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Efficient Algorithms for GCD and Cubic Residuosity
 IN THE RING OF EISENSTEIN INTEGERS, FCT ’03, LNCS 2751
, 2003
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Survey of Computational Assumptions Used in Cryptography Broken or Not by Shor's Algorithm
, 2001
"... We survey the computational assumptions of various cryptographic schemes, and discuss the security threat posed by Shor's quantum algorithm. ..."
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We survey the computational assumptions of various cryptographic schemes, and discuss the security threat posed by Shor's quantum algorithm.
Efficient Algorithms for Computing the Jacobi Symbol (Extended Abstract)
 JOURNAL OF SYMBOLIC COMPUTATION
, 1998
"... We present two new algorithms for computing the Jacobi Symbol: the rightshift and leftshift kary algorithms. For inputs of at most n bits in length, both algorithms take O(n 2 = log n) time and O(n) space. This is asymptotically faster than the traditional algorithm, which is based in Euclid' ..."
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We present two new algorithms for computing the Jacobi Symbol: the rightshift and leftshift kary algorithms. For inputs of at most n bits in length, both algorithms take O(n 2 = log n) time and O(n) space. This is asymptotically faster than the traditional algorithm, which is based in Euclid's algorithm for computing greatest common divisors. In practice, we found our new algorithms to be about two to three times faster for inputs of 100 to 1000 decimal digits in length. We also present parallel versions of both algorithms for the CRCW PRAM. One version takes O ffl (n= log log n) time using O(n 1+ffl ) processors, giving the first sublinear parallel algorithms for this problem, and the other version takes polylog time using a subexponential number of processors.
AN EFFICIENT SEVENTH POWER RESIDUE SYMBOL ALGORITHM
"... Communicated by xxx Power residue symbols and their reciprocity laws have applications not only in number theory, but also in other fields like cryptography. A crucial ingredient in certain public key cryptosystems is a fast algorithm for computing power residue symbols. Such algorithms have only be ..."
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Communicated by xxx Power residue symbols and their reciprocity laws have applications not only in number theory, but also in other fields like cryptography. A crucial ingredient in certain public key cryptosystems is a fast algorithm for computing power residue symbols. Such algorithms have only been devised for the Jacobi symbol as well as for cubic and quintic power residue symbols, but for no higher powers. In this paper, we provide an efficient procedure for computing 7th power residue symbols. The method employs arithmetic in the field Q(ζ), with ζ a primitive 7th root of unity, and its ring of integers Z[ζ]. We give an explicit characterization for an element in Z[ζ] to be primary, and provide an algorithm for finding primary associates of integers in Z[ζ]. Moreover, we formulate explicit forms of the complementary laws to Kummer’s 7th degree reciprocity law, and use Lenstra’s normEuclidean algorithm in the cyclotomic field.
UNIVERSITY OF CALGARY On Residue Symbols and Kummer’s Reciprocity Law of Degree Seven
, 2009
"... Reciprocity laws and their residue symbols have applications not only in number theory but also in other fields like cryptography. In their effort to develop cryptosystems with security equivalent to the difficulty of integer factorization, mathematicians utilized these objects to develop such schem ..."
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Reciprocity laws and their residue symbols have applications not only in number theory but also in other fields like cryptography. In their effort to develop cryptosystems with security equivalent to the difficulty of integer factorization, mathematicians utilized these objects to develop such schemes in cyclotomic fields of degree λ−1, where λ = 2, 3, and 5. A crucial part of such schemes is an efficient and fast residue symbol algorithm. Such algorithms were devised for λ = 2, 3, 5 but not for 7. Here we develop a fast and efficient residue symbol algorithm for λ = 7. We accomplish this by giving explicit conditions on integers in Q(ζ) (with ζ a primitive 7th root of unity) to be primary, by formulating explicit forms of the complementaries to Kummer’s 7th degree reciprocity law, and by using the normEuclidean algorithm in Q(ζ). We also reformulate the complementaries we obtain using Dickson’s system of quadratic Diophantine equations. ii Acknowledgements I am greatly indebted to the people who helped with the completion of this thesis. I would like to express my deepest gratitude to my supervisor, Dr. Renate Scheidler
Efficient Cryptosystems From 2 kth Power Residue Symbols ⋆
"... Abstract. Goldwasser and Micali (1984) highlighted the importance of randomizing the plaintext for publickey encryption and introduced the notion of semantic security. They also realized a cryptosystem meeting this security notion under the standard complexity assumption of deciding quadratic resid ..."
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Abstract. Goldwasser and Micali (1984) highlighted the importance of randomizing the plaintext for publickey encryption and introduced the notion of semantic security. They also realized a cryptosystem meeting this security notion under the standard complexity assumption of deciding quadratic residuosity modulo a composite number. The GoldwasserMicali cryptosystem is simple and elegant but is quite wasteful in bandwidth when encrypting large messages. A number of works followed to address this issue and proposed various modifications. This paper revisits the original GoldwasserMicali cryptosystem using 2 kth power residue symbols. The soobtained cryptosystems appear as a very natural generalization for k ≥ 2 (the case k = 1 corresponds exactly to the GoldwasserMicali cryptosystem). Advantageously, they are efficient in both bandwidth and speed; in particular, they allow for fast decryption. Further, the cryptosystems described in this paper inherit the useful features of the original cryptosystem (like its homomorphic property) and are shown to be secure under a similar complexity assumption. As a prominent application, this paper describes an efficient lossy trapdoor function based thereon.