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132 Chapter 5. Complex multiplication and elliptic curves5.3.4 Computing the Hilbert class polynomial . . . . . . . . . . . . . . . . . 1525.3.5 Shimura’s reciprocity law and class invariants . . . . . . . . . . . . . . 1535.4 Elliptic curves in cryptography . . . . . . . . . . . . . . . . . . . . . 1535.4.1 Curves with a given number of points . . . . . . . . . . . . . . . . . . 1535.4.2 The MOV/Frey–Rück attack . . . . . . . . . . . . . . . . . . . . . . . 1555.4.3 Identity-based cryptography . . . . . . . . . . . . . . . . . . . . . . . . 156In Part II (hyper)elliptic curves were introduced through the connections between their numbersof points and values of different exponential sums over finite fields. A particular emphasis was puton the structure of their groups of rational points and on the efficiency of point counting on suchcurves, thus leading to the design of efficient tests for (hyper-)bentness and efficient algorithms togenerate (hyper-)bent functions. It was also brought to light that elliptic curves with complexmultiplication were an important theoretical tool; the number of such curves with a given numberof points was used to count the number of exponential sums with a given value.It is a fact that (hyper)elliptic curves with complex multiplication enjoy a number of verypleasant properties, not only for the application mentioned above, but also for several applicationsin asymmetric cryptography. Therefore, we will now depart from the world of Boolean functionsto solely concentrate our efforts on the explicit construction of such curves and the computationof class polynomials.The theory of elliptic curves with complex multiplication is quite well understood nowadays.This chapter, which builds essentially upon a course taught at Télécom ParisTech, aims at givingan overview of this theory and the construction of class polynomials for general orders throughthe complex analytic method, and so at introducing the unfamiliar reader to the more generaland involved theory of abelian varieties with complex multiplication which will be treated inChapter 6. For the most of this chapter we will omit the proofs of the very classical results weintroduce, but provide references to classical textbooks where they can be found. If one referencehad to be chosen, we would refer the reader to the classical textbooks 1 of Silverman [244, 245]which were more than a source of inspiration for the material presented here.Section 5.1 gives further background on the algebraic theory of elliptic curves, whereasSection 5.2 develops a completely different point of view: over the complex numbers, ellipticcurves can indeed be described as complex tori. Such a description is especially useful in Section 5.3when studying the endomorphism rings and the isomorphism classes of complex elliptic curveswith complex multiplication by a given order. Moreover, this approach will not only lead to thepowerful main theorem of complex multiplication, but also to the explicit construction of classpolynomials and elliptic curves with complex multiplication. To conclude this chapter, someapplications of elliptic curves — and particularly of elliptic curves with complex multiplication —in asymmetric cryptography are presented in Section 5.4.5.1 Further background on elliptic curvesIn this section we give further background on algebraic curves and elliptic curves. Unless statedotherwise, all curves we consider are projective.5.1.1 Morphisms between algebraic curvesIt is a basic property of projective varieties, or more generally of complete varieties, that theimage of a morphism is closed. For morphisms between projective curves, this fact gives the1 We could not resist choosing two books.