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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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Cited by 37 (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.
Height zeta functions of toric varieties
, 1996
"... 2. Algebraic tori................................................ 6 ..."
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Cited by 18 (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 16 (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 13 (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 10 (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.
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 10 (4 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.
Rationality problem of GL4 group actions
 Adv. Math
"... Let K be any field which may not be algebraically closed, V be a fourdimensional vector space over K; sAGLðVÞ where the order of s may be finite or infinite, f ðTÞAKT be the characteristic polynomial of s: Let a; ab1; ab2; ab3 be the four roots of f ðTÞ 0 in some extension field of K: Theorem 1. ..."
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Let K be any field which may not be algebraically closed, V be a fourdimensional vector space over K; sAGLðVÞ where the order of s may be finite or infinite, f ðTÞAKT be the characteristic polynomial of s: Let a; ab1; ab2; ab3 be the four roots of f ðTÞ 0 in some extension field of K: Theorem 1. Both KðVÞ/sS and KðPðVÞÞ/sS are rational ð purely transcendental) over K if at least one of the following conditions is satisfied: (i) char K 2; (ii) f ðTÞ is a reducible or inseparable polynomial in KT ; (iii) not all of b1; b2; b3 are roots of unity, (iv) if f ðTÞ is separable irreducible, then the Galois group of f ðTÞ over K is not isomorphic to the dihedral group of order 8 or the Klein four group. Theorem 2. Suppose that all bi are roots of unity and f ðTÞAK T is separable irreducible. (a) If the Galois group of f ðTÞ is isomorphic to the dihedral group of order 8, then both KðVÞ/sS and KðPðVÞÞ/sS are not stably rational over K: (b) When the Galois group of f ðTÞ is isomorphic to the Klein four group, then a necessary and sufficient condition for rationality of KðVÞ/sS and KðPðVÞÞ/sS is provided. (See Theorem 1.5. for details.)