Whitepaper > Theoretical Foundations

Theoretical Foundations

Balanced Ternary System

3¹ = 3 triangles

3² = 9 triangles

3³ = 27 triangles

Implementation Status

Ternary Arithmetic100%

Galois Field Operations95%

T/ECC Implementation90%

Key Properties

△

Geometric Growth

3ⁿ progression in spatial representation

Natural Balance

Perfect symmetry around zero

Quantum Ready

Optimized for quantum resistance

3.1 Balanced Ternary Mathematics

Core Implementation

Our balanced ternary system is implemented using TypeScript for maximum type safety:

typescript

1// Core types for balanced ternary operations
2type Trit = -1 | 0 | 1;
3type TritString = string; // Sequence of 'T' (-1), '0', '1'
4
5interface TritNumber {
6  readonly value: TritString;
7
8  add(other: TritNumber): TritNumber;
9  multiply(other: TritNumber): TritNumber;
10  negate(): TritNumber;
11  toString(): string;
12}
13
14// React component for trit visualization
15const TritDisplay: React.FC<{ value: TritNumber }> = ({ value }) => {
16  return (
17    <StyledTritDisplay>
18      {value.toString().split('').map((trit, index) => (
19        <TritSymbol key={index} value={trit} />
20      ))}
21    </StyledTritDisplay>
22  );
23};

Our balanced ternary implementation leverages TypeScript's type system for compile-time safety while maintaining efficient computation.

Arithmetic Operations

Implemented with strict TypeScript typing and Next.js API routes:

typescript

1// API route for ternary arithmetic
2export default function handler(
3  req: NextApiRequest,
4  res: NextApiResponse<TritOperationResponse>
5) {
6  const { operation, a, b } = req.body as TritOperationRequest;
7
8  try {
9    const result = performTritOperation(operation, a, b);
10    res.status(200).json({ result });
11  } catch (error) {
12    res.status(400).json({ error: error.message });
13  }
14}

3.2 Ternary Galois Fields

Field Construction

Implementation in our TypeScript environment:

typescript

1class TernaryField {
2  private readonly dimension: number;
3  private readonly modulus: bigint;
4
5  constructor(dimension: number) {
6    this.dimension = dimension;
7    this.modulus = BigInt(Math.pow(3, dimension));
8  }
9
10  // Field operations with TypeScript type safety
11  add(a: TritFieldElement, b: TritFieldElement): TritFieldElement {
12    return new TritFieldElement(
13      (a.value + b.value) % this.modulus
14    );
15  }
16
17  multiply(a: TritFieldElement, b: TritFieldElement): TritFieldElement {
18    return new TritFieldElement(
19      (a.value * b.value) % this.modulus
20    );
21  }
22}
23
24// React component for field visualization
25const FieldOperationView: React.FC<{
26  field: TernaryField;
27  operation: FieldOperation;
28}> = ({ field, operation }) => {
29  // Component implementation
30};

All field operations are implemented with constant-time algorithms to prevent timing attacks, leveraging TypeScript's type system for additional security.

3.3 Ternary Elliptic Curve Cryptography

Curve Implementation

Built on our Next.js architecture:

typescript

1interface TritECPoint {
2  x: TritFieldElement;
3  y: TritFieldElement;
4  isInfinity(): boolean;
5}
6
7class TritCurve {
8  private readonly a: TritFieldElement;
9  private readonly b: TritFieldElement;
10  private readonly field: TernaryField;
11
12  constructor(
13    field: TernaryField,
14    a: TritFieldElement,
15    b: TritFieldElement
16  ) {
17    this.field = field;
18    this.a = a;
19    this.b = b;
20  }
21
22  // Point addition with strict typing
23  addPoints(P: TritECPoint, Q: TritECPoint): TritECPoint {
24    // Implementation
25  }
26}

3.4 Isogeny-Based Cryptography

Quantum Resistance

Implementation using our React/TypeScript stack:

typescript

1interface IsogenyParameters {
2  sourceCurve: TritCurve;
3  targetCurve: TritCurve;
4  kernel: TritECPoint[];
5}
6
7class TritIsogeny {
8  static async computeIsogeny(
9    params: IsogenyParameters
10  ): Promise<IsogenyResult> {
11    // Implementation using Web Workers for heavy computation
12    return new Promise((resolve) => {
13      const worker = new Worker('/isogeny-worker.ts');
14      worker.postMessage(params);
15      worker.onmessage = (e) => resolve(e.data);
16    });
17  }
18}
19
20// React component for isogeny visualization
21const IsogenyView: React.FC<{
22  isogeny: TritIsogeny;
23}> = ({ isogeny }) => {
24  // Component implementation using styled-components
25};

3.5 Ternary Merkle Trees

Tree Implementation

Optimized for our Next.js environment:

typescript

1interface TritMerkleNode {
2  hash: TritHash;
3  left?: TritMerkleNode;
4  middle?: TritMerkleNode;
5  right?: TritMerkleNode;
6}
7
8class TritMerkleTree {
9  private readonly root: TritMerkleNode;
10
11  constructor(data: TritData[]) {
12    this.root = this.buildTree(data);
13  }
14
15  // Tree operations with TypeScript safety
16  verifyProof(proof: TritMerkleProof): boolean {
17    // Implementation
18  }
19}
20
21// React component for tree visualization
22const MerkleTreeView: React.FC<{
23  tree: TritMerkleTree;
24}> = ({ tree }) => {
25  // Component implementation
26};

Our Merkle tree implementation is optimized for server-side rendering with Next.js, ensuring efficient proof verification and visualization.

Implementation Notes

All theoretical components are implemented following our monorepo structure:

Strict TypeScript typing throughout
React components for visualization
Next.js API routes for heavy computation
Styled-components for consistent UI
Full test coverage with Jest
Comprehensive documentation with TSDoc

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