The purpose of this research was to determine if recognizable error types exist in the work of preservice teachers required to create story problems for specific fraction operations. Students were given a particular single-operation fraction expression and asked to do the calculation and then create a story problem that would require the use of both the expression and calculation to answer the story problem. Distinct error types, determined by mathematical and grammatical characteristics, were identified by the researchers. These error types were then used to analyze an additional set of work samples to determine frequency rates of error types. Future research will involve evaluating the efficacy of specific instructional methods with the goal of reducing the frequency of certain error types. 88 Volume 112 (2)

Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy under Contract DE-AC04-94AL85000.

We present a VHDL design that incorporates optimizations intended to provide digital signature generation with as little power, space, and time as possible. These three primary objectives of power, size, and speed must be balanced along with other important goals, including flexibility of the hardware and ease of use. The highest-level function offered by our hardware design is Elliptic Curve Optimal El Gamal digital signature generation. Our parameters are defined over the finite field GF (2 178), which gives security that is roughly equivalent to that provided by 1500-bit RSA signatures. Our optimizations include using the point-halving algorithm for elliptic curves, field towers to speed up the finite field arithmetic in general, and further enhancements of basic finite field arithmetic operations. The result is a synthesized VHDL digital signature design (using a CMOS 0.5µm, 5V , 25 • C library) of 191,000 gates that generates a signature in 4.4 ms at 20 MHz.

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