BFV, CKKS, LWE FHE Ciphertext Noise Budget, Relin and Bootstrapping Explainer

BFV, CKKS, LWE FHE Ciphertext Noise Budget, Relin and Bootstrapping Explainer | Online Free Tool | Elysia Tools

Tool usage guide

Learn when to use this tool, what it supports, and how real users apply it.

Key facts

Category: Security
Input types: select, number, text, checkbox
Output type: html
Sample coverage: 4
API ready: Yes

Overview

The BFV, CKKS, and LWE FHE Explainer is an interactive educational tool that models fully homomorphic encryption pipelines over the polynomial ring Z_q[x]/(xⁿ+1). It demonstrates exact invariant noise budget consumption in bits, homomorphic multiplication with digit-decomposition relinearization, SIMD slot packing over finite fields, and bootstrapping refreshes on toy parameters.

When to use

Analyzing how homomorphic multiplications and tensor operations reduce the invariant noise budget in BFV/CKKS ciphertexts.
Understanding the mathematical impact of relinearization and modulus switching across level chains.
Evaluating SIMD batching mechanics and slot-wise polynomial packing for batch encrypted computations.

How it works

1Select an evaluation scheme, ring degree (n = 32 or 64), and define slot values to be packed into plaintext polynomials.
2Configure evaluation depth parameters, modulus levels, and optional bootstrapping simulation to observe noise growth across squarings.
3Execute the BigInt pipeline to compute polynomial rings, evaluate invariant noise bits per the SEAL specification, and generate an interactive HTML report.

Use cases

Demonstrating homomorphic noise growth, decryption failures, and bootstrapping recovery in applied cryptography courses.
Verifying SIMD slot packing behavior and batch polynomial multiplications under BFV and CKKS paradigms.
Comparing ciphertext size growth and performance trade-offs between raw tensor products and relinearized ciphertexts.

Examples

1. Observing Noise Exhaustion During Repeated BFV Squarings

Cryptography Student

Background

Studying the mathematical limits of leveled homomorphic encryption without noise refreshing.

Problem

Needs to see the exact multiplication depth where invariant noise budget drops to zero and produces decoding corruption.

How to use

Select 'bfv-deep-multiply', set ring degree to 32, enter slot values '12, 34, 56, 78', set multiplication depth to 4, and enable bootstrapping simulation.

scheme: bfv-deep-multiply, ringDegree: 32, slotValues: 12, 34, 56, 78, multiplyDepth: 4, showBootstrap: true

Outcome

Generates a budget bar chart tracking the bit drop per multiplication, shows corrupted values at budget zero, and demonstrates slot restoration after bootstrapping.

2. Batch SIMD Element-Wise Multiplication

Privacy Engineer

Background

Designing an encrypted vector processing workflow using Chinese Remainder Theorem slot packing.

Problem

Wants to verify single-instruction multiple-data (SIMD) packing behavior across finite field evaluation slots.

How to use

FAQ

What does invariant noise budget measure in BFV?

It measures the remaining noise tolerance in bits before homomorphic error corrupts plaintext message decoding.

Why is relinearization necessary after ciphertext multiplication?

Multiplying two 2-element ciphertexts produces a 3-element ciphertext; relinearization reduces it back to 2 elements using evaluation keys.

What condition enables SIMD slot packing in this tool?

Plaintext modulus t must satisfy t ≡ 1 (mod 2n), allowing the plaintext polynomial to split into n independent evaluation slots.

Are these ring parameters secure for production cryptography?

No, parameters like n = 32 and n = 64 are toy sizes strictly intended for fast visual teaching and algorithm inspection.

How does simulated bootstrapping restore the noise budget?

It evaluates the decryption circuit homomorphically to output a fresh ciphertext with a renewed noise budget at the base level.

scheme: simd-batching, ringDegree: 32, levels: 1, slotValues: 3, 5, 7, 9, 11, 13, multiplyDepth: 1, showBootstrap: false

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Key facts

Overview

When to use

How it works

Use cases

Examples

1. Observing Noise Exhaustion During Repeated BFV Squarings

2. Batch SIMD Element-Wise Multiplication

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