| Summary: | Modular arithmetic operations (i.e., inversion, multiplication and exponentiation)
are used in several cryptography applications, such as decipherment operation of RSA
algorithm, Diffie-Hellman key exchange algorithm, elliptic curve cryptography, and the
Digital Signature Standard including the Elliptic Curve Digital Signature Algorithm.
The most important of these arithmetic operations is the modular multiplication operation
since it is the core operation in many cryptographic functions.
Given the increasing demands on secure communications, cryptographic algorithms
will be embedded in almost every application involving exchange of information. Some
of theses applications such as smart cards and hand-helds require hardware restricted in
area and power resources.
Cryptographic applications use a large number of bits in order to be considered
secure. While some of these applications use 256-bit precision operands, others use
precision values up to 2048 or 4096 such as in some exponentiation-based cryptographic
applications. Based on this characteristics, a scalable multiplier that operates on any
bit-size of the input values (variable precision) was recently proposed. It is replicated
in order to generate long-precision results independently of the data path precision for
which it was originally designed.
The multiplier presented in this work is based on the Montgomery multiplication
algorithm. This thesis work contributes by presenting a modified radix-4 Montgomery
multiplication algorithm with new encoding technique for the multiples of the modulus.
This work also describes the scalable hardware design and analyzes the synthesis results
for a 0.5 ��m CMOS technology. The results are compared with two other proposed scalable
Montgomery multiplier designs, namely, the radix-2 design, and the radix-8 design.
The comparison is done in terms of area, total computational time and complexity.
Since modular exponentiation can be generated by successive multiplication, we
include in this thesis an analysis of the boundaries for inputs and outputs. Conditions
are identified to allow the use of one multiplication output as the input of another one
without adjustments (or reduction).
High-radix multipliers exhibit higher complexity of the design. This thesis shows
that radix-4 hardware architectures does not add significant complexity to radix-2 design
and has a significant performance gain. === Graduation date: 2003
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