Modulo Multiplier by using Radix-8 Modified Booth Algorithm

Modular arithmetic operations (inversion, multiplication and exponentiation) are used in several cryptography applications. RSA and elliptic curve cryptography (ECC) are two of the most well established and widely used public key cryptographic (PKC) algorithms. The encryption and decryption of these PKC algorithms are performed by repeated modulo multiplications. These multiplications differ from those encountered in signal processing and general computing applications in their sheer operand size. Key sizes in the range of 512~1024 bits and 160~512 bits are typical in RSA and ECC, respectively. Hence, the long carry propagation of large integer multiplication is the bottleneck in hardware implementation of PKC. The residue number system (RNS) has emerged as a promising alternative number representation for the design of faster and low power multipliers owing to its merit to distribute a long integer multiplication into several shorter and independent modulo multiplications. RNS has also been successfully employed to design fault tolerant digital circuits. A special moduli set of forms {2n-1, 2n, 2n +1} are preferred over the generic moduli due to the ease of hardware implementation of modulo arithmetic functions as well as system-level intermodulo operations, such as RNS-to-binary conversion and sign detections. To facilitate design of highspeed full-adder based modulo arithmetic units, it is worthwhile to keep the moduli of a high-DR RNS in forms of {2n-1, 2n, 2n +1}.The modulo 2n-1 multiplier is usually the noncritical datapath among all modulo multipliers in such high-DR RNS multiplier. With this precept, a family of radix-8 Booth encoded modulo 2n1 multipliers, with delay adaptable to the RNS multiplier delay, is proposed. The modulo 2n-1 multiplier delay is made scalable by controlling the word-length of the ripple carry adder, employed for radix-8 hard multiple generation.