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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...
Implementation Options for Finite Field Arithmetic for Elliptic Curve Cryptosystems
, 1999
"... Contents 1. Motivation 2. Overview on Finite Field Arithmetic 3. Arithmetic in GF(p) 4. Arithmetic in GF(2 m ) 5. Arithmetic in GF(p m ) 6. Open Problems ECC '99 WPI Why PublicKey Algorithms? Traditional tool for data security: Privatekey (or symmetric) cryptography Main applications: ffl En ..."
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Cited by 11 (0 self)
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Contents 1. Motivation 2. Overview on Finite Field Arithmetic 3. Arithmetic in GF(p) 4. Arithmetic in GF(2 m ) 5. Arithmetic in GF(p m ) 6. Open Problems ECC '99 WPI Why PublicKey Algorithms? Traditional tool for data security: Privatekey (or symmetric) cryptography Main applications: ffl Encryption ffl Message Authentication Traditional shortcomings: 1. Key distribution, especially with large, dynamic user population (Internet) 2. How to assure sender authenticity and nonrepudiation? Solution: Publickey schemes, e.g., DiffieHellman key exchange or digital signatures. ECC '99 WPI Practical PublicKey Algorithms There are three families of PK algorithms of practical relevance: Integer Factorization Schemes Exp: RSA, Rabin, etc. required ope
A SuperSerial Galois Fields Multiplier for FPGAs and its Application to PublicKey Algorithms
 In Seventh Annual IEEE Symposium on FieldProgrammable Custom Computing Machines, FCCM '99
, 1999
"... This contribution introduces a scalable multiplier architecture for Galois field GF (2 k ) amenable for field programmable gate arrays (FPGAs) implementations. This architecture is well suited for the implementation of publickey cryptosystems which require programmable multipliers in large Galois ..."
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Cited by 11 (2 self)
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This contribution introduces a scalable multiplier architecture for Galois field GF (2 k ) amenable for field programmable gate arrays (FPGAs) implementations. This architecture is well suited for the implementation of publickey cryptosystems which require programmable multipliers in large Galois fields. The architecture trades a reduction in resources with an increase in the number of clock cycles. This architecture is also fine grain scalable in both the time and the area (or logic) dimensions thus facilitating implementations that maximize their use of finite FPGA resources while achieving fast computational speed. This leads to an architecture that requires less resources than traditional bit serial multipliers, which we demonstrated with implementations of multipliers in the field GF (2 167 ). Our results demonstrate that for this field one can realize superserial multipliers that use 2.76 times fewer function generators and 6.84 times fewer flipflops than their serial mult...
Efficient Implementation of Rijndael Encryption With Composite Field Arithmetic
"... We explore the use of subfield arithmetic for efficient implementations Galois Field arithmetic in the context of Rijndael cipher. ..."
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Cited by 4 (2 self)
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We explore the use of subfield arithmetic for efficient implementations Galois Field arithmetic in the context of Rijndael cipher.
Efficient Galois Field Arithmetic on SIMD Architectures
"... We propose techniques to utilize the data parallelism capabilities of a SIMD architecture in computations involving Galois Field arithmetic. Galois Field arithmetic nds wide use in engineering applications, including errorcorrecting codes and cryptography. Often these applications involve exten ..."
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Cited by 3 (0 self)
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We propose techniques to utilize the data parallelism capabilities of a SIMD architecture in computations involving Galois Field arithmetic. Galois Field arithmetic nds wide use in engineering applications, including errorcorrecting codes and cryptography. Often these applications involve extensive arithmetic on small (8bit) numbers, and straightforward implementations may highly underutilize the wideword capabilities of a SIMD processor.
On the Circuit Complexity of Isomorphic Galois Field Transformations
"... We study the circuit complexity of linear transformations between Galois fields GF(2 and their isomorphic composite fields GF((2 ). For such a transformation, we show a lower bound of mn) on the number of gates required in any circuit consisting of constantfanin XOR gates, except for a class of tr ..."
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Cited by 1 (0 self)
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We study the circuit complexity of linear transformations between Galois fields GF(2 and their isomorphic composite fields GF((2 ). For such a transformation, we show a lower bound of mn) on the number of gates required in any circuit consisting of constantfanin XOR gates, except for a class of transformations between representations of such fields which are nicely characterized. The exceptions show that the polynomials representing the fields must be of a regular form, which may be of independent interest. We characterize a family of transformations which can be implemented as crosswires (permutations), without using any gates, which is very useful in designing hardware implementations  and through bitslicing, software implementations  of computations based on Galois Field arithmetic. We also show that our lower bound is tight, by demonstrating a class of transformations which only require a linear number of gates.