Analysis of Parallel Montgomery Multiplication in CUDA

For a given level of security, elliptic curve cryptography (ECC) offers improved efficiency over classic public key implementations. Point multiplication is the most common operation in ECC and, consequently, any significant improvement in perfor- mance will likely require accelerating point multiplication. In ECC, the Montgomery algorithm is widely used for point multiplication. The primary purpose of this project is to implement and analyze a parallel implementation of the Montgomery algorithm as it is used in ECC. Specifically, the performance of CPU-based Montgomery multiplication and a GPU-based implementation in CUDA are compared

Granatstein on Montgomery

Review of Stephen Brooks, ed. Montgomery and the Eighth Army: A Selection from the Diaries, Correspondence and Other Papers of Field Marshal the Viscount Montgomery of Alamein, August 1942 to December 1943. London: The Army Records Society, 1991

Realizing arbitrary-precision modular multiplication with a fixed-precision multiplier datapath

Within the context of cryptographic hardware, the term scalability refers to the ability to process operands of any size, regardless of the precision of the underlying data path or registers. In this paper we present a simple yet effective technique for increasing the scalability of a fixed-precision Montgomery multiplier. Our idea is to extend the datapath of a Montgomery multiplier in such a way that it can also perform an ordinary multiplication of two n-bit operands (without modular reduction), yielding a 2n-bit result. This conventional (nxn->2n)-bit multiplication is then used as a “sub-routine” to realize arbitrary-precision Montgomery multiplication according to standard software algorithms such as Coarsely Integrated Operand Scanning (CIOS). We show that performing a 2n-bit modular multiplication on an n-bit multiplier can be done in 5n clock cycles, whereby we assume that the n-bit modular multiplication takes n cycles. Extending a Montgomery multiplier for this extra functionality requires just some minor modifications of the datapath and entails a slight increase in silicon area

Children's bodies: the battleground for their rights?

The UNCRC has changed profoundly ideas about adult/child relationships and there is now an acknowledgment in both law and policy that children have a right to be consulted and to participate in decisions made about their lives. This has been widely discussed and critiqued and one of the most significant battlegrounds for debate has been children’s rights to consent or refuse medical treatment and the issue of exactly who has the right to control children’s bodies. This article will compare several cases where the English and Scottish courts have made various decisions and rulings about the extent to which children do have rights to control their bodies. It will question why, twenty years after the UK ratified the UNCRC, children are still considered incompetent in matters concerning their own bodies, unless proved otherwise, while adults are automatically considered competent unless shown not to be and will analyse whether this situation is compatible with a children’s rights agenda

Edwards curves and CM curves

Edwards curves are a particular form of elliptic curves that admit a fast, unified and complete addition law. Relations between Edwards curves and Montgomery curves have already been described. Our work takes the view of parameterizing elliptic curves given by their j-invariant, a problematic that arises from using curves with complex multiplication, for instance. We add to the catalogue the links with Kubert parameterizations of X0(2) and X0(4). We classify CM curves that admit an Edwards or Montgomery form over a finite field, and justify the use of isogenous curves when needed