A clipped product is a specified range of digits or coefficients of a product, computed without necessarily forming the remaining parts (Norman and Watt 2024). Suppose
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(1)
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For a finite interval 
of integer indices, define the clipped value and clipped product by
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(2)
|
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(3)
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For integers, 
is the radix and the 
are base-
digits. For polynomials,
is an indeterminate and the 
are coefficients in the underlying polynomial
ring.
For example, let 
and 
.
Then
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(4)
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(5)
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For polynomial products, a requested coefficient range depends only on selected coefficient pairs. For integer products, a carry from discarded lower digits can enter the requested range, so additional lower product columns may be needed. Algorithms for clipped products can be based on classical multiplication, Karatsuba multiplication, or the fast Fourier transform. Products retaining only a low, middle, or high portion are special cases of clipped products (Norman and Watt 2024).
