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The Number Theoretic Transform is the discrete Fourier transform (DFT) equivalent over a finite field Zqthe integers sub q A prime number where Root of Unity ( ): An element such that

Ak=∑j=0n−1aj⋅ωjk(modq)cap A sub k equals sum from j equals 0 to n minus 1 of a sub j center dot omega raised to the j k power space open paren mod space q close paren 3. Calculate Each Point For an input sequence A0cap A sub 0 : A1cap A sub 1 : A2cap A sub 2 : A3cap A sub 3 : (and so on). 4. Polynomial Multiplication Once transformed, you multiply the results point-wise:

The is a critical optimization for modular arithmetic in cryptography, enabling faster multiplication by moving from the coefficient domain to a point-value domain using roots of unity.

ωi≢1omega to the i-th power modified triple bar with not overlay 1

It converts polynomials from coefficient representation to point-value representation, allowing multiplication in time instead of Procedural Step-by-Step: Computing a 4-point NTT

. Then, apply the to return to coefficients. Visualization of Polynomial Transformation

The following graph demonstrates how a polynomial's behavior changes when transformed into the frequency domain via NTT-like operations. ✅ Result Summary

The Number Theoretic Transform is the discrete Fourier transform (DFT) equivalent over a finite field Zqthe integers sub q A prime number where Root of Unity ( ): An element such that

Ak=∑j=0n−1aj⋅ωjk(modq)cap A sub k equals sum from j equals 0 to n minus 1 of a sub j center dot omega raised to the j k power space open paren mod space q close paren 3. Calculate Each Point For an input sequence A0cap A sub 0 : A1cap A sub 1 : A2cap A sub 2 : A3cap A sub 3 : (and so on). 4. Polynomial Multiplication Once transformed, you multiply the results point-wise:

The is a critical optimization for modular arithmetic in cryptography, enabling faster multiplication by moving from the coefficient domain to a point-value domain using roots of unity.

ωi≢1omega to the i-th power modified triple bar with not overlay 1

It converts polynomials from coefficient representation to point-value representation, allowing multiplication in time instead of Procedural Step-by-Step: Computing a 4-point NTT

. Then, apply the to return to coefficients. Visualization of Polynomial Transformation

The following graph demonstrates how a polynomial's behavior changes when transformed into the frequency domain via NTT-like operations. ✅ Result Summary

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