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Karatsuba-like formulae and their associated techniques

Efficient polynomial multiplication formulae are required for cryptographic computation. From elliptic curve cryptography to homomorphic encryption, many cryptographic systems need efficient multiplication formulae. The most widely used multiplication formulae for cryptographic systems are the Karatsuba-like polynomial multiplication formulae. In this paper, these formulae and Montgomery's work yielding more efficient such formulae are introduced. Moreover, recent efforts to improve these results are discussed by presenting associated techniques. The state of art for this area is also discussed.