By Ian F. Blake, Gadiel Seroussi, Nigel P. Smart

Because the visual appeal of the authors' first quantity on elliptic curve cryptography in 1999 there was large development within the box. In a few subject matters, relatively element counting, the growth has been striking. different subject matters comparable to the Weil and Tate pairings were utilized in new and significant how one can cryptographic protocols that carry nice promise. Notions comparable to provable protection, aspect channel research and the Weil descent procedure have additionally grown in significance. This moment quantity addresses those advances and brings the reader modern. well-known participants to the examine literature in those components have supplied articles that mirror the present country of those vital subject matters. they're divided into the components of protocols, implementation recommendations, mathematical foundations and pairing established cryptography. all of the themes is gifted in an available, coherent and constant demeanour for a large viewers that might contain mathematicians, machine scientists and engineers.

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**Example text**

Adaptive Semi-Logarithm Resistance : For a conversion function f , group G , and e ∈ Z/qZ with e = 1, an e-shifted semi-logarithm of a group element P to the base G is a pair of integers (t, u) such that t = f ([u−1 ]([e]G+ [t]P )). The adaptive semi-logarithm problem is to ﬁnd a semi-logarithm of a point P to the base G given access to an oracle for e-shifted semi-logarithms. Adaptive semi-logarithm resistance means that the adaptive semi-logarithm problem is intractable. The adaptive semi-logarithm resembles an active forgery of ECDSA, except that no hash functions are involved.

A KEM encryption function takes as input a public key and outputs a session key and the encryption of the session key under the given public key. The KEM decryption operation takes as input a private key and the output from a KEM encryption and produces the associated session key. 16 I. ECC PROTOCOLS As mentioned before, the deﬁnition of ECIES-KEM in the draft ISO standard is slightly diﬀerent from earlier versions of ECIES. In particular, the way the ephemeral secret is processed to deal with small subgroup attacks and how chosen ciphertext attacks are avoided is changed in the following scheme.

2. (k1 k2 ) ← KD(T, l). 3. Decrypt the message m ← Dk1 (c). 4. If r = M ACk2 (c) then output ‘‘Invalid Ciphertext’’. 5. Output m. Notice that the T computed in the decryption algorithm is the same as the T computed in the encryption algorithm since Tdecryption = [x]U = [x]([k]G) = [k]([x]G) = [k]Y = Tencryption . One can show that, assuming various properties of the block cipher, key derivation function and keyed hash function, the ECIES scheme is secure against adaptive chosen ciphertext attack, assuming a variant of the Diﬃe– Hellman problem in the elliptic curve group is hard; see [1] and Chapter III.