## Security of an Identity-Based Cryptosystem and the Related Reductions (1998)

Venue: | In Advances in Cryptology, Eurocrypt'98, LNCS 1403 |

Citations: | 6 - 0 self |

### BibTeX

@INPROCEEDINGS{Okamoto98securityof,

author = {Tatsuaki Okamoto and Shigenori Uchiyama},

title = {Security of an Identity-Based Cryptosystem and the Related Reductions},

booktitle = {In Advances in Cryptology, Eurocrypt'98, LNCS 1403},

year = {1998},

pages = {546--560},

publisher = {Springer Verlag}

}

### OpenURL

### Abstract

Abstract. Recently an efficient solution to the discrete logarithm prob-lem on elliptic curves over F, with p points (p: prime), so-called anorna-lous curues, was independently discovered by Semaev [14], Smart [17], and Satoh and Araki [12]. Since the solution is very efficient, i.e., 0(lpl3), the Semaev-Smart-Satoh-Araki (SSSA) algorithm implies the possibil-ity of realizing a trapdoor for the discrete logarithm problem, and we have tried to utilize the SSSA algorithm for constructing a cryptographic scheme. One of our trials was to realize an identity-based cryptosystem (key-distribution) which has been proven to be as secure as a prim-itive problem, called the Diffie-Hellman problem on an elliptic curve over Z/nZ (n = pq, p and q are primes) where Ep and E, are anoma-lous curves (anomalous En-Diffie-Hellman problem). Unfortunately we have found that the anomalous En-Diffie-Hellman problem is not secure (namely, our scheme is not secure). First, this paper introduces our trial of realizing an identity-based cryptosystem based on the SSSA algorithm, and then shows why the anomalous En-Diffie-Hellman problem is not se-cure. In addition, we generalize the observation of our breaking algorithm and present reductions of factoring n to computing the order ’ of an el-liptic curve over Z/nZ. (These reductions roughly imply the equivalence of intractability between factoring and computing elliptic curve’s order.) The algorithm of breaking our identity-based cryptosystem is considered to be a special case of these reductions, and the essential reason why our system was broken can be clarified through these reductions: En in our system is a very specific curve such that the order of En (i.e., n) is trivially known.

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Citation Context ...an elliptic curve, En, over Z/nZ [5]. Hence our Reduction 2 corresponds to the KK reduction. In addition Reduction 1 corresponds to the reduction as a variant of the KK reduction described in Remarks =-=(3)-=-, in [5]. Both the KK reduction and our Reduction 2 work for non-negligible fractions of #En. The difference between these reductions are the failure cases of reduction. The KK reduction does not work... |

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Citation Context .... Recently an efficient solution to the discrete logarithm problem on elliptic curves over F, with p points (p: prime), so-called anornalous curues, was independently discovered by Semaev [14], Smart =-=[17]-=-, and Satoh and Araki [12]. Since the solution is very efficient, i.e., 0(lpl3), the Semaev-Smart-Satoh-Araki (SSSA) algorithm implies the possibility of realizing a trapdoor for the discrete logarith... |

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Citation Context ... the Euler function 4(n) [9, 61. Kunihiro and Koyama have recently presented the reduction (say the “KK reduction”) of the factoring problem to computing the order of an elliptic curve, En, over Z/nZ =-=[5]-=-. Hence our Reduction 2 corresponds to the KK reduction. In addition Reduction 1 corresponds to the reduction as a variant of the KK reduction described in Remarks (3), in [5]. Both the KK reduction a... |

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Citation Context ...ns (2, y) E Ii' x K to the equation E : y2=z3+az+b (a,bEIi,4a3+27b2#O), together with a special point O,, called the point at infinity. The set forms a finite Abelian group, and the group law formula =-=[16]-=-(usually we call it the addition, and use the notation +) is defined over the points on I<. Let p be a prime (p > 5), and F, be a finite field with p elements. Now, let #E(F,) be the number of points ... |