Cryptography Reference
In-Depth Information
Big number and modular arithmetic : In cryptography, arithmetic operations are
performed on numbers that are hundreds of bits long. Such numbers are not sup-
ported natively in many computing languages and must be implemented externally.
Appropriate handling of big numbers and modular arithmetic is a key to efficient cryp-
tographic operations.
Elliptic curve arithmetic : Cryptographic mechanisms based on elliptic curves depend
on arithmetic involving the points of the curve. Elliptic curve operations rely on arith-
metic in the finite field, but the performance of point addition and doubling is also
important. The efficiency of curve arithmetic depends on the type of the curve, the
size of the underlying field, the point representation system, and the implementation
of the algorithms.
ECC primitives : One of the main cryptographic primitives in ECC is scalar point multi-
plication. This operation is the basis for many ECC schemes, such as ECDSA, Elliptic
Curve Diffie-Hellman, and other elliptic curve encryption schemes. The performance
of point multiplication is crucial in ECC, as it dominates the execution time of the
above schemes. Point multiplication is an important primitive, not only in standard
ECC, but also in IBC. Efficient implementation of point multiplication is difficult due
to the variety of existing algorithms and different types of elliptic curves.
Cryptographic pairings : The calculation of cryptographic pairings is currently the most
computationally intensive operation in IBC schemes. Software implementations of
pairings are considered complex and computationally expensive for embedded devices,
and the application of Pairing-Based Cryptography (PBC) on sensor networks is an
emerging field in WSN security. For instance, for an 80-bit security level, PBC works
with 1024-bit numbers, as opposed to the conventional 160-bit numbers used in ECC.
9.8 Algorithms for Finite Field Arithmetic
Finite field arithmetic forms the fundamental basis for building identity-based crypto-
graphic protocols (Szczechowiak 2010). In general, finite fields portray special proper-
ties that could be utilized for cryptography and especially ECC, as elliptic curves used
in cryptography are defined over finite fields. In addition, curve operations such as
addition, subtraction, multiplication, squaring, and inversion are performed on the
underlying field elements. Due to the repeated use of these operations, the algorithms
used for implementing field arithmetic plays a crucial role in increasing the efficiency
of the system. Additionally, the type of field (prime fields [Section 3.3]; binary fields
[Section 3.4]; or optimal extension fields) and its size also influence the overall perfor-
mance of the system. Among the arithmetic operations, multiprecision multiplication
of field elements is the most time-critical operation, and the efficiency of a low-end
processor (8-bit Atmega 128 CPU, 16-bit MSP430 microcontroller, and 32-bit ARM
processor) is largely dependent on its effective implementation.
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