Hardware elliptic curve cryptographic processor over GF(p)

A novel hardware architecture for elliptic curve cryptography (ECC) over GF(p) is introduced. This can perform the main prime field arithmetic functions needed in these cryptosystems including modular inversion and multiplication. This is based on a new unified modular inversion algorithm that offers considerable improvement over previous ECC techniques that use Fermat's Little Theorem for this operation. The processor described uses a full-word multiplier which requires much fewer clock cycles than previous methods, while still maintaining a competitive critical path delay. The benefits of the approach have been demonstrated by utilizing these techniques to create a field-programmable gate array (FPGA) design. This can perform a 256-bit prime field scalar point multiplication in 3.86 ms, the fastest FPGA time reported to date. The ECC architecture described can also perform four different types of modular inversion, making it suitable for use in many different ECC applications. © 2006 IEEE.

An adaptable and scalable asymmetric cryptographic processor

In this paper a novel scalable public-key processor architecture is presented that supports modular exponentiation and Elliptic Curve Cryptography over both prime GF(p) and binary GF(2) extension fields. This is achieved by a high performance instruction set that provides a comprehensive range of integer and polynomial basis field arithmetic. The instruction set and associated hardware are generic in nature and do not specifically support any cryptographic algorithms or protocols. Firmware within the device is used to efficiently implement complex and data intensive arithmetic. A firmware library has been developed in order to demonstrate support for numerous exponentiation and ECC approaches, such as different coordinate systems and integer recoding methods. The processor has been developed as a high-performance asymmetric cryptography platform in the form of a scalable Verilog RTL core. Various features of the processor may be scaled, such as the pipeline width and local memory subsystem, in order to suit area, speed and power requirements. The processor is evaluated and compares favourably with previous work in terms of performance while offering an unparalleled degree of flexibility. © 2006 IEEE.

FPGA montgomery modular multiplication architectures suitable for ECCs over GF(p)

New FPGA architectures for the ordinary Montgomery multiplication algorithm and the FIOS modular multiplication algorithm are presented. The embedded 18×18-bit multipliers and fast carry look-ahead logic located on the Xilinx Virtex2 Pro family of FPGAs are used to perform the ordinary multiplications and additions/subtractions required by these two algorithms. The architectures are developed for use in Elliptic Curve Cryptosystems over GF(p), which require modular field multiplication to perform elliptic curve point addition and doubling. Field sizes of 128-bits and 256-bits are chosen but other field sizes can easily be accommodated, by rapidly reprogramming the FPGA. Overall, the larger the word size of the multiplier, the more efficiently it performs in terms of area/time product. Also, the FIOS algorithm is flexible in that one can tailor the multiplier architecture is to be area efficient, time efficient or a mixture of both by choosing a particular word size. It is estimated that the computation of a 256-bit scalar point multiplication over GF(p) would take about 4.8 ms.

Generic Architecture and Semiconductor Intellectual Property Cores for Avanced Encryption Standard Cryptography

A generic architecture for implementing the advanced encryption standard (AES) encryption algorithm in silicon is proposed. This allows the instantiation of a wide range of chip specifications, with these taking the form of semiconductor intellectual property (IP) cores. Cores implemented from this architecture can perform both encryption and decryption and support four modes of operation: (i) electronic codebook mode; (ii) output feedback mode; (iii) cipher block chaining mode; and (iv) ciphertext feedback mode. Chip designs can also be generated to cover all three AES key lengths, namely 128 bits, 192 bits and 256 bits. On-the-fly generation of the round keys required during decryption is also possible. The general, flexible and multi-functional nature of the approach described contrasts with previous designs which, to date, have been focused on specific implementations. The presented ideas are demonstrated by implementation in FPGA technology. However, the architecture and IP cores derived from this are easily migratable to other silicon technologies including ASIC and PLD and are capable of covering a wide range of modem communication systems cryptographic requirements. Moreover, the designs produced have a gate count and throughput comparable with or better than the previous one-off solutions.

Bronsted-Evans-Polanyi relation of multistep reactions and volcano curve in heterogeneous catalysis

Multistep surface processes involving a number of association reactions and desorption processes may be considered as hypothetical one-step desorption processes. Thus, heterogeneous catalytic reactions can be treated kinetically as consisting of two steps: adsorption and desorption. It is also illustrated that the hypothetical one-step desorption process follows the BEP relation. A volcano curve can be obtained from kinetic analysis by including both adsorption and desorption processes.