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Large integer Modular Multiplication and Exponentiation (MM and ME) are the foundation of most publickey cryptosystems, specifically RSA, Diffie-Helleman, ElGamal and the Elliptic Curve Cryptosystems. Thus MM algorithms have been studied widely and extensively. Most of the work is based on the well known Montgomery Multiplication Method and its variants, which require standard multiplication operations. Despite their better complexity orders, Karatsuba and FFT algorithms seem to be rarely used for hardware implementation. In this paper, we review their hardware complexity and propose original implementations of MM and ME that become useful for 24-bit operators (Karatsuba algorithm) or 373-bit operators (FFT algorithm).
Cryptography, Multiplication, Modular Arithmetic, Hardware Complexity

K. Kalach, J. P. David and N. Tittley, "Hardware Complexity of Modular Multiplication and Exponentiation," in IEEE Transactions on Computers, vol. 56, no. , pp. 1308-1319, 2007.
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