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By Dupre M.J., Glazebrook J.F., Prevlato E.

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51. Olsen, J. , Scholtz, R. A. and L. R. Welch. Bent function sequences, IEEE Trans. on Inf. Theory , vol IT- 28, n◦ 6, 1982. 52. E. Pasalic. Degree optimized resilient Boolean functions from Maiorana-McFarland class. In 9th IMA Conference on Cryptography and Coding, 2003. 53. O. S. Rothaus. On “bent” functions, J. Comb. Theory, 20A, 300-305, 1976. 54. E. Pasalic, T. Johansson, S. Maitra and P. Sarkar. New constructions of resilient and correlation immune Boolean functions achieving upper bounds on nonlinearity.

Set r = H3 (σ, m) and check whether C1 = g r holds. If not, rejects the decryption. Otherwise, output m. , assuming H1 , . . , H4 are truly random functions) under the BDH assumption. 42 T. 3 Boneh-Boyen IBE Scheme There are three Boneh-Boyen IBE schemes that are secure in the standard model (two are in [7] and one is in [9]). One of the two schemes in [7] is IND-sID-CPA secure, and the other is INDsID-CCA2 secure. The IND-sID-CCA2 secure scheme [7] is constructed by converting from the IND-sID-CPA secure basic scheme through the conversion technique of [19].

For example, the typical key size of ECC that guarantees the security comparable to that of 1024 bit key size with the RSA cryptosystems is considered to be just 160 bits. Therefore, several of the most eﬃcient public-key encryption schemes and digital signatures are ECC such as EC-ElGamal (the elliptic curve version of ElGamal) and EC-DSA. The reason why ECC has such short key lengths is that the index calculus technique is considered to be ineﬀective for computing the discrete logarithm (DL) of the elliptic curve group over ﬁnite ﬁelds, while it can eﬀectively compute integer factoring and DL of the multiplicative group of a ﬁnite ﬁeld.

### A Banach Algebra Version of the Sato Grassmannian and Commutative Rings of Differential Operators by Dupre M.J., Glazebrook J.F., Prevlato E.

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