FPGA based Hardware Accleration for Elliptic Curve Cryptography ...

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1.3. GOALS OF THIS STUDY 3 specific and synthesizable VHDL descriptions from a superior, generic coprocessor-model has also been adopted from this previous design. 1.3 Goals of this Study The main goal of this work is the design and the implementation of a arithmetic processor kernel which can be embedded into the previously described EC coprocessor design. After some research on existing algorithms and implementations it was decided that the main work should concentrate on the finite field arithmetic while the EC level algorithms should be adopted from the literature. The main reason for this decision was the fact, that an efficient hardware architecture has a much greater influence on the efficiency of the data flow dominated finite field algorithms than on the control flow oriented EC level algorithms. The final design should be widely scalable in terms of different types of resource usage. To provide this scalability and flexibility a generator program should be used to produce the VHDL hardware models, out of which the FPGA programming bitstream is synthesized subsequently. While it was clear that the most important parameter for scalability will be the bitwidth of the design, the complete parameter set and the resulting degree of flexibility caused by the generator approach was specified during the implementation progress. A minor goal has been the compatibility to existing modules. The interface of the new developed design should correspond to that of the existing ONB implementation. Using this goal it was possible to reuse the already optimized EC Controller from the ONB implementation without modifications. A new family of attacks against cryptographic hardware implementations is currently gaining much importance. These so called Side-Channel-Attacks use additional information the hardware provides beside the cryptographic functions to extract knowledge of the secret key. Examples for this additional information are the runtime of an operation that might depend on the secret key or the power consumption of a chip during the computation. Though it was not a main goal of this work to provide resistance against such attacks, the problem should be reminded during implementation. Where possible, simple countermeasures should be implemented. To evaluate the functionality of the hardware implementation, the results should be compared against results of a pure software implementation. To provide a framework for this evaluation process, the existing C software implementation has been extended to support- elements in polynomial representation. 1.4 Content of this work The mathematical background of elliptic curves and finite fields is introduced in the following chapter. Furthermore, the multi-segment Karatsuba multiplication scheme is described in detail. Chap. 3 focuses on the architecture and the implementation of the proposed ECC coprocessor. Special attention is given to the arithmetic processor kernel. Implementation results and some performance numbers are given in Chap. 4. Finally, Chap. 5 summarizes the conclusions and gives an outlook on work that might follow.
Chapter 2 Mathematical Background There are several cryptographic schemes based on elliptic curves. These schemes work on a subgroup of points of an EC over a finite field. Arbitrary finite fields are approved to be suitable for ECC. This work concentrates on elliptic curves over the finite field- and their arithmetics only. For further information see [11] and [12]. There are several bases known for . The most common bases, which are also proposed by the leading standards concerning ECC (IEEE 1363 [13] and ANSI X9.62 [14]) are polynomial bases and normal bases. Please remark, that the design detailed in the following is exclusively treating with polynomial basis representation. Sec. 2.1 introduces some basic facts and algorithms of elliptic curves. In Sec. 2.2 a short review on the finite field based on polynomial basis representation is given. Sec. 2.3 presents several multiplication schemes in- and leads to the multi-segment Karatsuba multiplication algorithm which is one main contribution of this work. 2.1 Elliptic Curve Arithmetic 2.1.1 Affine Coordinates An elliptic curve over is defined as the cubic equation ),5 176 598:) (2.1) /¢02143 1FE and>HG 6!I . The set of solutions J=K5D@ 1 ML 1 3 )?5 1N6 5 8 )O;P5 3 )Q>SR is called the points of the elliptic curve/ . By defining an appropriate addition operation and an extra pointT , called the with;9@A>B@C5D@ point at infinity, these points become an additive, abelian withT group the neutral element. Fig. 2.1 depicts an example of an elliptic curve over the reals. Here, a geometric interpretation of the addition can be given: Find the third intersection (-U point ) of a straight line through V and with the elliptic curve. resultU 6 Q)WV The is found by -U mirroring at the x-axis.
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