A method for obtaining digital signatures and public-key cryptosystems
Communications of the ACM
Handbook of Applied Cryptography
Handbook of Applied Cryptography
Low-Cost Double-Size Modular Exponentiation or How to Stretch Your Cryptoprocessor
PKC '99 Proceedings of the Second International Workshop on Practice and Theory in Public Key Cryptography
Increasing the Bitlength of a Crypto-Coprocessor
CHES '02 Revised Papers from the 4th International Workshop on Cryptographic Hardware and Embedded Systems
CARDIS '08 Proceedings of the 8th IFIP WG 8.8/11.2 international conference on Smart Card Research and Advanced Applications
Unbridle the bit-length of a crypto-coprocessor with montgomery multiplication
SAC'06 Proceedings of the 13th international conference on Selected areas in cryptography
Double-size bipartite modular multiplication
ACISP'07 Proceedings of the 12th Australasian conference on Information security and privacy
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A technique for computing the quotient ($\lfloor ab/n \rfloor$) of Euclidean divisions from the difference of two remainders $(ab \pmod{n} - ab \pmod{n+1})$ was proposed by Fischer and Seifert. The technique allows a 2ℓ-bit modular multiplication to work on most ℓ-bit modular multipliers. However, the cost of the quotient computation rises sharply when computing modular multiplications larger than 2ℓ bits with a recursive approach. This paper addresses the computation cost and improves on previous 2ℓ-bit modular multiplication algorithms to return not only the remainder but also the quotient, resulting in an higher performance in the recursive approach, which becomes twice faster in the quadrupling case and four times faster in the octupling case. In addition to Euclidean multiplication, this paper proposes a new 2ℓ-bit Montgomery multiplication algorithm to return both of the remainder and the quotient.