T. Satoh and K. Araki, "Fermat quotients and the polynomial time discrete log algorithm for anomalous elliptic curves," Commentarii Mathematici Universitatis Sancti Pauli, 47, pp.81-92, 1998.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....a subexponential time algorithm for the DLP in JC (F q ) For example, if C is a supersingular elliptic curve de ned over a nite eld, then k 2 f1; 2; 3; 4; 6g. If C is an algebraic curve de ned over a prime eld F p such that #JC (F p ) p, then the DLP in JC (F p ) can be eciently solved [66, 69, 74, 64]. The curves that appear to be the most attractive for cryptographic applications are low genus hyperelliptic curves. This is because very ecient algorithms are known for the group law and because of the absence of subexponential time algorithms for the discrete logarithm problem for carefully ....

T. Satoh and K. Araki, \Fermat quotients and the polynomial time discrete log algorithm for anomalous elliptic curves", Commentarii Mathematici Universitatis Sancti Pauli, 47 (1998), 81-92.

....has been supported by the European Commission through the IST Programme under Contract IST 1999 12324. 1 cryptosystems such as ECIES or ECDSA. Some of these attacks rely on a peculiarity of a curve or a family of curves, such as supersingular elliptic curves [13] or elliptic curves of trace one [22, 18, 19]. Nevertheless, in general, such elliptic curves can be easily avoided, except in the case when the field F h n is intrinsically weak, and this may happen when h = 2 with l 4 [23] Indeed Galbraith Gaudry HessSmart devised a practical implementation of the Weil descent to compute discrete ....

T. Satoh and K. Araki. Fermat Quotients and the Polynomial Time Discrete Log Algorithm for Anomalous Elliptic Curves. Commentarii Math. Univ. St. Pauli, 47:81--92, 1998.

....over the J(C; F q ) to the logarithm problem in an extended field k . It is e#cient especially for supersingular curves, see [5] 3. When q is prime, there should be no subgroup of order q in J(C; F q ) Because there is an attack on anomalous curves investigated by Semaev [19] Satoh and Araki [18],Smart [21] for elliptic and generalized by Ruck for hyperelliptic curves in [16] 4. 2g 1 log q. When 2g 1 log q, Adleman, DeMarrais and Huang gave a sub exponential time algorithm to solve HCDLP in [1] Further study by Gaudry in [7] suggested that g 4. Therefore, We will consider ....

T. Satoh, and K. Araki, Fermat quotients and the polynomial time discrete log algorithm for anomalous elliptic curves, Commentari Math. Univ. St. Pauli 47 (1998), 81-92.

....not apply to a particular curve, one only needs to check that r, the order of the point P , does not divide q B 1 for all small B for which the DLP in F q B is intractable (1 B 2000= log 2 q ) su ces) This is known as the MOV condition. 2. Anomalous curves. Smart [53] and Satoh and Araki [47] independently showed that the ECDLP for the special class of anomalous elliptic curves is easy to solve. An anomalous elliptic curve over F q is an elliptic curve over F q which has exactly q points. This attack does not extend to any other classes of elliptic curves. Consequently, by verifying ....

T. Satoh and K. Araki. Fermat quotients and the polynomial time discrete log algorithm for anomalous elliptic curves. 1997.

.... by P has smooth order or when there is a mapping from E into a small degree extension of the base eld F q (such a mapping can arise from the Weil or Tate pairings, see Menezes, Okamoto and Vanstone [24] Frey and R uck [12] or from taking p adic logarithms, see Semaev [32] Satoh and Araki [29], Smart [35] or R uck [28] For the rest of this section we will assume that all elliptic curves in question do not belong to one of these special cases. For these remaining elliptic curves, the only known methods for solving the elliptic curve discrete logarithm problem involve reducing to ....

T. Satoh, K. Araki, Fermat quotients and the polynomial time discrete log algorithm for anomalous elliptic curves, Commentarii Mathematici, Vol. 47 No. 1 (1998) 81-92

....avoid special curves one only has to look at the history of elliptic curve cryptography. In the past various authors proposed using supersingular or anomalous curves as they offered some advantages over other more general curves. However, both types of curves are now known to be weak, see [14] [22], 24] and [25] Hence it is probably worth adopting the principle of always avoiding special curves of any shape or form. In the current authors opinion this is the major open problem with using hyperelliptic curves for cryptographic purposes: How to choose a suitable curve efficiently. 6 3. ....

....due to Frey and Ruck [9] 2. Anomalous curves over F p and in general curves which have a large subgroup of order p in a field of characteristic p. This attack uses a generalization due to Ruck [21] of the anomalous curve attack for elliptic curves due to Semaev, Satoh, Araki and Smart, see [22], 24] and [25] However, such cases are easy to check for and only eliminate a small fraction of all possible curves. For hyperelliptic curves the most interesting case, from a theoretical standpoint, is when the genus is large in comparison to the size of the field of definition of the ....

T. Satoh and K. Araki. Fermat quotients and the polynomial time discrete log algorithm for anomalous elliptic curves. Comm. Math. Univ. Sancti Pauli, 47, 81--92, 1998.

....method by n 2 . For genus 2 and 3 this does not a ect the security of our system. But for genus 4 we need to be aware of that e ect and either avoid these curves or choose a larger exponent. Furthermore there is an attack on anomalous curves investigated by Semaev [48] see also Satoh and Araki [47], and Smart [50] for elliptic and by R uck [46] for hyperelliptic curves. This works for groups of order a multiple of p r where p is the characteristic of the ground eld. But the hyperelliptic Koblitz curves we use do not lead to a curve which is weak under that attack since we work in the ....

T. Satoh, K. Araki, Fermat quotients and the polynomial time discrete log algorithm for anomalous elliptic curves, Commentari Math. Univ. St. Pauli 47 (1998), 81-92.

.... MIYAJI , Member, Masaki NAKABAYASHI , and Shunzou TAKANO , Nonmembers SUMMARY Ellipticcurv e cryptosystems( 19] 25] are based on the ellipticcurv e discrete logarithm problem (EC LP) If ellipticcurv e cryptosystems av oid FRreduction ( 11] 17] and anomalous ellipticcurv e ov er #q ([3], 33] 35] then with current knowledge we can construct ellipticcurv e cryptosystems ov er a smaller definition field. EC LP has an interesting property that the security deeply depends on ellipticcurv e traces rather than definition fields, which does not occur in the case of the discrete ....

....1. Introduction Koblitzand Miller proposed ind end tly a public key cryptosystembased on an elliptic curve E d2EO8 over a finitefield F q (q = p r ) 19] 5] If elliptic curve cryptosystems satisfy socalled FRcondO0GO2 ( 11] 17] 4] and avoid anomalous elliptic curve over F q ([3], 33] 35] then the only known attacks are the Pollard # method ( 7] and the Pohlig Hellman method ( 6] Hence with current knowledEL we can construct elliptic curve cryptosystems over a smallerdaller2L field than thede2OAOS logarithm problem(DLP) based cryptosystems like the ElGamal ....

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K. Araki and T. Satoh "Fermat quotients and the polynomial time discrete log algorithm for anomalous ellipticcurv es", Commentarii Math. Univ. St. Pauli., v ol. 47 (1998), 81-92.

...., and the results in Theorem 3 and Corollary 3 will still hold. Therefore, can be considered as a map from F q to F 2 . Remark 5 If q = p, that is, the curve de ned over F p has order p, in this case the discrete logarithm problem over such curves is solved by Samaev, Smart, and Satoh and Araki [17,18,16]. 4 Binary Sequences Obtained from Supersingular Elliptic Curves with f(x) x 1 For the case n = 1, we take f(x) x 1; c 1 = 1 and Q = P 0 . Then we have P k = P k 1 P 0 ; k = 1; 2; Thus P k = k 1)P 0 ; k = 0; 1; 12) Linear Recursive Sequences over Elliptic Curves 13 ....

T. Satoh and K. Araki, Fermat quotients and the polynomial time discrete log algorithm for anomalous elliptic curves, Commentarii Mathematici Universitatis Sancti Paul, 47 (1998), 81-92.

.... goal was to trapdoor the elliptic curve discrete logarithm problem on an elliptic curve over a ring Z=NZwhere N is a product pq of primes, by using the fact that it is easy to solve the elliptic curve discrete logarithm problem on an anomalous curve over a prime field F p using the methods of [12], 13] 15] However, Okamoto and Uchiyama realised that their attempts to build such a system were unsuccessful, in that it was possible to derive the factorisation of N from the public information of their system. In general, it seems unlikely that cryptosystems based on anomalous elliptic ....

T. Satoh, K. Araki, Fermat quotients and the polynomial time discrete logarithm algorithm for anomalous elliptic curves, Commentarii Mathematici, 47, No. 1 (1998), pp. 81--92.

....properties that are useful for counting points or for accelerating arithmetic operations occurring in the protocols. However choosing such curves can be dangerous. Perhaps the most striking example is trace 1 curves. The number of points over F q is simply q. However Smart [Sma99] Satoh Araki [SA98] and Semaev [Sem98] independently discovered a polynomial time attack. Another attack due to Menezes Okamoto Vanstone [MOV91] and generalised by Frey R uck [FR94] reduces discrete logs on supersingular and trace 2 curves to discrete logs in a small degree extension of F q . This yields an ....

T. Satoh and K. Araki. Fermat quotients and the polynomial time discrete log algorithm for anomalous elliptic curves. Comment. Math. Univ. St. Paul., 47:81-92, 1998.

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T. Satoh and K. Araki, "Fermat quotients and the polynomial time discrete log algorithm for anomalous elliptic curves," Commentarii Mathematici Universitatis Sancti Pauli, 47, pp.81-92, 1998.

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T. Satoh and K. Araki, "Fermat quotients and the polynomial time discrete log algorithm for anomalous elliptic curves", Commentarii Mathematici Universitatis Sancti Pauli, 47 (1998), 81-92.

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T. Satoh and K. Araki. Fermat quotients and the polynomial time discrete log algorithm for anomalous elliptic curves. Commentarii Mathematici Universitatis Sancti Pauli, 47(1), 1998.

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T. Satoh and K. Araki. Fermat Quotient and the Polynomial Time Discrete Log Algorithm for Anomalous Elliptic Curves. Preprint, 1997.

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T. Satoh and K. Araki, Fermat quotients and the polynomial time discrete log algorithm for anomalous elliptic curves, Commentarii Mathematici Universitatis Sancti Pauli, 47 (1998), pp. 81-92. 45

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T. Satoh and K. Araki. Fermat quotients and the polynomial time discrete log algorithm for anomalous elliptic curves. Comment. Math. Helv., 47(1):81--92, 1998.

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T. Satoh and K. Araki, Fermat quotients and the polynomial time discrete log algorithm for anomalous elliptic curves, preprint (1997).

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T. Satoh and K. Araki. Fermat quotients and the polynomial time discrete log algorithm for anomalous elliptic curves. Comment. Math. Univ. St. Paul., 47(1):81-92, 1998.

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T. Satoh and K. Araki. Fermat Quotient and the Polynomial Time Discrete Log Algorithm for Anomalous Elliptic Curves. Preprint, 1997.

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T. Satoh and K. Araki, Fermat quotients and the polynomial time discrete log algorithm for anomalous elliptic curves, Comment. Math. Univ. St. Pauli 47 (1998), no. 1, 81--92.

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T. Satoh and K. Araki, Fermat quotients and the polynomial time discrete log algorithm for anomalous elliptic curves, Math. Univ. St. Paul. 47 (1998), 81-92.

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T. Satoh and K. Araki. Fermat quotients and the polynomial time discrete log algorithm for anomalous elliptic curves. Commentarii Mathematici Universitatis Sancti Pauli, (47):8192, 1998.

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T. Satoh and K. Araki. Fermat Quotient and the Polynomial Time Discrete Log Algorithm for Anomalous Elliptic Curves. Preprint, 1997.

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T. Satoh and K. Araki, \Fermat quotients and the polynomial time discrete log algorithm for anomalous elliptic curves", Commentarii Mathematici Universitatis Sancti Pauli, 47 (1998), 81-92.

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