Abstract
This chapter consists of two sections that cover algorithms and hardware architectures for modular addition and multiplication: (x + y) mod m and xy mod m. Subtraction and division are also included—as the addition of an inverse and as multiplication by an inverse. The underlying algorithms and hardware structures are those of Chap. 1, modified for modular arithmetic. For both operations we shall consider generic algorithms and hardware structures for arbitrary moduli and also those for special moduli.
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Notes
- 1.
Strictly, the first addition may be in n bits.
- 2.
Recall that such a compressor can be built to have a delay that is slightly larger than that through a 3:2 compressor.
- 3.
In ordinary multiplication the 2i factor is taken care of by shifting the partial product instead of the multiplicand multiple. That is possible because the lower order i bits of the ith partial product are not included in the corresponding addition. On the other hand, with modular multiplication—specifically the required reductions—all bits of a partial product must be included in the arithmetic; therefore, 2ixy i is taken in its entirety.
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R. Omondi, A. (2020). Modular Addition and Multiplication. In: Cryptography Arithmetic. Advances in Information Security, vol 77. Springer, Cham. https://doi.org/10.1007/978-3-030-34142-8_5
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