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Class Field Theory in Characteristic p, its Origin and Development
 the Proceedings of the International Conference on Class Field Theory (Tokyo
, 2002
"... Today's notion of "global field" comprises number fields (algebraic, of finite degree) and function fields (algebraic, of dimension 1, finite base field). They have many similar arithmetic properties. The systematic study of these similarities seems to have been started by Dedekind (1 ..."
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Today's notion of "global field" comprises number fields (algebraic, of finite degree) and function fields (algebraic, of dimension 1, finite base field). They have many similar arithmetic properties. The systematic study of these similarities seems to have been started by Dedekind (1857). A new impetus was given by the seminal thesis of E. Artin (1921, published in 1924). In this exposition I shall report on the development during the twenties and thirties of our century, with emphasis on the emergence of class field theory for function elds. The names of F.K.Schmidt, H. Hasse, E. Witt, C. Chevalley (among others) are closely connected with that development.
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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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
Harbingers of Artin’s Reciprocity Law, I. The Continuing Story of the Auxiliary Primes, preprint 2011
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ON ARTIN LFUNCTIONS
"... Artin spent the first 15 years of his career in Hamburg. Weil characterized this period of Artin’s career as a “love affair with the zeta function ” [72]. Chevalley, in his obituary of Artin [12], pointed out that Artin’s use of zeta functions was to discover exact algebraic facts as opposed to esti ..."
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Artin spent the first 15 years of his career in Hamburg. Weil characterized this period of Artin’s career as a “love affair with the zeta function ” [72]. Chevalley, in his obituary of Artin [12], pointed out that Artin’s use of zeta functions was to discover exact algebraic facts as opposed to estimates or approximate evaluations. In particular, it seems clear to me that during this period Artin was quite interested in using the Artin Lfunctions as a tool for finding a nonabelian class field theory, expressed as the desire extend results from relative abelian extensions to general extensions of number fields. Artin introduced his Lfunctions attached to characters of the Galois group in 1923 in hopes of developing a nonabelian class field theory. Instead, through them he was led to formulate and prove the Artin Reciprocity Law the crowning achievement of abelian class field theory. But Artin never lost interest in pursuing a nonabelian class field theory. At the Princeton University Bicentennial Conference on the Problems of Mathematics held in 1946 “Artin stated that “My own belief is that we know it already, though no one will believe me – that whatever can be said about nonAbelian class field theory follows from what we know now, since it depends on the behavior of the broad field over the intermediate fields – and
Weak Composite DiffieHellman ∗
, 2005
"... In1985, Shmuley proposed a theorem about intractability of Composite DiffieHellman. The theorem of Shmuley may be paraphrased as saying that if there exist a probabilistic polynomial time oracle machine which solves the DiffieHellman modulo an RSAnumber with oddorder bases then there exist a pro ..."
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In1985, Shmuley proposed a theorem about intractability of Composite DiffieHellman. The theorem of Shmuley may be paraphrased as saying that if there exist a probabilistic polynomial time oracle machine which solves the DiffieHellman modulo an RSAnumber with oddorder bases then there exist a probabilistic algorithm which factors the modulo. In the other hand Shmuely proved the theorem only for oddorder bases and left the evenorder case as an open problem. In this paper we show that the theorem is also true for evenorder bases. Precisely speaking we prove that even if there exist a probabilistic polynomial time oracle machine which can solve the problem only for evenorder bases still a probabilistic algorithm can be constructed which factors the modulo in polynomial time for more than 98 % of RSAnumbers.
Deciding Properties of Polynomials without Factoring
, 1997
"... The polynomial time algorithm of Lenstra, Lenstra, and Lovasz [17] for ..."