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28
The XTR public key system
, 2000
"... This paper introduces the XTR public key system. XTR is based on a new method to represent elements of a subgroup of a multiplicative group of a finite field. Application of XTR in cryptographic protocols leads to substantial savings both in communication and computational overhead without compromis ..."
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Cited by 82 (12 self)
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This paper introduces the XTR public key system. XTR is based on a new method to represent elements of a subgroup of a multiplicative group of a finite field. Application of XTR in cryptographic protocols leads to substantial savings both in communication and computational overhead without compromising security.
The function field sieve in the medium prime case
 Advances in Cryptology – EUROCRYPT 2006, LNCS 4004 (2006
"... Abstract. In this paper, we study the application of the function field sieve algorithm for computing discrete logarithms over finite fields of the form Fqn when q is a mediumsized prime power. This approach is an alternative to a recent paper of Granger and Vercauteren for computing discrete logar ..."
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Cited by 29 (9 self)
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Abstract. In this paper, we study the application of the function field sieve algorithm for computing discrete logarithms over finite fields of the form Fqn when q is a mediumsized prime power. This approach is an alternative to a recent paper of Granger and Vercauteren for computing discrete logarithms in tori, using efficient torus representations. We show that when q is not too large, a very efficient L(1/3) variation of the function field sieve can be used. Surprisingly, using this algorithm, discrete logarithms computations over some of these fields are even easier than computations in the prime field and characteristic two field cases. We also show that this new algorithm has security implications on some existing cryptosystems, such as torus based cryptography in T30, short signature schemes in characteristic 3 and cryptosystems based on supersingular abelian varieties. On the other hand, cryptosystems involving larger basefields and smaller extension degrees, typically of degree at most 6, such as LUC, XTR or T6 torus cryptography, are not affected. 1
TorusBased Cryptography
 In Advances in Cryptology (CRYPTO 2003), Springer LNCS 2729
, 2003
"... We introduce cryptography based on algebraic tori, give a new public key system called CEILIDH, and compare it to other discrete log based systems including LUC and XTR. Like those systems, we obtain small key sizes. While LUC and XTR are essentially restricted to exponentiation, we are able to perf ..."
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Cited by 28 (2 self)
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We introduce cryptography based on algebraic tori, give a new public key system called CEILIDH, and compare it to other discrete log based systems including LUC and XTR. Like those systems, we obtain small key sizes. While LUC and XTR are essentially restricted to exponentiation, we are able to perform multiplication as well. We also disprove the open conjectures from [2], and give a new algebrogeometric interpretation of the approach in that paper and of LUC and XTR.
Key improvements to XTR
 PROCEEDINGS OF ASIACRYPT 2000, LNCS 1976
, 2000
"... This paper describes improved methods for XTR key representation and parameter generation (cf. [4]). If the field characteristic is properly chosen, the size of the XTR public key for signature applications can be reduced by a factor of three at the cost of a small one time computation for the reci ..."
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Cited by 14 (5 self)
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This paper describes improved methods for XTR key representation and parameter generation (cf. [4]). If the field characteristic is properly chosen, the size of the XTR public key for signature applications can be reduced by a factor of three at the cost of a small one time computation for the recipient of the key. Furthermore, the parameter setup for an XTR system can be simplified because the trace of a proper subgroup generator can, with very high probability, be computed directly, thus avoiding the probabilistic approach from [4]. These nontrivial extensions further enhance the practical potential of XTR.
Using Primitive Subgroups to Do More with Fewer Bits
, 2004
"... This paper gives a survey of some ways to improve the ef ciency of discrete logbased cryptography by using the restriction of scalars and the geometry and arithmetic of algebraic tori and abelian varieties. ..."
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Cited by 13 (3 self)
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This paper gives a survey of some ways to improve the ef ciency of discrete logbased cryptography by using the restriction of scalars and the geometry and arithmetic of algebraic tori and abelian varieties.
Looking beyond XTR
 IN ADVANCES IN CRYPTOLOGY — ASIACRYPT 2002, LECT. NOTES IN COMP. SCI. 2501
, 2002
"... XTR is a general methodthat can be appliedto discrete logarithm based cryptosystems in extension fields of degree six, providing a compact representation of the elements involved. In this paper we present a precise formulation of the BrouwerPellikaanVerheul conjecture, originally posedin [4], con ..."
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Cited by 13 (0 self)
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XTR is a general methodthat can be appliedto discrete logarithm based cryptosystems in extension fields of degree six, providing a compact representation of the elements involved. In this paper we present a precise formulation of the BrouwerPellikaanVerheul conjecture, originally posedin [4], concerning the size of XTRlike representations of elements in extension fields of arbitrary degree. If true this conjecture wouldprovide even more compact representations of elements than XTR in extension fields of degree thirty. We test the conjecture by experiment, showing that in fact it is unlikely that such a compact representation of elements can be achieved in extension fields of degree thirty.
Asymptotically optimal communication for torusbased cryptography
 In Advances in Cryptology (CRYPTO 2004), Springer LNCS 3152
, 2004
"... Abstract. We introduce a compact and efficient representation of elements of the algebraic torus. This allows us to design a new discretelog based publickey system achieving the optimal communication rate, partially answering the conjecture in [4]. For n the product of distinct primes, we construct ..."
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Cited by 11 (1 self)
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Abstract. We introduce a compact and efficient representation of elements of the algebraic torus. This allows us to design a new discretelog based publickey system achieving the optimal communication rate, partially answering the conjecture in [4]. For n the product of distinct primes, we construct efficient ElGamal signature and encryption schemes in a subgroup of F ∗ qn in which the number of bits exchanged is only a φ(n)/n fraction of that required in traditional schemes, while the security offered remains the same. We also present a DiffieHellman key exchange protocol averaging only φ(n) log2 q bits of communication per key. For the cryptographically important cases of n = 30 and n = 210, we transmit a 4/5 and a 24/35 fraction, respectively, of the number of bits required in XTR [14] and recent CEILIDH [24] cryptosystems. 1
On the Discrete Logarithm Problem on Algebraic Tori
 In Advances in Cryptology (CRYPTO 2005), Springer LNCS 3621, 66–85
, 2005
"... Abstract. Using a recent idea of Gaudry and exploiting rational representations of algebraic tori, we present an index calculus type algorithm for solving the discrete logarithm problem that works directly in these groups. Using a prototype implementation, we obtain practical upper bounds for the di ..."
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Cited by 11 (3 self)
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Abstract. Using a recent idea of Gaudry and exploiting rational representations of algebraic tori, we present an index calculus type algorithm for solving the discrete logarithm problem that works directly in these groups. Using a prototype implementation, we obtain practical upper bounds for the difficulty of solving the DLP in the tori T2(Fpm)and T6(Fpm) for various p and m. Our results do not affect the security of the cryptosystems LUC, XTR, or CEILIDH over prime fields. However, the practical efficiency of our method against other methods needs further examining, for certain choices of p and m in regions of cryptographic interest. 1
An Overview of the XTR Public Key System
 IN PUBLICKEY CRYPTOGRAPHY AND COMPUTATIONAL NUMBER THEORY, VERLAGES WALTER DE GRUYTER
, 2000
"... XTR is a new method to represent elements of a subgroup of a multiplicative group of a finite field. Application of XTR in cryptographic protocols leads to substantial savings both in communication and computational overhead without compromising security. This paper describes and explains the techn ..."
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Cited by 10 (1 self)
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XTR is a new method to represent elements of a subgroup of a multiplicative group of a finite field. Application of XTR in cryptographic protocols leads to substantial savings both in communication and computational overhead without compromising security. This paper describes and explains the techniques and properties that are relevant for the XTR cryptosystem and its implementation. It is based on the material from [10,?,?,?].
Practical Cryptography in High Dimensional Tori
 In Advances in Cryptology (EUROCRYPT 2005), Springer LNCS 3494
, 2004
"... At Crypto 2004, van Dijk and Woodruff introduced a new way of using the algebraic tori Tn in cryptography, and obtained an asymptotically optimal n/φ(n) savings in bandwidth and storage for a number of cryptographic applications. However, the computational requirements of compression and dec ..."
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Cited by 8 (5 self)
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At Crypto 2004, van Dijk and Woodruff introduced a new way of using the algebraic tori Tn in cryptography, and obtained an asymptotically optimal n/&phi;(n) savings in bandwidth and storage for a number of cryptographic applications. However, the computational requirements of compression and decompression in their scheme were impractical, and it was left open to reduce them to a practical level. We give a new method that compresses orders of magnitude faster than the original, while also speeding up the decompression and improving on the compression factor (by a constant term). Further, we give the first efficient implementation that uses T30 , compare its performance to XTR, CEILIDH, and ECC, and present new applications. Our methods achieve better compression than XTR and CEILIDH for the compression of as few as two group elements. This allows us to apply our results to ElGamal encryption with a small message domain to obtain ciphertexts that are 10% smaller than in previous schemes.