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and polynomial operations in finite fields","homepage":"https://github.com/GuildOfWeavers/galois#readme","keywords":["finite fields","fast fourier transform","lagrange interpolation","cryptography"],"repository":{"type":"git","url":"git+https://github.com/GuildOfWeavers/galois.git"},"author":{"name":"Bobbin Threadbare","email":"bobbinth@protonmail.com"},"bugs":{"url":"https://github.com/GuildOfWeavers/galois/issues"},"license":"MIT","readme":"# Galois\r\nArithmetic and polynomial operations in finite fields.\r\n\r\n## Install\r\n```bash\r\n$ npm install @guildofweavers/galois --save\r\n```\r\n\r\n## Example\r\n```TypeScript\r\nimport * as galois from '@guildofweavers/galois';\r\n\r\n// create a prime field with a large modulus\r\nconst field = galois.createPrimeField(2n ** 256n - 351n * 2n ** 32n + 1n);\r\n\r\nconst a = field.rand();     // generate a random field element\r\nconst b = field.rand();     // generate another random element\r\nconst c = field.exp(a, b);  // do some math\r\n```\r\n\r\n## API\r\nYou can find complete API definitions in [galois.d.ts](/galois.d.ts). Here is a quick overview of the provided functionality:\r\n\r\n### Creating Finite Fields\r\nTo perform operations in a finite field, you'll first need to create a `FiniteField` object. Currently, only prime fields are supported. To create a prime field you can use the `createPrimeField` function. The function has the following signature:\r\n\r\n* **createPrimeField**(modulus: `bigint`, useWasm?: `boolean`): `FiniteField` <br />\r\n  Creates a prime field for the specified `modulus`. If `useWasm` is set to true, will try to instantiate a WebAssembly-optimized version of the field. If WASM optimization is not available for the specified modulus, will create a regular JavaScript-based field.\r\n\r\n* **createPrimeField**(modulus: `bigint`, wasmOptions: `WasmOptions`): `FiniteField`\r\n  Tries to create a WebAssembly-optimized prime field for the specified `modulus` and pass the provided options to it. If WASM optimization is not available for the specified modulus, will create a regular JavaScript-based field.\r\n\r\nWhen provided, `wasmOptions` object must have the following form:\r\n\r\n| Property | Description |\r\n| ---------| ----------- |\r\n| memory   | A WebAssembly Memory object with which the WASM module will be instantiated. |\r\n\r\nOnce you've created a `FiniteField` object, you can use the methods described in the following sections to do math.\r\n\r\n#### WASM optimization\r\nVector, matrix, and polynomial operations for certain types of fields have been optimized to make use of WebAssembly. This can speed up such operations by a factor of 6x - 10x (depending on the operation, see [performance](#Performance) for more details). The optimization is currently available for the following fields:\r\n\r\n* Prime fields with modulus of the form 2<sup>128</sup>-k, where k < 2<sup>64</sup>\r\n\r\n**Note:** there is currently no way to free memory consumed by WASM modules, so, you might have to periodically re-create `FiniteField` objects to avoid memory leaks. This will be addressed in the future versions of this library by relying on [finalizers](https://v8.dev/features/weak-references), which should be available in major JS engines soon.\r\n\r\n### Basic arithmetics\r\nThese methods are self-explanatory. `inv` computes a multiplicative inverse using the [Extended Euclidean algorithm](https://en.wikipedia.org/wiki/Extended_Euclidean_algorithm), `neg` computes an additive inverse.\r\n\r\n* **add**(a: `bigint`, b: `bigint`): `bigint`\r\n* **sub**(a: `bigint`, b: `bigint`): `bigint`\r\n* **mul**(a: `bigint`, b: `bigint`): `bigint`\r\n* **div**(a: `bigint`, b: `bigint`): `bigint`\r\n* **exp**(b: `bigint`, p: `bigint`): `bigint`\r\n* **inv**(a: `bigint`): `bigint`\r\n* **neg**(a: `bigint`): `bigint`\r\n\r\n### Vectors\r\nVectors are 1-dimensional data structures similar to arrays. Vectors are immutable: once created, their contents cannot be modified. To read values from a vector you can use the following methods:\r\n\r\n* **getValue**(index: `number`): `bigint`\r\n* **toValues**(): `bigint[]`\r\n\r\n**Note:** for WASM-optimized fields `toValues()` is a relatively expensive operation since all vector elements have to be copied out of WASM memory into a new JavaScript array.\r\n\r\n#### Creating vectors\r\nA `FiniteField` object exposes two methods which you can use to create new vectors:\r\n\r\n* **newVector**(length: `number`): `Vector`<br />\r\n  Creates a new vector with the specified length (all values initialized to `0`).\r\n\r\n* **newVectorFrom**(values: `bigint[]`): `Vector`<br />\r\n  Creates a new vector and populates it with the provided values.\r\n\r\nYou can also create new vectors by transforming existing vectors using the following methods:\r\n\r\n* **pluckVector**(v: `Vector`, skip: `number`, times: `number`): `Vector`<br />\r\n  Creates a new vector by selecting values from the source vector by skipping over the specified number of elements. If `skip*times` is greater than length of the source vector, the operation will \"wrap around\" and start plucking from the beginning of the vector.\r\n\r\n* **truncateVector**(v: `Vector`, newLength: `number`): `Vector`<br />\r\n  Creates a new vector by selecting the number of elements specified by `newLength` parameter from the head of the source vector.\r\n\r\n* **duplicateVector**(v: `Vector`, times?: `number`): `Vector`<br />\r\n  Creates a new vector by duplicating the existing vector the specified number of times. For example, duplicating `[1, 2]` two times will result in `[1, 2, 1, 2, 1, 2, 1, 2]`.\r\n\r\n#### Vector operations\r\nBasic operations can be applied to vectors in the _element-wise_ fashion. For example, `addVectorElements` computes a new vectors `v` such that `v[i] = a[i] + b[i]` for all elements. When the second argument is a scalar, uses that scalar as the second operand in the operation.\r\n\r\n* **addVectorElements**(a: `Vector`, b: `bigint` | `Vector`): `Vector`\r\n* **subVectorElements**(a: `Vector`, b: `bigint` | `Vector`): `Vector`\r\n* **mulVectorElements**(a: `Vector`, b: `bigint` | `Vector`): `Vector`\r\n* **divVectorElements**(a: `Vector`, b: `bigint` | `Vector`): `Vector`\r\n* **expVectorElements**(a: `Vector`, b: `bigint` | `Vector`): `Vector`\r\n\r\n* **invVectorElements**(v: `Vector`): `Vector`<br />\r\n  Computes multiplicative inverses of all vector elements using Montgomery batch inversion.\r\n\r\n* **negVectorElements**(v: `Vector`): `Vector`<br />\r\n  Computes additive inverses of all vector elements.\r\n\r\nBesides the element-wise operations, the following operations can be applied to vectors:\r\n\r\n* **combineVectors**(a: `Vector`, b: `Vector`): `bigint`<br />\r\n  Computes a linear combination of two vectors.\r\n\r\n* **combineManyVectors**(v: `Vector[]`, k: `Vector`): `Vector`<br />\r\n  Computes linear combinations of vector rows using specified coefficients. For example, `v[0][i]*k[0] + v[1][i]*k[1]` for all `i`.\r\n\r\n### Matrixes\r\nMatrixes are 2-dimensional data structures similar to 2-dimensional arrays. Matrixes are immutable: once created, their contents cannot be modified. Values in matrixes are assumed to be in row-major form. To read values from a matrix you can use the following methods:\r\n\r\n* **getValue**(row: `number`, column: `number`): `bigint`\r\n* **toValues**(): `bigint[][]`\r\n\r\n**Note:** for WASM-optimized fields `toValues()` is a relatively expensive operation since all matrix elements have to be copied out of WASM memory into new JavaScript arrays.\r\n\r\n#### Creating matrixes\r\nA `FiniteField` object exposes two methods which you can use to create new matrixes:\r\n\r\n* **newMatrix**(rows: `number`, columns: `number`): `Matrix`<br />\r\n  Creates a new matrix with the specified number of rows and columns.\r\n\r\n* **newMatrixFrom**(values: `bigint[][]`): `Matrix`<br />\r\n  Creates a new matrix with the same dimensions as `values` and populates it with the provided values.\r\n\r\nYou can also create new matrixes by transforming existing vectors using the following methods:\r\n\r\n* **vectorToMatrix**(v: `Vector`, columns: `number`): `Matrix`<br />\r\n  Transposes the provided vector into a matrix with the specified number of columns. For example, converting vector `[1, 2, 3, 4, 5, 6]` into a matrix with 2 columns would result in `[[1, 3], [2, 5], [3, 6]]` matrix.\r\n\r\n#### Matrix operations\r\nBasic operations can be applied to matrixes in the _element-wise_ fashion. For example, `addMatrixElements` computes a new matrix `m` such that `m[i][j] = a[i][j] + b[i][j]` for all elements. When the second argument is a scalar, uses that scalar as the second operand in the operation.\r\n\r\n* **addMatrixElements**(a: `Matrix`, b: `bigint` | `Matrix`): `Matrix`\r\n* **subMatrixElements**(a: `Matrix`, b: `bigint` | `Matrix`): `Matrix`\r\n* **mulMatrixElements**(a: `Matrix`, b: `bigint` | `Matrix`): `Matrix`\r\n* **divMatrixElements**(a: `Matrix`, b: `bigint` | `Matrix`): `Matrix`\r\n* **expMatrixElements**(a: `Matrix`, b: `bigint` | `Matrix`): `Matrix`\r\n\r\n* **invMatrixElements**(m: `Matrix`): `Matrix`<br />\r\n  Computes multiplicative inverses of all matrix elements using Montgomery batch inversion.\r\n\r\n* **negMatrixElements**(m: `Matrix`): `Matrix`<br />\r\n  Computes additive inverses of all matrix elements.\r\n\r\nBesides the element-wise operations, the following operations can be applied to matrixes:\r\n\r\n* **mulMatrixes**(a: `Matrix`, b: `Matrix`): `Matrix`<br />\r\n  Computes a [product of two matrixes](https://en.wikipedia.org/wiki/Matrix_multiplication) such that given input matrix dimensions [*m*,*p*] and [*p*,*n*], the output matrix will have dimensions of [*m*,*n*].\r\n\r\n* **mulMatrixByVector**(m: `Matrix`, v: `Vector`): `Vector`<br />\r\n  Similar to matrix multiplication but the second parameter is a vector. Given a matrix with dimensions [*m*,*n*] and a vector with length *n*, the output vector will have length *m*.\r\n\r\n* **mulMatrixRows**(m: `Matrix`, v: `Vector`): `Matrix`<br />\r\n  Performs an *element-wise* multiplication of the vector with each row of the matrix.\r\n\r\n* **matrixRowsToVectors**(m: `Matrix`): `Vector[]`<br />\r\n  Creates an array of vectors corresponding to rows of the source matrix.\r\n\r\n### Basic polynomial operations\r\nPolynomials are `Vectors` with coefficients of the polynomial encoded in reverse order. For example, a polynomial `x^3 + 2x^2 + 3x + 4` would be encoded as `[4n, 3n, 2n, 1n]`.\r\n\r\nThese methods can be used to perform basic polynomial arithmetic:\r\n\r\n* **addPolys**(a: `Vector`, b: `Vector`): `Vector`\r\n* **subPolys**(a: `Vector`, b: `Vector`): `Vector`\r\n* **mulPolys**(a: `Vector`, b: `Vector`): `Vector`\r\n* **divPolys**(a: `Vector`, b: `Vector`): `Vector`\r\n* **mulPolyByConstant**(p: `Vector`, c: `bigint`): `Vector`\r\n\r\n### Polynomial evaluation and interpolation\r\n\r\n* **evalPolyAt**(p: `Vector`, x: `bigint`): `bigint`<br />\r\n  Evaluates a polynomial at the provided x-coordinate.\r\n\r\n* **evalPolyAtRoots**(p: `Vector`, rootsOfUnity: `Vector`): `Vector`<br />\r\n  Uses Fast Fourier Transform to evaluate polynomials at all provided roots of unity.\r\n\r\n* **evalPolysAtRoots**(p: `Matrix`, rootsOfUnity: `Vector`): `Matrix`<br />\r\n  Uses Fast Fourier Transform to evaluate polynomials at all provided roots of unity. Each row of the matrix is assumed to be a polynomial, and the result will be a matrix of values.\r\n\r\n* **evalQuarticBatch**(polys: `Matrix`, x: `bigint` | `Vector`): `Vector`<br />\r\n  Evaluates a batch of degree 3 polynomials at the provided x coordinate(s). Each row on the `poly` matrix is assumed to be a polynomial represented by 4 values.\r\n\r\n* **interpolate**(xs: `Vector`, ys: `Vector`): `Vector`<br />\r\n  Uses Lagrange Interpolation to compute a polynomial from the provided points (x and y coordinates).\r\n\r\n* **interpolateRoots**(rootsOfUnity: `Vector`, ys: `Vector` | `Matrix`): `Vector` | `Matrix`<br />\r\n  Uses Fast Fourier Transform to compute polynomials from the provided points (roots of unity are as x coordinates). If the second parameter is a matrix, each row of the matrix is assumed to be a separate set of y coordinates. In this case, a matrix will be returned with each row representing a separate polynomial.\r\n\r\n* **interpolateQuarticBatch**(xSets: `Matrix`, ySets: `Matrix`): `Matrix`<br />\r\n  Uses an optimized version of Lagrange Interpolation for degree 3 polynomials. x and y coordinates should be provided in matrixes with 4 values per row.\r\n\r\n### Other miscellaneous operations\r\n\r\n* **rand**(): `bigint`<br />\r\n  Generate a cryptographically-secure random field element.\r\n\r\n* **prng**(seed: `bigint` | `Buffer`, length?: `number`): `Vector` | `bigint`<br />\r\n  Generates pseudorandom field elements from the provided seed. If the `length` parameter is provided, a sequence of elements is returned; otherwise, the returned value is a single field element.\r\n\r\n* **getRootOfUnity**(order: `number`): `bigint`<br />\r\n  Computes a primitive root of unity such that `root**order = 1`.\r\n\r\n* **getPowerSeries**(base: `bigint`, length: `number`): `Vector`<br />\r\n  Computes a series of powers for the provided base element. More specifically: `[1n, base**1, base**2, ..., base**(length - 1)]`.\r\n\r\n## Performance\r\nSome very informal benchmarks run on Intel Core i5-7300U @ 2.60GHz (single thread). These show approximate number of **operations/second**:\r\n\r\n| Operation       | JS BigInt (256-bit) | JS BigInt (128-bit) | WASM (128-bit) |\r\n| --------------- | ------------------: | ------------------: | -------------: |\r\n| Additions       | 3,200,000           | 5,000,000           | 44,000,000     |\r\n| Multiplications | 950,000             | 1,850,000           | 16,300,000     |\r\n| Exponentiations | 3,200               | 10,500              | 97,000\r\n\r\n## References\r\n\r\n* Wikipedia article on [finite fields](https://en.wikipedia.org/wiki/Finite_field).\r\n* Many algorithms in this library have been copied (with minimal changes) from Vitalik Buterin's [MIMC STARK project](https://github.com/ethereum/research/tree/master/mimc_stark), including: Fast Fourier Transform, Lagrange Interpolation, and Montgomery batch inversion.\r\n* Other algorithms, including modular multiplication and binary GCD algorithm have been copied from the [Handbook of Applied Cryptography](http://cacr.uwaterloo.ca/hac/about/chap14.pdf).\r\n\r\n# License\r\n[MIT](/LICENSE) © 2019 Guild of Weavers","readmeFilename":"README.md"}