Usage
The polynomial basis is frequently used in cryptographic applications that are based on the discrete logarithm problem such as elliptic curve cryptography.
The advantage of the polynomial basis is that multiplication is relatively easy. For contrast, the normal basis is an alternative to the polynomial basis and it has more complex multiplication but squaring is very simple. Hardware implementations of polynomial basis arithmetic usually consume more power than their normal basis counterparts.
Read more about this topic: Polynomial Basis
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