Advances in Cryptology — ASIACRYPT 2001: 7th International by Craig Gentry, Jakob Jonsson, Jacques Stern, Michael Szydlo

By Craig Gentry, Jakob Jonsson, Jacques Stern, Michael Szydlo (auth.), Colin Boyd (eds.)

The origins of the Asiacrypt sequence of meetings should be traced again to 1990, while the ?rst Auscrypt convention used to be held, even though the identify Asiacrypt was once ?rst used for the 1991 convention in Japan. beginning with Asiacrypt 2000, the convention is now one in every of 3 annual meetings equipped by means of the Inter- tional organization for Cryptologic learn (IACR). the ongoing good fortune of Asiacrypt is in no small half as a result of the e?orts of the Asiacrypt guidance C- mittee (ASC) and the powerful aid of the IACR Board of administrators. there have been 153 papers submitted to Asiacrypt 2001 and 33 of those have been permitted for inclusion in those lawsuits. The authors of each paper, even if approved or no longer, made a valued contribution to the luck of the convention. Sending out rejection noti?cations to such a lot of not easy operating authors is without doubt one of the such a lot disagreeable projects of this system Chair. The assessment approach lasted a few 10 weeks and consisted of an preliminary refe- eing part via an intensive dialogue interval. My heartfelt thank you visit all contributors of this system Committee who installed severe quantities of time to offer their professional research and reviews at the submissions. All papers have been reviewed by means of not less than 3 committee individuals; in lots of situations, relatively for these papers submitted by means of committee contributors, extra experiences have been obt- ned. professional studies have been supplied through a military of exterior reviewers with out whom our judgements could were even more di?cult.

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Extra info for Advances in Cryptology — ASIACRYPT 2001: 7th International Conference on the Theory and Application of Cryptology and Information Security Gold Coast, Australia, December 9–13, 2001 Proceedings

Example text

The (i, j) entry in this matrix is just the coefficient in the j’th relation of the term corresponding to row i. The diagonal entries of the matrix P are all equal to p, and the diagonal entries of the matrix E correspond to the bounds on the terms associated with each row. Specifically, if the term which is associated with row i is bounded by B, then entry (i, i) in E is equal to 1/B. That is, the row corresponding to the constant term has diagonal entry 1, rows corresponding to i have diagonal entries 1/2m−k , and rows corresponding to 0 j have diagonal entries 1/22(m−k) .

This is because we can set β βr yi = (xi + s) · r−1 and then we have α+y = (αr+s)+x . 4, it follows that this problem is infeasible to solve when the number of “missing bits” is more than m/2 (as opposed to 2m/3 for the original MIHNP). This means that we may be able to output as many The Modular Inversion Hidden Number Problem 49 as m/2 − r bits for security level of 2r . Assuming that we still set r = k, this argument suggests that we must set m (the size of our prime) so that k ≤ m/2−k, or m ≥ 4k.

Dm−1 ∈ [ϕ(N )] arbitrarily, we obtain a congruence of the form fm dm ≡ D (mod ϕ(N )) which has at most gcd(fm , ϕ(N )) ≤ fm < 2 solutions dm ∈ [ϕ(N )]. Finally, for any of ϕ(N )m−2 possible choices of dm+1 , . . , d2m−2 ∈ m−2 the equation (7) gives at most one value for dm−1 and then the second [ϕ(N )] congruence in (4) gives us at most gcd(f2m , ϕ(N )) ≤ f2m < 2 possible values for d2m . So the total number of solutions for such u is at most 22 ϕ(N )2m−3 . The total number of such vectors is at most U 2m .

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