Synthesis Optimization on Galois-Field Based Arithmetic Operators for Rijndael Cipher

A  series  of  experiments  has  been  conducted  to  show  that  FPGA synthesis  of  Galois-Field  (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 GF-based  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

Implementaciones hardware de circuitos aritméticos sobre cuerpos finitos (Hardwareimolementations of arithmetic circuits over finite field)

La aritmética sobre cuerpos finitos ha recibido mucho interés debido a su importancia en criptografía, control de errores de codificación y procesado de señales digitales. Una gran parte del tiempo de las rutinas criptográficas se dedica al cálculo de operaciones aritméticas sobre cuerpos finitos. Los sistemas que usan esta aritmética deben ser rápidos debido a los rendimientos requeridos en los sistemas de comunicación actuales. La suma en GF(2^m) es una operación XOR binaria independiente, puede ser realizada de forma rápida y sin retardo. Sin embargo otras operaciones son mucho más complejas y con mayor retardo. La eficiencia de las implementaciones hardware se mide en términos del número de puertas (XOR y AND) y del retardo total debido a esas puertas del circuito. El objetivo de este documento es hacer un estudio comparativo de diferentes circuitos aritméticos sobre GF(2^m), se utilizarán los cuerpos recomendados por el NIST y el SECG. Por su importancia, se han estudiado diferentes implementaciones para los algoritmos de multiplicación, tanto multiplicación serie como paralela junto con multiplicación dígito serie. Para el estudio de toras operaciones aritméticas, también se estudian algoritmos para obtener el cuadrado y el inverso de elementos pertencientes a GF(2^m). Para realizar este trabajo se implentarán los algoritmos mencionados en VHDL para FPGAs estudiando el consumo de área y tiempo de las operaciones comparando los resultados entre sí y con los obtenidos por otros autores. [ABSTRACT]Finite field arithmetic has received much attention due to its importance in cryptography, error control coding and digital signal processing. A large portion of time from the routines of the cryptographies algorithms is used in the calculation of arithmetic operations on finite fields. Systems using this arithmetic must be faster because of performance required in current communication systems. Addition in GF(2^m) is bit independent XOR operation, it can be implemented in fast and inexpensive ways. Nevertheless other operations are much more complex and expensive. The efficiency of the hardware implementations is measured in terms of the numbers of gates (XOR and AND) and of the total gate delay of the circuit. The aim of this document is to make a comparative study of different arithmetic circuits over GF(2^m), NIST and SECG recommended fields will be used. Due to multiplication is one of the most complex and important operation in finite field arithmetic, different implementations will be treated, parallel and serial along with digit-serial algorithms. To perform other operations, also inversion and square algorithms over GF(2^m) have been discussed. VHDL implementations of these algorithms for FPGAs have been realized to study time and area consumption and to compare the result each other and with other authors'results

A new approach in building parallel finite field multipliers

A new method for building bit-parallel polynomial basis finite field multipliers is proposed in this thesis. Among the different approaches to build such multipliers, Mastrovito multipliers based on a trinomial, an all-one-polynomial, or an equally-spacedpolynomial have the lowest complexities. The next best in this category is a conventional multiplier based on a pentanomial. Any newly presented method should have complexity results which are at least better than those of a pentanomial based multiplier. By applying our method to certain classes of finite fields we have gained a space complexity as n2 + H - 4 and a time complexity as TA + ([ log2(n-l) ]+3)rx which are better than the lowest space and time complexities of a pentanomial based multiplier found in literature. Therefore this multiplier can serve as an alternative in those finite fields in which no trinomial, all-one-polynomial or equally-spaced-polynomial exists

Low Complexity Finite Field Multiplier for a New Class of Fields

Finite fields is considered as backbone of many branches in number theory, coding theory, cryptography, combinatorial designs, sequences, error-control codes, and algebraic geometry. Recently, there has been considerable attention over finite field arithmetic operations, specifically on more efficient algorithms in multiplications. Multiplication is extensively utilized in almost all branches of finite fields mentioned above. Utilizing finite field provides an advantage in designing hardware implementation since the ground field operations could be readily converted to VLSI design architecture. Moreover, due to importance and extensive usage of finite field arithmetic in cryptography, there is an obvious need for better and more efficient approach in implementation of software and/or hardware using different architectures in finite fields. This project is intended to utilize a newly found class of finite fields in conjunction with the Mastrovito algorithm to compute the polynomial multiplication more efficiently

The construction of self-dual normal polynomials over GF(2) and their applications to the Massey-Omura algorithm

Gaussian periods are used to locate a normal element of the finite field GF(2e) of odd degree e and an algorithm is presented for the construction of self-dual normal polynomials over GF(2) for any odd degree. This gives a new constructive proof of the existence of a self-dual basis for odd degree. The use of such polynomials in the Massey-Omura multiplier improves the efficiency and decreases the complexity of the multiplie

Long period pseudo random number sequence generator

A circuit for generating a sequence of pseudo random numbers, (A sub K). There is an exponentiator in GF(2 sup m) for the normal basis representation of elements in a finite field GF(2 sup m) each represented by m binary digits and having two inputs and an output from which the sequence (A sub K). Of pseudo random numbers is taken. One of the two inputs is connected to receive the outputs (E sub K) of maximal length shift register of n stages. There is a switch having a pair of inputs and an output. The switch outputs is connected to the other of the two inputs of the exponentiator. One of the switch inputs is connected for initially receiving a primitive element (A sub O) in GF(2 sup m). Finally, there is a delay circuit having an input and an output. The delay circuit output is connected to the other of the switch inputs and the delay circuit input is connected to the output of the exponentiator. Whereby after the exponentiator initially receives the primitive element (A sub O) in GF(2 sup m) through the switch, the switch can be switched to cause the exponentiator to receive as its input a delayed output A(K-1) from the exponentiator thereby generating (A sub K) continuously at the output of the exponentiator. The exponentiator in GF(2 sup m) is novel and comprises a cyclic-shift circuit; a Massey-Omura multiplier; and, a control logic circuit all operably connected together to perform the function U(sub i) = 92(sup i) (for n(sub i) = 1 or 1 (for n(subi) = 0)

Novel Single and Hybrid Finite Field Multipliers over GF(2m) for Emerging Cryptographic Systems

With the rapid development of economic and technical progress, designers and users of various kinds of ICs and emerging embedded systems like body-embedded chips and wearable devices are increasingly facing security issues. All of these demands from customers push the cryptographic systems to be faster, more efficient, more reliable and safer. On the other hand, multiplier over GF(2m) as the most important part of these emerging cryptographic systems, is expected to be high-throughput, low-complexity, and low-latency. Fortunately, very large scale integration (VLSI) digital signal processing techniques offer great facilities to design efficient multipliers over GF(2m). This dissertation focuses on designing novel VLSI implementation of high-throughput low-latency and low-complexity single and hybrid finite field multipliers over GF(2m) for emerging cryptographic systems. Low-latency (latency can be chosen without any restriction) high-speed pentanomial basis multipliers are presented. For the first time, the dissertation also develops three high-throughput digit-serial multipliers based on pentanomials. Then a novel realization of digit-level implementation of multipliers based on redundant basis is introduced. Finally, single and hybrid reordered normal basis bit-level and digit-level high-throughput multipliers are presented. To the authors knowledge, this is the first time ever reported on multipliers with multiple throughput rate choices. All the proposed designs are simple and modular, therefore suitable for VLSI implementation for various emerging cryptographic systems

Highly Efficient GF(2^8) Inversion Circuit Based on Redundant GF Arithmetic and Its Application to AES Design

This paper proposes a compact and efficient GF(2^8) inversion circuit design based on a combination of non-redundant and redundant Galois Field (GF) arithmetic. The proposed design utilizes redundant GF representations, called Polynomial Ring Representation (PRR) and Redundantly Represented Basis (RRB), to implement GF(2^8) inversion using a tower field GF((2^4)^2). In addition to the redundant representations, we introduce a specific normal basis that makes it possible to map the former components for the 16th and 17th powers of input onto logic gates in an efficient manner. The latter components for GF(2^4) inversion and GF(2^4) multiplication are then implemented by PRR and RRB, respectively. The flexibility of the redundant representations provides efficient mappings from/to the GF(2^8). This paper also evaluates the efficacy of the proposed circuit by means of gate counts and logic synthesis with a 65 nm CMOS standard cell library and comparisons with conventional circuits, including those with tower fields GF(((2^2)^2)^2). Consequently, we show that the proposed circuit achieves approximately 40% higher efficiency in terms of area-time product than the conventional best GF(((2^2)^2)^2) circuit excluding isomorphic mappings. We also demonstrate that the proposed circuit achieves the best efficiency (i.e., area-time product) for an AES encryption S-Box circuit including isomorphic mappings