By Alfred J. Menezes

Elliptic curves were intensively studied in algebraic geometry and quantity conception. lately they've been utilized in devising effective algorithms for factoring integers and primality proving, and within the building of public key cryptosystems.

*Elliptic Curve Public Key Cryptosystems* offers an updated and self-contained therapy of elliptic curve-based public key cryptology. Elliptic curve cryptosystems probably offer identical safeguard to the present public key schemes, yet with shorter key lengths. Having brief key lengths skill smaller bandwidth and reminiscence necessities and will be a vital consider a few purposes, for instance the layout of shrewdpermanent card structures. The booklet examines numerous matters which come up within the safe and effective implementation of elliptic curve systems.

*Elliptic Curve Public Key Cryptosystems* is a helpful reference source for researchers in academia, govt and who're all for problems with facts safeguard. end result of the complete therapy, the ebook is usually appropriate to be used as a textual content for complex classes at the subject.

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There are several efficient polynomial time algorithms for finding the roots of a polynomial over F2mj for example, see [10]. 7 Notes The work of Waterhouse is based on Deuring's classic paper [32). Deuring considers two elliptic curves defined over Fq to be isomorphic over Fq if they are isomorphic, in our sense, over Fq. Some of Waterhouse's work was generalized by Ruck [133) to Jacobians of algebraic curves of genus 2 over finite fields. 6 is taken from [94). Jogarithm Problem There are many public-key cryptosystems whose security lies in the presumed intractability of the discrete logarithm problem in some group G.

1 Introduction Let (%) denote the usual Jacobi symbol. We also define (i) ~: if a == ±1 mod 8, if a· == 0 mod 2, = { -1, if a == ±3 mod 8. Waterhouse [152] (see also [137]) counted the number of isomorphism classes of elliptic curves defined over the finite field Fq by first determining which rings can occur as the endomorphism ring of some elliptic curve, and then counting the number of isomorphism classes of elliptic curves with a given endomorphism ring. He also proceeded to determine Nq(t), the number of isomorphism cla~ses of elliptic curves over Fq such that #E(Fq ) = q + 1 - t.

Type II: a3 is a cube, and Te(a4) f; O. Type III: a3 is a cube, and Te( a4) = O. Type I Curves We call a Type I curve with the coefficient of x being 0, a Type Ia curve. 11) Since a3 = a3/ u 3 and a3 is a non-cube, a3 is also a non-cube. Hence E2 is also a Type I curve. 11) in F2m. 9) has exactly 3 solutions. namely Ut,CIU}, and C2Ul. 10) has exactly one solution for each choice of u. 10) are 8 = 81,CI8},C281 respectively. 11), namely tl and tl +a3' Thus there are 6 admissible changes of variables which transform El to E 2 • Since the total number of admissible changes of variables is (q -1 )q2, the number of curves isomorphic to El is (q - 1)q2/6.