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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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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.
Height zeta functions of toric varieties
, 1996
"... 2. Algebraic tori................................................ 6 ..."
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Cited by 14 (0 self)
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2. Algebraic tori................................................ 6
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.
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
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.
Density of integral points on algebraic varieties, Rational points on algebraic varieties
 169–197, Progr. Math. 199
, 2001
"... Let K be a number field, S a finite set of valuations of K, including the archimedean valuations, and OS the ring of Sintegers. Let X be an algebraic variety defined over K and D a divisor on X. We will use X and D to denote models over Spec(OS). ..."
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Cited by 7 (3 self)
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Let K be a number field, S a finite set of valuations of K, including the archimedean valuations, and OS the ring of Sintegers. Let X be an algebraic variety defined over K and D a divisor on X. We will use X and D to denote models over Spec(OS).
ESSENTIAL DIMENSION OF ALGEBRAIC TORI
"... The essential dimension is a numerical invariant of an algebraic group G which may be thought of as a measure of complexity of Gtorsors over fields. A recent theorem of N. Karpenko and A. Merkurjev gives a simple formula for the essential dimension of a finite pgroup. We obtain similar formulas fo ..."
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Cited by 6 (3 self)
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The essential dimension is a numerical invariant of an algebraic group G which may be thought of as a measure of complexity of Gtorsors over fields. A recent theorem of N. Karpenko and A. Merkurjev gives a simple formula for the essential dimension of a finite pgroup. We obtain similar formulas for the essential pdimension of a broad class of groups, which includes all algebraic tori.
Algebraic Tori in Cryptography
 In High Primes and Misdemeanours: Lectures in Honour of the 60th birthday of Hugh Cowie Williams, Fields Institute Communications Series 41, American Mathematical Society
, 2004
"... Abstract. We give a mathematical interpretation in terms of algebraic tori of Lucasbased cryptosystems, XTR, and the conjectural generalizations in [2]. We show that the varieties underlying these systems are quotients of algebraic tori by actions of products of symmetric groups. Further, we use th ..."
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Abstract. We give a mathematical interpretation in terms of algebraic tori of Lucasbased cryptosystems, XTR, and the conjectural generalizations in [2]. We show that the varieties underlying these systems are quotients of algebraic tori by actions of products of symmetric groups. Further, we use these varieties to disprove conjectures from [2]. 1