We reduced the unsolved problem of Evans triangles to the problem of nding rational points on a class of elliptic curves, the twisted Edwards curves En. We proved that the set of all Evans triangles with a given ratio is isomorphic to a quotient group of En(Q). After this, using Sage, we can solve the Evans problem for an amount of cases within the capacity of computer. Even without the help of a computer, we can still obtain a large class of new Evans triangles, and nd an eective way to test if the rank of an Edwards curve is positive or not.

Nowadays, pictures have more and more applications in various fields. Because different situations may have different commands of the ratios of pictures, we present a program which can resize the pictures according to their contents. Compared with the traditional method which only cares about the geometric constraints, our program can reserve the important contents of the pictures, which means that the visual effect of the pictures will not be damaged after the resizing. In order to change the ratio of pictures while avoiding distorting their main contents, we use a method called seam carving. Based on the energy function used for finding useless pixels and dynamic programming used for finding some continuous seams which will be deleted later, our program can finally change the image to a new ratio perfectly. Due to some deficiencies of the basic program, we also make some improvements. The first one is the improvement of the algorithm: we add face detection to the basic program. Not only does this improvement can solve many problems of the basic program automatically, but also it is our own innovation of the study of content-aware resizing. The second one is the addition of customization: we allow users to select the specific contents that they want to reserve, which can meet the different needs of users. It’s also our own innovation which will be very useful in practical situations. Our program can have many practical applications. We have some ideas about how to put this algorithm into real use, which is one of our practical innovations of this topic. In terms of different ratios of photo printing, graphic designing and screen displaying, this program will have various uses, which will make a groundbreaking progress in practice.

In this report, we have mainly developed an abstract framework of k bit (t,n) secret sharing framework for cloud storage. Such framework is clean and it can be implemented. A successful implementation of the framework would provide users with protection when the system is under the attack on its confidentiality,integrity and reliability. Furthermore, such system has its own encryption by using permutations and tailor made error detection, location and data rescue. We make use of Lagrange polynomials and take the advantages of the algebraic property “t distinct points on the plane can uniquely determine a polynomial function of degree t-1” to design a k bit (t,n) -secret sharing distributed storage. We employ the set with unique factorization property (UFP) so that we simply need to calculate the y intercept of a Lagrange polynomial and then use a look up table to recover a secret.Moreover, the set which has minimum UFP would help us to design storage with smallest containers. In addition to the algebraic methods, we can utilize the geometric facts that three non collinear points determine a unique circle and four non coplanar points determine a unique sphere to construct k bit (3,n) and k bit (4,n) secret sharing storage respectively. To generalize to arbitrary case, it is straight forward if we have defined the Haar measure on the higher dimensional unit sphere. The last method is an application of Chinese Remainder Theorem (CRT) and we have designed k bit (t,n) secret sharing distributed storage and one of the designs can produce containers with the half size of the original secret. However, such k is no longer unrestricted and it has to be chosen from an interval. We have developed a C program for implementing both algebraic, geometric k bit (3,n) secret sharing distributed storages as well as the CRT method. The performances of both algebraic and geometric designs are satisfactory in term of processing time and compressed container size. The container size is even half of the size of the original secret in the CRT case and it is also very speedy.

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A topic for study derives from an interesting problem on the internet.Some significative conclusions about the problem of number ring transformation are made in the paper. Firstly, an assumption that a given starting number ring could be transformed to the end ring in finite degree is proved. Secondly, transformation degree is proved to be unconcerned with the order of the transformation. Furthermore, a function of computing transformation degree is worked out. Finally, the same end ring is gained with a given starting number ring transformed in different sequence.