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A New Architecture for a Parallel Finite Field Multiplier with Low Complexity Based on Composite Fields
 IEEE Transactions on Computers
, 1996
"... In this paper a new bitparallel structure for a multiplier with low complexity in Galois fields is introduced. The multiplier operates over composite fields GF ((2 n ) m ), with k = nm. The KaratsubaOfman algorithm is investigated and applied to the multiplication of polynomials over GF (2 n ..."
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Cited by 27 (1 self)
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In this paper a new bitparallel structure for a multiplier with low complexity in Galois fields is introduced. The multiplier operates over composite fields GF ((2 n ) m ), with k = nm. The KaratsubaOfman algorithm is investigated and applied to the multiplication of polynomials over GF (2 n ). It is shown that this operation has a complexity of order O(k log 2 3 ) under certain constraints regarding k. A complete set of primitive field polynomials for composite fields is provided which perform modulo reduction with low complexity. As a result, multipliers for fields GF (2 k ) up to k = 32 with low gate counts and low delays are listed. The architectures are highly modular and thus well suited for VLSI implementation. This paper was presented in part at the SwedishRussian Workshop on Information Theory, August 2227, 1993, Molle, Sweden y The author is with the Electrical and Computer Engineering Department, Worcester Polytechnic Institute, Worcester, MA 01609. Email: ...
Fast Arithmetic for PublicKey Algorithms in Galois Fields with Composite Exponents
 IEEE Transactions on Computers
, 1999
"... This contribution describes a new class of arithmetic architectures for Galois fields GF (2 k ). The main applications of the architecture are publickey systems which are based on the discrete logarithm problem for elliptic curves. The architectures use a representation of the field GF (2 k ..."
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Cited by 24 (2 self)
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This contribution describes a new class of arithmetic architectures for Galois fields GF (2 k ). The main applications of the architecture are publickey systems which are based on the discrete logarithm problem for elliptic curves. The architectures use a representation of the field GF (2 k ) as GF ((2 n ) m ), where k = n \Delta m. The approach explores bit parallel arithmetic in the subfield GF (2 n ), and serial processing for the extension field arithmetic. This mixed parallelserial (hybrid) approach can lead to fast implementations. As the core module, a hybrid multiplier is introduced and several This paper is an extension of [1]. The bit parallel squarer architectures have been completely revised. 1 optimizations are discussed. We provide two different approaches to squaring. We develop exact expressions for the complexity of parallel squarers in composite fields which can have a surprisingly low complexity. The hybrid architectures are capable of explori...
A generalized method for constructing subquadratic complexity GF(2 k ) multipliers
 IEEE Transactions on Computers
, 2004
"... We introduce a generalized method for constructing subquadratic complexity multipliers for even characteristic field extensions. The construction is obtained by recursively extending short convolution algorithms and nesting them. To obtain the short convolution algorithms the Winograd short convolu ..."
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Cited by 22 (0 self)
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We introduce a generalized method for constructing subquadratic complexity multipliers for even characteristic field extensions. The construction is obtained by recursively extending short convolution algorithms and nesting them. To obtain the short convolution algorithms the Winograd short convolution algorithm is reintroduced and analyzed in the context of polynomial multiplication. We present a recursive construction technique that extends any d point multiplier into an n = d k point multiplier with area that is subquadratic and delay that is logarithmic in the bitlength n. We present a thorough analysis that establishes the exact space and time complexities of these multipliers. Using the recursive construction method we obtain six new constructions, among which one turns out to be identical to the Karatsuba multiplier. All six algorithms have subquadratic space complexities and two of the algorithms have significantly better time complexities than the Karatsuba algorithm. Keywords: Bitparallel multipliers, finite fields, Winograd convolution 1
Elliptic curve cryptosystems on reconfigurable hardware
 MASTER’S THESIS, WORCESTER POLYTECHNIC INST
, 1998
"... Security issues will play an important role in the majority of communication and computer networks of the future. As the Internet becomes more and more accessible to the public, security measures will have to be strengthened. Elliptic curve cryptosystems allow for shorter operand lengths than other ..."
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Cited by 19 (0 self)
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Security issues will play an important role in the majority of communication and computer networks of the future. As the Internet becomes more and more accessible to the public, security measures will have to be strengthened. Elliptic curve cryptosystems allow for shorter operand lengths than other publickey schemes based on the discrete logarithm in finite fields and the integer factorization problem and are thus attractive for many applications. This thesis describes an implementation of a crypto engine based on elliptic curves. The underlying algebraic structures are composite Galois fields GF((2 n) m) in a standard base representation. As a major new feature, the system is developed for a reconfigurable platform based on Field Programmable Gate Arrays (FPGAs). FPGAs combine the flexibility of software solutions with the security of traditional hardware implementations. In particular, it is possible to easily change all algorithm parameters such as curve coefficients, field order, or field representation. The thesis deals with the design and implementation of elliptic curve point multiplicationarchitectures. The architectures are described in VHDL and mapped to Xilinx FPGA devices. Architectures over Galois fields of different order and representation were implemented and compared. Area and timing measurements are provided for all architectures. It is shown that a full point multiplication on elliptic curves of realworld size can be implemented on commercially available FPGAs.
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 ..."
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Cited by 17 (2 self)
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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
Efficient Multiplier Architectures for Galois Fields GF(2 4n )
 IEEE Transactions on Computers
, 1998
"... This contribution introduces a new class of multipliers for finite fields GF ((2 n ) 4 ). The architecture is based on a modified version of the KaratsubaOfman algorithm (KOA). By determining optimized field polynomials of degree four, the last stage of the KOA and the modulo reduction can b ..."
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Cited by 13 (0 self)
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This contribution introduces a new class of multipliers for finite fields GF ((2 n ) 4 ). The architecture is based on a modified version of the KaratsubaOfman algorithm (KOA). By determining optimized field polynomials of degree four, the last stage of the KOA and the modulo reduction can be combined. This saves computation and area in VLSI implementations. The new algorithm leads to architectures which show a considerably improved gate complexity compared to traditional approaches and reduced delay if compared with KOAbased architectures with separate modulo reduction. The new multipliers lead to highly modular architectures an are thus well suited for VLSI implementations. Three types of field polynomials are introduced and conditions for their existence are established. For the small fields where n = 2; 3; : : : ; 8, which are of primary technical interest, optimized field polynomials were determined by an exhaustive search. For each field order, exact space and ti...
Comparison of Arithmetic Architectures for ReedSolomon Decoders in Reconfigurable Hardware
 IEEE Transactions on Computers
, 1997
"... ReedSolomon (RS) error correction codes are being widely used in modern communication systems such as compact disk players or satellite communication links. RS codes rely on arithmetic in finite, or Galois fields. The specific field GF (2 8 ) is of central importance for many practical systems. T ..."
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Cited by 13 (2 self)
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ReedSolomon (RS) error correction codes are being widely used in modern communication systems such as compact disk players or satellite communication links. RS codes rely on arithmetic in finite, or Galois fields. The specific field GF (2 8 ) is of central importance for many practical systems. The most costly, and thus most critical, elementary operations in RS decoders are multiplication and inversion in Galois fields. Although there have been considerable efforts in the area of Galois field arithmetic architectures, there appears to be very little reported work for Galois field arithmetic for reconfigurable hardware. This contribution provides a systematic comparison of two promising arithmetic architecture classes. The first one is based on a standard base representation, and the second one is based on composite fields. For both classes a multiplier and an inverter for GF (2 8 ) are described and theoretical gate counts are provided. Using a design entry based on a VHDL descr...
Normal Bases over Finite Fields
, 1993
"... Interest in normal bases over finite fields stems both from mathematical theory and practical applications. There has been a lot of literature dealing with various properties of normal bases (for finite fields and for Galois extension of arbitrary fields). The advantage of using normal bases to repr ..."
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Cited by 9 (0 self)
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Interest in normal bases over finite fields stems both from mathematical theory and practical applications. There has been a lot of literature dealing with various properties of normal bases (for finite fields and for Galois extension of arbitrary fields). The advantage of using normal bases to represent finite fields was noted by Hensel in 1888. With the introduction of optimal normal bases, large finite fields, that can be used in secure and e#cient implementation of several cryptosystems, have recently been realized in hardware. The present thesis studies various theoretical and practical aspects of normal bases in finite fields. We first give some characterizations of normal bases. Then by using linear algebra, we prove that F q n has a basis over F q such that any element in F q represented in this basis generates a normal basis if and only if some groups of coordinates are not simultaneously zero. We show how to construct an irreducible polynomial of degree 2 n with linearly i...
Some Remarks on Efficient Inversion in Finite Fields
 In 1995 IEEE International Symposium on Information Theory
, 1995
"... This contribution is concerned with bit parallel inverters over finite fields. Two alternative approaches for inversion with low complexity which were proposed in the late nineteen eighties will be reviewed. Previously they seem to have received relatively little attention in the scientific communit ..."
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Cited by 8 (1 self)
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This contribution is concerned with bit parallel inverters over finite fields. Two alternative approaches for inversion with low complexity which were proposed in the late nineteen eighties will be reviewed. Previously they seem to have received relatively little attention in the scientific community. Both methods are based on multiple field extension of GF (2). We will try to restate the two algorithms in a clear fashion. It will be shown that one architecture is a generalization of the other's architecture core algorithm. As an impressive example of the advantage of inverters operating over extension fields, the optimized complexity of a bit parallel inverter in the important field GF (2 8 ) will be computed, resulting in a surprisingly low gate count. 1 Introduction Galois field arithmetic has wide spread applications in contemporary communication systems, in particular in cryptography and in channel coding. Modern applications in many cases call for VLSI implementations of the a...
Fast Arithmetic Architectures for PublicKey Algorithms over Galois Fields GF((2 n ) m )
 in Advances in Cryptography  EUROCRYPT '97
, 1997
"... This contribution describes a new class of arithmetic architectures for Galois fields GF (2 k ). The main applications of the architecture are publickey systems which are based on the discrete logarithm problem for elliptic curves. The architectures use a representation of the field GF (2 k ) ..."
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Cited by 7 (3 self)
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This contribution describes a new class of arithmetic architectures for Galois fields GF (2 k ). The main applications of the architecture are publickey systems which are based on the discrete logarithm problem for elliptic curves. The architectures use a representation of the field GF (2 k ) as GF ((2 n ) m ), where k = n \Delta m. The approach explores bit parallel arithmetic in the subfield GF (2 n ), and serial processing for the extension field arithmetic. This mixed parallelserial (hybrid) approach can lead to very fast implementations. The principle of these approach was initially suggested by Mastrovito. As the core module, a hybrid multiplier is introduced and several optimizations are discussed. We provide two different approaches to squaring which, in conjunction with the multiplier, yield fast exponentiation architectures. The hybrid architectures are capable of exploring the timespace tradeoff paradigm in a flexible manner. In particular, the number of clock...