describe in this paper an algorithm for computing the zeta function of Ē¯γ, with ¯γ in a degree n extension field of F, which has as time complexity Õ(n3) bit operations and memory requirements O(n2) bits. With a slightly different algorithm we can get time O(n2.667) and memory O(n2.5), and the computation for n curves of the family can be done in time Õ(n3.376). All of these algorithms are polynomialtime in the genus.
Computing Zeta Functions of Hyperelliptic Curves over Finite Fields of Characteristic 2
 Advances in CryptologyCRYPTO 2002, LNCS 2442
, 2002
"... We present an algorithm for computing the zeta function of an arbitrary hyperelliptic curve over a finite field Fq of characteristic 2, thereby extending the algorithm of Kedlaya for small odd characteristic. For a genus g hyperelliptic curve over n , the asymptotic running time of the algorithm ..."
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Cited by 10 (1 self)
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We present an algorithm for computing the zeta function of an arbitrary hyperelliptic curve over a finite field Fq of characteristic 2, thereby extending the algorithm of Kedlaya for small odd characteristic. For a genus g hyperelliptic curve over n , the asymptotic running time of the algorithm is O(g ) and the space complexity is O(g ).
Counting Points for Hyperelliptic Curves of type y 2
 x 5 + ax over Finite Prime Fields, Selected Areas in Cryptography(SAC2003), Springer LNCS
, 2004
"... Abstract. Counting rational points on Jacobian varieties of hyperelliptic curves over finite fields is very important for constructing hyperelliptic curve cryptosystems (HCC), but known algorithms for general curves over given large prime fields need very long running times. In this article, we prop ..."
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Cited by 10 (2 self)
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Abstract. Counting rational points on Jacobian varieties of hyperelliptic curves over finite fields is very important for constructing hyperelliptic curve cryptosystems (HCC), but known algorithms for general curves over given large prime fields need very long running times. In this article, we propose an extremely fast point counting algorithm for hyperelliptic curves of type y 2 = x 5 + ax over given large prime fields �p, e.g. 80bit fields. For these curves, we also determine the necessary condition to be suitable for HCC, that is, to satisfy that the order of the Jacobian group is of the form l · c where l is a prime number greater than about 2 160 and c is a very small integer. We show some examples of suitable curves for HCC obtained by using our algorithm. We also treat curves of type y 2 = x 5 + a where a is not square in �p. 1
Elliptic & hyperelliptic curves on embedded µp
 ACM Transactions in Embedded Computing Systems (TECS), 2003. Special Issue on Embedded Systems and Security
"... To appear in the special issue on Embedded Systems and Security of the ..."
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Cited by 10 (4 self)
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To appear in the special issue on Embedded Systems and Security of the
Computing endomorphism rings of jacobians of genus 2 curves
 In Symposium on Algebraic Geometry and its Applications, Tahiti
, 2006
"... Abstract. We present probabilistic algorithms which, given a genus 2 curve C defined over a finite field and a quartic CM field K, determine whether the endomorphism ring of the Jacobian J of C is the full ring of integers in K. In particular, we present algorithms for computing the field of definit ..."
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Cited by 9 (5 self)
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Abstract. We present probabilistic algorithms which, given a genus 2 curve C defined over a finite field and a quartic CM field K, determine whether the endomorphism ring of the Jacobian J of C is the full ring of integers in K. In particular, we present algorithms for computing the field of definition of, and the action of Frobenius on, the subgroups J[ℓ d] for prime powers ℓ d. We use these algorithms to create the first implementation of Eisenträger and Lauter’s algorithm for computing Igusa class polynomials via the Chinese Remainder Theorem [EL], and we demonstrate the algorithm for a few small examples. We observe that in practice the running time of the CRT algorithm is dominated not by the endomorphism ring computation but rather by the need to compute p 3 curves for many small primes p. 1.
Fast computation with two algebraic numbers
 September
, 2002
"... We propose fast algorithms for computing composed products and composed sums, as well as diamond products of univariate polynomials. These operations correspond to special resultants, that we compute using power sums of roots of polynomials, by means of their generating series. ..."
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Cited by 8 (3 self)
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We propose fast algorithms for computing composed products and composed sums, as well as diamond products of univariate polynomials. These operations correspond to special resultants, that we compute using power sums of roots of polynomials, by means of their generating series.
Hyperelliptic Curve Coprocessors on a FPGA
 In Workshop on Information Security Applications  WISA, Jeju Island, Korea
, 2004
"... Abstract. Cryptographic algorithms are used in a large variety of different applications to ensure security services. It is, thus, very interesting to investigate various implementation platforms. Hyperelliptic curve schemes are cryptographic primitives to which a lot of attention was recently given ..."
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Cited by 8 (2 self)
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Abstract. Cryptographic algorithms are used in a large variety of different applications to ensure security services. It is, thus, very interesting to investigate various implementation platforms. Hyperelliptic curve schemes are cryptographic primitives to which a lot of attention was recently given due to the short operand size compared to other algorithms. They are specifically interesting for specialpurpose hardware. This paper provides a comprehensive investigation of highefficient HEC architectures. We propose a genus2 hyperelliptic curve cryptographic coprocessor using affine coordinates. We implemented a special class of hyperelliptic curves, namely using the parameter h(x) = x and f = x 5 + f1x + f0 and the base field GF(2 89). In addition, we only consider the most frequent case in our implementation and assume that the other cases are handled, e.g. by the protocol. We provide three different implementations ranging from high speed to moderate area. Hence, we provide a solution for a variety of applications. Our high performance HECC coprocessor is 78.5 % faster than the best previous implementation and our low area implementation utilizes only 22.7 % of the area that the smallest published design uses. Taking into account both area and latency, our coprocessor is an order of magnitude more efficient than previous implementations. We hope that the work at hand provides a step towards introducing HEC systems in practical applications.
Parallelizing Explicit Formula for Arithmetic in the Jacobian of Hyperelliptic Curves
 In Advances in Cryptology — Asiacrypt 2003, volume LNCS
, 2003
"... Abstract. 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 deve ..."
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Cited by 7 (0 self)
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Abstract. 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 addanddouble algorithm is an important countermeasure against side channel attacks, we develop parallel algorithms for encapsulated addanddouble 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.