Fast arithmetic for public-key algorithms in Galois fields withcomposite exponents
Paar, C.; Fleischmann, P.; Soria-Rodriguez, P.
Computers, IEEE Transactions on
Volume 48, Issue 10, Oct 1999 Page(s):1025 - 1034
Digital Object Identifier 10.1109/12.805153
Summary:The article describes a novel 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=n·m. 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 fast implementations. As
the core module, a hybrid multiplier is introduced and several
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 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 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 multiplication and
squaring in GF((2n)m), m variable
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