1 Montgomery Modular Multiplication in Hard- ware

FEI KEMT
data
inputs
clock
Look-up
Table
carry
input
Carry
Chain
carry
output
D
Flip
Flop
data
outputs
Figure 1 – 1 Typical architecture of the smallest functional unit in a FPGA.
graphic algorithms and protocols can be represented as sequence of algebraic func-
tions in chosen operational area. The operations in cryptography are often similar to
the ones used in the fields mentioned above. Therefore the optimised blocks in struc-
ture of FPGAs provide means for efficient realisation of cryptographic primitives,
too.
The additional property of cryptosystems - the security, is supported by vendors
of the FPGAs by enhancing the devices with hard-wired encryption cores and special
purpose memories. With raising importance of cryptography the FPGA vendors will
be pushed to provide more and more features supporting security of FPGA-based
cryptosystems as it was proposed in [93]. More information on FPGA features
and their relation to implementation of cryptosystems including analysis of possible
attacks can be found in [122].
1.2 RSA Algorithm
Nowadays the most popular asymmetric cryptosystem is RSA which was developed
by Ronald Rivest, Adi Shamir and Leonard Adleman in 1978 [96].
A private key for RSA algorithm consists of two large primes p and q with com-
parable sizes and a secret exponent D. A public key is represented by an exponent
E and modulus M, where
M = pq (1.1)
The Euler totien function φ(M) is defined as a number of positive integers smaller
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