# Primitives

- [NAF (Non-adjacent form)](https://fractalyze.gitbook.io/intro/primitives/naf-non-adjacent-form.md): Reduction of Hamming weight to ~1/3
- [Chinese Remainder Theorem (CRT)](https://fractalyze.gitbook.io/intro/primitives/chinese-remainder-theorem-crt.md)
- [Euclidean Algorithm](https://fractalyze.gitbook.io/intro/primitives/euclidean-algorithm.md)
- [Extended Euclidean Algorithm](https://fractalyze.gitbook.io/intro/primitives/euclidean-algorithm/extended-euclidean-algorithm.md)
- [Binary Euclidean Algorithm](https://fractalyze.gitbook.io/intro/primitives/euclidean-algorithm/binary-euclidean-algorithm.md)
- [Extended Binary Euclidean Algorithm](https://fractalyze.gitbook.io/intro/primitives/euclidean-algorithm/extended-binary-euclidean-algorithm.md)
- [Coding Theory](https://fractalyze.gitbook.io/intro/primitives/coding-theory.md)
- [Linear Code](https://fractalyze.gitbook.io/intro/primitives/coding-theory/linear-code.md)
- [Number Theoretic Transform](https://fractalyze.gitbook.io/intro/primitives/number-theoretic-transform.md): Presentation: https://youtu.be/jeHQtLduHN8
- [Parallel/Distributed FFT](https://fractalyze.gitbook.io/intro/primitives/number-theoretic-transform/parallel-distributed-fft.md)
- [NTT Over Extension Field](https://fractalyze.gitbook.io/intro/primitives/number-theoretic-transform/ntt-over-extension-field.md)
- [Abstract Algebra](https://fractalyze.gitbook.io/intro/primitives/abstract-algebra.md)
- [Group](https://fractalyze.gitbook.io/intro/primitives/abstract-algebra/group.md)
- [-Morphisms](https://fractalyze.gitbook.io/intro/primitives/abstract-algebra/group/morphisms.md)
- [Batch Inverse](https://fractalyze.gitbook.io/intro/primitives/abstract-algebra/group/batch-inverse.md): Reduction of the number of field inverse operations when computing batch operations.
- [Elliptic Curve](https://fractalyze.gitbook.io/intro/primitives/abstract-algebra/elliptic-curve.md)
- [Weierstrass Curve](https://fractalyze.gitbook.io/intro/primitives/abstract-algebra/elliptic-curve/weierstrass-curve.md)
- [Coordinate Forms](https://fractalyze.gitbook.io/intro/primitives/abstract-algebra/elliptic-curve/weierstrass-curve/coordinate-forms.md)
- [Fast Elliptic Curve Arithmetic and Improved WEIL Pairing Evaluation](https://fractalyze.gitbook.io/intro/primitives/abstract-algebra/elliptic-curve/weierstrass-curve/fast-elliptic-curve-arithmetic-and-improved-weil-pairing-evaluation.md)
- [Edwards Curve](https://fractalyze.gitbook.io/intro/primitives/abstract-algebra/elliptic-curve/edwards-curve.md)
- [Coordinate Forms](https://fractalyze.gitbook.io/intro/primitives/abstract-algebra/elliptic-curve/edwards-curve/coordinate-forms.md)
- [Twisted Edwards ↔ Short Weierstrass Transformation](https://fractalyze.gitbook.io/intro/primitives/abstract-algebra/elliptic-curve/edwards-curve/twisted-edwards-short-weierstrass-transformation.md)
- [Batch Inverse for Batch Point Additions](https://fractalyze.gitbook.io/intro/primitives/abstract-algebra/elliptic-curve/batch-inverse-for-batch-point-additions.md)
- [Scalar Multiplication](https://fractalyze.gitbook.io/intro/primitives/abstract-algebra/elliptic-curve/scalar-multiplication.md)
- [Double-and-add](https://fractalyze.gitbook.io/intro/primitives/abstract-algebra/elliptic-curve/scalar-multiplication/double-and-add.md): Reduction of scalar multiplication into double and add operations
- [GLV Decomposition](https://fractalyze.gitbook.io/intro/primitives/abstract-algebra/elliptic-curve/scalar-multiplication/glv-decomposition.md): Splitting a scalar multiplication into 2 and reducing count of double operations
- [MSM](https://fractalyze.gitbook.io/intro/primitives/abstract-algebra/elliptic-curve/msm.md)
- [Pippenger's Algorithm](https://fractalyze.gitbook.io/intro/primitives/abstract-algebra/elliptic-curve/msm/pippengers-algorithm.md): Reduction of multi-scalar multiplication (MSM) into double and add operations
- [Signed Bucket Index](https://fractalyze.gitbook.io/intro/primitives/abstract-algebra/elliptic-curve/msm/signed-bucket-index.md)
- [CycloneMSM](https://fractalyze.gitbook.io/intro/primitives/abstract-algebra/elliptic-curve/msm/cyclonemsm.md): Presentation: https://youtu.be/djA3mzn7BPg
- [EdMSM](https://fractalyze.gitbook.io/intro/primitives/abstract-algebra/elliptic-curve/msm/edmsm.md): ZPrize "Accelerating MSM on Mobile" winner
- [cuZK](https://fractalyze.gitbook.io/intro/primitives/abstract-algebra/elliptic-curve/msm/cuzk.md): Presentation: https://youtu.be/D8b4yi\_URJE
- [ICICLE MSM](https://fractalyze.gitbook.io/intro/primitives/abstract-algebra/elliptic-curve/msm/icicle-msm.md)
- [2-Chain and 2-Cycle of Elliptic Curves](https://fractalyze.gitbook.io/intro/primitives/abstract-algebra/elliptic-curve/2-chain-and-2-cycle-of-elliptic-curves.md)
- [Extension Field](https://fractalyze.gitbook.io/intro/primitives/abstract-algebra/extension-field.md)
- [Multiplication](https://fractalyze.gitbook.io/intro/primitives/abstract-algebra/extension-field/multiplication.md)
- [Karatsuba Multiplication](https://fractalyze.gitbook.io/intro/primitives/abstract-algebra/extension-field/multiplication/karatsuba-multiplication.md): Presentation: https://youtu.be/jtImWo3CmZM
- [Toom-Cook Multiplication](https://fractalyze.gitbook.io/intro/primitives/abstract-algebra/extension-field/multiplication/toom-cook-multiplication.md)
- [Inversion](https://fractalyze.gitbook.io/intro/primitives/abstract-algebra/extension-field/inversion.md): Presentation: https://youtu.be/9wpJk-wUa-w, https://www.youtube.com/watch?v=54I7Rd9\_9\_s
- [Optimized Inversion Algorithm](https://fractalyze.gitbook.io/intro/primitives/abstract-algebra/extension-field/inversion/optimized-inversion-algorithm.md): Presentation: https://www.youtube.com/watch?v=QNxgKYcc-A0
- [Encryption Scheme](https://fractalyze.gitbook.io/intro/primitives/encryption-scheme.md)
- [ElGamal Encryption](https://fractalyze.gitbook.io/intro/primitives/encryption-scheme/elgamal-encryption.md)
- [Modular Arithmetic](https://fractalyze.gitbook.io/intro/primitives/modular-arithmetic.md)
- [Modular Reduction](https://fractalyze.gitbook.io/intro/primitives/modular-arithmetic/modular-reduction.md)
- [Barrett Reduction](https://fractalyze.gitbook.io/intro/primitives/modular-arithmetic/modular-reduction/barrett-reduction.md)
- [Montgomery Reduction](https://fractalyze.gitbook.io/intro/primitives/modular-arithmetic/modular-reduction/montgomery-reduction.md)
- [Modular Inverse](https://fractalyze.gitbook.io/intro/primitives/modular-arithmetic/modular-inverse.md)
- [Bernstein-Yang's Inverse](https://fractalyze.gitbook.io/intro/primitives/modular-arithmetic/modular-inverse/bernstein-yangs-inverse.md)
- [Modular Square Root](https://fractalyze.gitbook.io/intro/primitives/modular-arithmetic/modular-square-root.md)
- [Shanks Algorithm](https://fractalyze.gitbook.io/intro/primitives/modular-arithmetic/modular-square-root/shanks-algorithm.md)
- [Tonelli-Shanks Algorithm](https://fractalyze.gitbook.io/intro/primitives/modular-arithmetic/modular-square-root/tonelli-shanks-algorithm.md)
- [Generalized Shanks Algorithm for Quadratic Extension Field](https://fractalyze.gitbook.io/intro/primitives/modular-arithmetic/modular-square-root/generalized-shanks-algorithm-for-quadratic-extension-field.md)
- [Multiset Check](https://fractalyze.gitbook.io/intro/primitives/multiset-check.md)
- [Sumcheck](https://fractalyze.gitbook.io/intro/primitives/sumcheck.md)
- [Speeding Up Sumcheck](https://fractalyze.gitbook.io/intro/primitives/sumcheck/speeding-up-sumcheck.md): Presentation: https://www.youtube.com/watch?v=YXANnCp5118
- [Commitment Scheme](https://fractalyze.gitbook.io/intro/primitives/commitment-scheme.md)
- [Fflonk](https://fractalyze.gitbook.io/intro/primitives/commitment-scheme/fflonk.md)
- [SHPlonk](https://fractalyze.gitbook.io/intro/primitives/commitment-scheme/shplonk.md)
- [Zeromorph](https://fractalyze.gitbook.io/intro/primitives/commitment-scheme/zeromorph.md)


---

# Agent Instructions: Querying This Documentation

If you need additional information that is not directly available in this page, you can query the documentation dynamically by asking a question.

Perform an HTTP GET request on the current page URL with the `ask` query parameter:

```
GET https://fractalyze.gitbook.io/intro/primitives.md?ask=<question>
```

The question should be specific, self-contained, and written in natural language.
The response will contain a direct answer to the question and relevant excerpts and sources from the documentation.

Use this mechanism when the answer is not explicitly present in the current page, you need clarification or additional context, or you want to retrieve related documentation sections.