Trinomial

Abstract: An efficient algorithm for the multiplication in GF(2/sup m/) was introduced by Mastrovito. The space complexity of the Mastrovito multiplier for the irreducible trinomial x/sup m/+x+1 was given as m/sup 2/-1 XOR and m/sup 2/ AND gales. In this paper, we describe an architecture based on a new formulation of the multiplication matrix and show that the Mastrovito multiplier for the generating trinomial x/sup m/+x/sup n/+1, where m/spl ne/2n, also requires m/sup 2/-1 XOR and m/sup 2/ AND gates, However, m/sup 2/-x/sup m/2/ XOR gates are sufficient when the generating trinomial is of the form x/sup m/+x/sup m/2/+1 for an even m. We also calculate the time complexity of the proposed Mastrovito multiplier and give design examples for the irreducible trinomials x/sup 7/+x/sup 4/+1 and x/sup 6/+x/sup 3/+1.

Abstract: We present a software implementation of arithmetic operations in a finite field GF(2n), based on an alternative representation of the field elements. An important application is in elliptic curve crypto-systems. Whereas previously reported implementations of elliptic curve cryptosystems use a standard basis or an optimal normal basis to perform field operations, we represent the field elements as polynomials with coefficients in the smaller field GF(216). Calculations in this smaller field are carried out using pre-calculated lookup tables. This results in rather simple routines matching the structure of computer memory very well. The use of an irreducible trinomial as the field polynomial, as was proposed at Crypto'95 by R. Schroeppel et al., can be extended to this representation. In our implementation, the resulting routines are slightly faster than standard basis routines.

Abstract: In options markets where there is a sign$cant or persistent volatility smile, implied tree models can ensure the consistency o f exotic options prices with the market prices o f liquid standard options. Implied trees can be constructed in a variety o f ways. Implied binomial trees are minimal. They have just enough parameters node prices and transition probabilities to f i t the smile. Trinomial trees inherently have more parameters than binomial trees. We can use these additional parameters to conveniently choose the “state space” o f all node prices in the trinomial tree, and let only the transition probabilities be constrained by market options prices. Thisjeedom ofstate space provides a flexibility that is advantageous in matching trees to smiles. A judiciously chosen state space is needed to obtain a reasonablefit to the smile. We discuss a simple method for building ‘?skewed” state spaces that f i t typical index option smiles rather well.

Abstract: New structures of bit-parallel weakly dual basis (WDB) multipliers over the binary ground field are proposed. An upper bound on the size complexity of bit-parallel multiplier using an arbitrary generating polynomial is given. When the generating polynomial is an irreducible trinomial x/sup m/+x/sup k/+1, 1/spl les/k/spl les/[m/2], the structure of the proposed bit-parallel multiplier requires only m/sup 2/ two-input AND gates and at most m/sup 2/-1 XOR gates. The time delay is no greater than T/sub A/+([log/sub 2/ m]+2)T/sub x/, where T/sub A/ and T/sub X/ are the time delays of an AND gate and an XOR gate, respectively.