Results 1 
8 of
8
Low Complexity Bit Parallel Architectures for Polynomial Basis Multiplication over GF(2 m
 IEEE Transactions on Computers
, 2004
"... Abstract—Representing the field elements with respect to the polynomial (or standard) basis, we consider bit parallel architectures for multiplication over the finite field GFð2 m Þ. In this effect, first we derive a new formulation for polynomial basis multiplication in terms of the reduction matri ..."
Abstract

Cited by 17 (2 self)
 Add to MetaCart
Abstract—Representing the field elements with respect to the polynomial (or standard) basis, we consider bit parallel architectures for multiplication over the finite field GFð2 m Þ. In this effect, first we derive a new formulation for polynomial basis multiplication in terms of the reduction matrix Q. The main advantage of this new formulation is that it can be used with any field defining irreducible polynomial. Using this formulation, we then develop a generalized architecture for the multiplier and analyze the time and gate complexities of the proposed multiplier as a function of degree m and the reduction matrix Q. To the best of our knowledge, this is the first time that these complexities are given in terms of Q. Unlike most other articles on bit parallel finite field multipliers, here we also consider the number of signals to be routed in hardware implementation and we show that, compared to the wellknown Mastrovito’s multiplier, the proposed architecture has fewer routed signals. In this article, the proposed generalized architecture is further optimized for three special types of polynomials, namely, equally spaced polynomials, trinomials, and pentanomials. We have obtained explicit formulas and complexities of the multipliers for these three special irreducible polynomials. This makes it very easy for a designer to implement the proposed multipliers using hardware description languages like VHDL and Verilog with minimum knowledge of finite field arithmetic. Index Terms—Finite or Galois field, Mastrovito multiplier, allone polynomial, polynomial basis, trinomial, pentanomial and equallyspaced polynomial. 1
FPLDImplementation of Computations over Finite Fields GF(2 m ) with Applications to Error Control Coding
, 1995
"... . This paper investigates the implementation of computations over finite fields GF(2 m ) using fieldprogrammable logic devices (FPLDs). Implementation details for addition/subtraction, multiplication, square, inversion, and division are given with mapping results for Xilinx LCAs, Altera CPLDs ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
. This paper investigates the implementation of computations over finite fields GF(2 m ) using fieldprogrammable logic devices (FPLDs). Implementation details for addition/subtraction, multiplication, square, inversion, and division are given with mapping results for Xilinx LCAs, Altera CPLDs and Actel ACT FPGAs. As an application example, mapping results for complete encoders for errorcorrecting codes are also presented. Finally, new opportunities emerging from FPLD technology for data transmission systems with dynamic code adaption are discussed. 1 Introduction Finite fields have seen an ongoing interest of the scientific community for more than three decades. This is due to the fact that computations over finite fields play a key role in many important applications such as cryptography [12] and errorcontrol coding [3], [4], [9]. Efficient implementations of algorithms for encoding and decoding usually make use of a considerable amount of computations in finite fields ...
Efficient VLSI implementation for Montgomery multiplication in GF(2 m
 Journal of Science and Engineering
"... The Montgomery multiplication algorithm without division operations is popular both in prime field GF(p) and Finite field GF(2 m). However, the Montgomery multiplication algorithm has the timedependent problem. We will present a timeindependent Montgomery multiplication algorithm. The results show ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
The Montgomery multiplication algorithm without division operations is popular both in prime field GF(p) and Finite field GF(2 m). However, the Montgomery multiplication algorithm has the timedependent problem. We will present a timeindependent Montgomery multiplication algorithm. The results show that our proposed timeindependent Montgomery multiplication algorithm not only saves about 50 % time complexity but also saves about 11 % space complexity as compared to the traditional Montgomery multiplication algorithm. Our proposed systolic array Montgomery multiplier has simplicity, regularity, modularity, and concurrency, and is very suitable for VLSI implementation.
410. ANALYTICAL REPRESENTATIONS OF mVALUED LOGICAL FUNCTIONS OVER THE RING OF INTEGERS MODULO m*
"... 1.1. This thesis consists of the following parts: ..."
Synthesis Optimization on GaloisField Based Arithmetic Operators for Rijndael Cipher
"... Abstract. A series of experiments has been conducted to show that FPGA synthesis of GaloisField (GF) based arithmetic operators can be optimized automatically to improve Rijndael Cipher throughput. Moreover, it has been demonstrated that efficiency improvement in GF operators does not directly corr ..."
Abstract
 Add to MetaCart
Abstract. A series of experiments has been conducted to show that FPGA synthesis of GaloisField (GF) based arithmetic operators can be optimized automatically to improve Rijndael Cipher throughput. Moreover, it has been demonstrated that efficiency improvement in GF operators does not directly correspond to the system performance at application level. The experiments were motivated by so many research works that focused on improving performance of GF operators. Each of the variants has the most efficient form in either time (fastest) or space (smallest occupied area) when implemented in FPGA chips. In fact, GF operators are not utilized individually, but rather integrated one to the others to implement algorithms. Contribution of this paper is to raise issue on GFbased application performance and suggest alternative aspects that potentially affect it. Instead of focusing on GF operator efficiency, system characteristics are worth considered in optimizing application performance. Keywords: FPGA; Galois Field; Rijndael Cipher; VHDL. 1
unknown title
"... The literature of cryptography has a curious history. Secrecy, of course, has always played a central role, but until the First World War, important developments appeared in print in a more or less timely fashion and the field moved forward in much the same way as other specialized disciplines. As l ..."
Abstract
 Add to MetaCart
The literature of cryptography has a curious history. Secrecy, of course, has always played a central role, but until the First World War, important developments appeared in print in a more or less timely fashion and the field moved forward in much the same way as other specialized disciplines. As late as 1918, one of the most influential cryptanalytic papers of the twentieth century, William F. Friedman’s monograph The Index of Coincidence and Its Applications in Cryptography, appeared as a research report of the private Riverbank Laboratories [577]. And this, despite the fact that the work had been done as part of the war effort. In the same year Edward H. Hebern of Oakland, California filed the first patent for a rotor machine [710], the device destined to be a mainstay of military cryptography for nearly 50 years. After the First World War, however, things began to change. U.S. Army and Navy organizations, working entirely in secret, began to make fundamental advances in cryptography. During the thirties and forties a few basic papers did appear in the open literature and several treatises on the subject were published, but the latter were farther and farther behind the state of the art. By the end of the war the transition was complete. With one notable exception, the public literature had died. That exception was Claude Shannon’s paper “The Communication Theory of Secrecy Systems, ” which