| This paper consists of three chapters. In chapter 1, we suppose n is a compos-ite, Zn is residue class ring mod n, r(x) ∈Zn[x] is a monic irreducible polynomialof degree k(k > 0). We give a definition for n is Generalized Carmichael numbersof order k modulo r(x) is denoted by Ck,r(x). So we give another definition for Ck:Ck = {∪Ck,r(x)| r(x) are all monic irreducible polynomial of degree k(k > 0) overZn}. Clearly, C1 is the ordinarily Carmichael numbers.We abtain a necessary and suf-ficient condition for n ∈Ck,r(x). Moreover, we get hold of a necessary and sufficientcondition for n ∈Ck. We get hold of an easily calculate necessary and sufficient con-dition for n ∈C2, a necessary condition and two easily calculate sufficient conditionfor n ∈C3 thorough researching on the property of C2 and C3. In addition, we proveC1(?)C2, C1(?) C3, C2(?) C3 and |C2| = ∞. So, we have answered a part of the twoproblems of R.Bhattacharjee and P.Pandey.In 2002, Agrawal, Kayal and Saxena presented a remarkable algorithm(the AKSalgorithm). It was the deterministic polynomial-time primality testing algorithm thatdetermines whether an input number is prime or composite. After the AKS algorithmwas posed, many researchers improved the AKS algorithm. One of the better improve-ments dued to Bernstein (the Bernstein algorithm). In charper 2, we center around theAKS algorithm and the Bernstein algorithm. The first task is showing the correctnessof the two algorithm. The second task is analyze the two algorithm in detail. For eachstep implementation we give a detailed algorithm. We demonstrate some ameliora-tion that we implement the AKS algorithm and the Bernstein algorithm by using Clanguage and assemble language on my machine.In chapter 3, we suppose n = pq and p,q are odd primes, Zn is residue class ringmod n. We discuss some fundamental properties of elliptic curves over ring Zn in de-tail. We propose a necessary and sufficient condition under which En(a,b) has a pointof order Mn = lcm{#Ep(a,b), #Eq(a,b)} and show by an example that En(a,b) mayhave a point G of order Mn even if Ep(a,b) is a cyclic group and Eq(a,b) is not. QVdigital signature scheme, SOM key exchange protocol and QV key exchange protocolwere based on an elliptic curve En(a,b) over the ring Zn with a point G of order Mn(Gis base point). They pointed out that such a base point G exists if Ep(a,b) and Eq(a,b)are both cyclic groups. This restricts the choice of elliptic curves used to implementtheir digital signature scheme and key exchange protocols. And we give a new digitalsignature scheme and a new three or more users key exchange protocol with a point oforder lcm{n1,m1} as base point, where n1, m1 are respectively the order of the maxi-mal cyclic subgroups of Ep(a,b) and Eq(a,b). Our digital signature scheme improvesKMOV and QV scheme, our key exchange protocol improves SOM and QV protocols.Our generalization makes it possible to choose more elliptic curves over the ring Zn toestablish digital signature scheme and key exchange protocol.
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