Efficient Algorithms for Multi-Scalar Multiplications

Since the internet was made accessible for the public, it is used more and more for the exchange of confidential data. Over the past few years, the number of electronic frauds increased and nowadays poses a serious threat. It is therefore extremely important to have methods to secure transactions and communications made via the internet, or in general in any electronic environment. The science that made security in electronic environments its business is called cryptography. Nowadays, there exist several methods, so-called cryptosystems, which enable users to secure their communications. Due to their tamper resistance and mobility, cryptosystems are often implemented on smart cards. However, since smart cards have only the size of a credit card, their computational power and memory is very limited. It is therefore crucial to compute the operations required by a cryptosystem as efficient as possible. The basic mathematic operation in cryptosystems are scalar multiplications and more general, sums of scalar multiplications, so-called multi-scalar multiplications. This thesis analyzes several methods to compute a multi-scalar multiplication in an efficient way. Here efficient means not only as fast as possible, but also using as little memory as possible. In detail, there exist several basic algorithms to compute a multi-scalar multiplication. The runtime of those algorithms can be decreased if special representations of the scalars are deployed. The emphasis of this thesis is on such representations. This thesis is organized as follows: Chapter 1 introduces the basic concept of cryptography and smart cards. Chapter 2 discusses elliptic curves and their application to cryptography. Chapter 3 introduces the basics about integer representations. Chapter 4 reviews several basic algorithms to compute a multiscalar multiplication and Chapters 5 and 6 introduce special representations of the scalars to speed up those algorithms. In Chapter 7, the author compares those representations and in Chapter 8, he estimates the total computational costs for computing a multi-scalar multiplication. Finally, Chapter 9 states the authors conclusion.