Daniel V. Bailey. Optimal Extension Fields. Major Qualifying Project (Senior Thesis), 1998. Computer Science Department, Worcester Polytechnic Institute, Worcester, MA, USA.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... space prevents a detailed comparison, but see [Sew00] There is a large body of semantic work on concurrent and distributed algorithms. Crudely, it can be subdivided into work taking an automata theoretic approach and work on encodings of high level primitives. The former includes [AP98, JNW98] addressing Mobile IP, and [Mor99] which studies an infrastructure providing a similar abstraction to that of this paper. All involve more or less idealised models of algorithms rather than directly executable code. The latter includes encodings of choice [NP96] # join communication [FG96] ....

Daniel Jackson, Yuchung Ng, and Jeannette Wing. A Nitpick analysis of mobile IPv6. Technical Report CMU-CS-98-113, Computer Science Department, CMU, March 1998.

....others space prevents a detailed comparison, but see [Sew00] There is a large body of semantic work on concurrent and distributed algorithms. Crudely, it can be subdivided into work taking an automata theoretic approach and work on encodings of high level primitives. The former includes [AP98, JNW98] addressing Mobile IP, and [Mor99] which studies an infrastructure providing a similar abstraction to that of this paper. All involve more or less idealised models of algorithms rather than directly executable code. The latter includes encodings of choice [NP96] join communication [FG96] ....

Daniel Jackson, Yuchung Ng, and Jeannette Wing. A Nitpick analysis of mobile IPv6. Technical Report CMU-CS-98-113, Computer Science Department, CMU, March 1998.

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Daniel V. Bailey. Optimal Extension Fields. Major Qualifying Project (Senior Thesis), 1998. Computer Science Department, Worcester Polytechnic Institute, Worcester, MA, USA.

....signature schemes using the fields GF (p) with p a 192 bit prime and GF (2 191 ) For ECCs over prime fields, their construction uses projective coordinates to eliminate the need for inversion, along with a 4 Previous Work 5 balanced ternary representation of the multiplicand. The work in [Bai98] and [BP98] marks a departure from these methods and serves as a starting point for this new research. A great deal of work has been done in studying aspects of inversion in a finite field especially since inversion is the most costly of the four basic operations. In the case of prime fields, in ....

....the necessary and su#cient conditions on these parameters. For simplicity of presentation, we present an algorithm to construct a Type II OEF, 44 OEF Construction and Statistics 45 fixing # = 2. Even with this restriction, OEFs are plentiful. This algorithm is an improvement over that found in [Bai98] since Algorithm 3 can be used to exhaustively find all Type II OEFs. The algorithm proceeds by finding pseudo Mersenne primes and then checking possible extension degrees m for the existence of a binomial. For our application, word size n will be chosen based on the attributes of the target ....

Daniel V. Bailey. Optimal Extension Fields. Major Qualifying Project (Senior Thesis), 1998. Computer Science Department, Worcester Polytechnic Institute, Worcester, MA, USA.

....coordinates to eliminate the need for inversion, along with a balanced ternary representation of the multiplier. Claus Schnorr presents a digital signature algorithm based on the finite field discrete logarithm problem in [23] The algorithm is especially suited for smart cards. The work in [1, 2] introduces OEFs and provides performance statistics on high end RISC workstations. A paper extending the work on OEFs appears in [16] In this paper, sub millisecond performance on high end RISC workstations is reported. Further, the authors achieve an ECC performance of 1.95 msec on a 400 MHz ....

....extension field multiplication requires m 2 inner products a i b j , and m 1 multiplications by # when the schoolbook method for polynomial multiplication is used. These m 2 m 1 subfield multiplicationsform the performance critical part of a field multiplication. In the earlier OEF work [1], 2] a subfield multiplication was performed as single precision integer multiplication resulting in a double precision product with a subsequent reduction modulo p. For OEFs with p = 2 n c, c 1, this approach requires 2 integer multiplications and several shifts and adds using Algorithm ....

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Daniel V. Bailey. Optimal Extension Fields. Major Qualifying Project (Senior Thesis), 1998. Computer Science Department, Worcester Polytechnic Institute, Worcester, MA, USA.

....arithmetic on a desktop PC, with a focus on its application to digital signature schemes. For ECCs over prime fields, their construction uses projective coordinates to eliminate the need for inversion, along with a balanced ternary representation of the multiplicand. The authors previous work in [2] and [3] marks a departure from these methods and serves as a starting point for this new research. A great deal of work has been done in studying aspects of inversion in a finite field especially since inversion is the most costly of the four basic operations. In the case of prime fields, in ....

....in fields GF (p m ) of which an OEF is a special case. The operation of inversion is the most costly of the four basic operations, and is thus treated separately in Section 5. In Section 6, improved multiplication algorithms are introduced. The material of this section is described in [2] and [3] and appears here solely for completeness of presentation. An OEF GF (p m ) is isomorphic to GF (p) x] P (x) where P (x) x m P m 1 i=0 p i x i , p i # GF (p) is a monic irreducible polynomial of degree m over GF (p) In the following, a residue class will be identified ....

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D. V. Bailey. Optimal Extension Fields. Major Qualifying Project (Senior Thesis), 1998. Computer Science Department, Worcester Polytechnic Institute, Worcester, MA, USA.

....of a primitive element on our search for fields, there are still enough Type II OEFs to construct fields for any application. Our computational experiments indicate that for n = 32 and n = 64 there are hundreds of fields that satisfy these criteria. Tables of OEFs for all 7 n 63 are found in [1]. For example, suppose we wish to construct a field for use on a modern workstation with 64 bit integer arithmetic for use in an elliptic curve key exchange algorithm. We set n 63, c 1, low 120, high 260. Then we apply a probabilitstic primality test for the integers 2 n Gamma c, ....

Daniel V. Bailey. Optimal extension fields. Major Qualifying Project (Senior Thesis), 1998. Computer Science Department, Worcester Polytechnic Institute, Worcester, MA, USA.

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