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This contribution describes a new class of arithmetic architectures for Galois fields GF(2k). The main applications of the architecture are public-key systems which are based on the discrete logarithm problem for elliptic curves. The architectures use a representation of the field GF(2k) as GF((2n)m), where k = nċm. The approach explores bit parallel arithmetic in the subfield GF(2n), and serial processing for the extension field arithmetic. This mixed parallel-serial (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 time-space trade-off paradigm in a flexible manner. In particular, the number of clock cycles for one field multiplication, which is the atomic operation in most public-key schemes, can be reduced by a factor of n compared to all other known realizations. The acceleration is achieved at the cost of an increased computational complexity. We describe a proof-of-concept implementation of an ASIC for exponentiation in GF((2n)m), m variable.

Fast Arithmetic Architectures for Public-Key Algorithms over Galois Fields GF((2n)m)