By Ueli Maurer (auth.), Nigel P. Smart (eds.)
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"This ebook includes chosen refereed lawsuits of the 10th Institute of arithmetic and its purposes (IMA) foreign convention. … The publication covers a few vital study components in coding and cryptography. … i feel somebody who desires to research or layout a process for crypto-protecting delicate information will locate worthwhile details during this publication. … Books reminiscent of this are important to making sure the growth of theoretical and useful reports on cryptography, probably the most very important applied sciences within the box of computing device and community security." (Patriciu Victor-Valeriu, Computing experiences, August, 2006)
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Extra resources for Cryptography and Coding: 10th IMA International Conference, Cirencester, UK, December 19-21, 2005. Proceedings
In the k = 1 case, to avoid the point multiplication by h one might want to use a diﬀerent identity-based encryption scheme, such as the one in  or , where Alice’s public key is an integer rather than a point. Pairing-Based Cryptography at High Security Levels 9 33 Open Problems (1) Prove Verheul’s theorem for class-VI supersingular elliptic curves, which, as we saw at the end of §3, contain subgroups isomorphic to the multiplicative groups of all ﬁnite ﬁelds. (2) To what extent can the special number ﬁeld sieve be applied to Fp for Solinas primes p?
3 9 9 In the supersingular case (4) with k = 2, where a = 0 rather than −3, a multiplication can be replaced by a squaring in the point-duplication part of both the Miller lite and full Miller computations. Of course, this has no eﬀect on Table 3. 32 N. Koblitz and A. Menezes These formulas give us the time estimates in Table 3. Notice that for nonsupersingular curves Table 3 suggests that even at the 80-bit security level the choice k = 2 is less eﬃcient than higher k, and that, more generally, for k ≥ 2 large k has an advantage.
To see that Q can be chosen in this way, we consider the “twisted” elliptic curve E with equation βy 2 = x3 + ax + b. It is easy to show that if E has pk/2 + 1 − t points over the ﬁeld Fpk/2 , then E has pk/2 + 1 + t points over Fpk/2 . Over the big ﬁeld Fpk the number of points on E is equal to the product of the orders of E and its twist E over Fpk/2 . 5 Thus, there √ is a point Q ∈ E(Fpk/2 ) of order n. 3 Operation Count for k ≥ 2 When computing the Tate pairing, major savings can be obtained by ignoring terms that are contained in a proper subﬁeld of Fpk (see [18,4,48]).
Cryptography and Coding: 10th IMA International Conference, Cirencester, UK, December 19-21, 2005. Proceedings by Ueli Maurer (auth.), Nigel P. Smart (eds.)