# learning with errors

Published articles for learning with errors.

This is one page of public article previews, not the complete archive. Follow Next page to continue. Summaries are not the original full articles.

## Encoding Schemes in FHE

DevFeed: [Encoding Schemes in FHE](<https://devfeed.tech/articles/encoding-schemes-in-fhe-40476.md>)

Original publisher: [Read original article](<https://www.jeremykun.com/2023/09/18/encoding-schemes-in-fhe/>)

Published: 2023-09-18T11:08:39Z

Content type: tutorial

Language: en

Sources: [Jeremy Kun](<https://devfeed.tech/sources/jeremy-kun.md>)

Topics: [Encoding](<https://devfeed.tech/topics/encoding.md>), [FHE](<https://devfeed.tech/topics/fhe.md>), [homomorphic encryption](<https://devfeed.tech/topics/homomorphic-encryption.md>), [Cryptography](<https://devfeed.tech/topics/cryptography.md>), [Decoding](<https://devfeed.tech/topics/decoding.md>)

Tags: [bits](<https://devfeed.tech/tags/bits.md>), [complex-numbers](<https://devfeed.tech/tags/complex-numbers.md>), [cryptography](<https://devfeed.tech/tags/cryptography.md>), [decoding](<https://devfeed.tech/tags/decoding.md>), [encoding](<https://devfeed.tech/tags/encoding.md>), [fhe](<https://devfeed.tech/tags/fhe.md>), [homomorphic-encryption](<https://devfeed.tech/tags/homomorphic-encryption.md>), [learning-with-errors](<https://devfeed.tech/tags/learning-with-errors.md>), [lwe](<https://devfeed.tech/tags/lwe.md>), [mathematics](<https://devfeed.tech/tags/mathematics.md>), [number-theory](<https://devfeed.tech/tags/number-theory.md>), [programming](<https://devfeed.tech/tags/programming.md>), [residue-number-system](<https://devfeed.tech/tags/residue-number-system.md>), [rlwe](<https://devfeed.tech/tags/rlwe.md>), [scaling](<https://devfeed.tech/tags/scaling.md>), [scheme](<https://devfeed.tech/tags/scheme.md>)

### AI overview

This article explains how cleartexts are transformed into plaintexts through encoding, and reversed through decoding, in homomorphic encryption. It catalogs encoding approaches for different HE schemes, including bit-field encoding for LWE and considerations for noise, scaling, bit positions, and programmable bootstrapping in CGGI/TFHE.

### Source excerpt

In cryptography, we need a distinction between a cleartext and a plaintext. A cleartext is a message in its natural form. A plaintext is a cleartext that is represented in a specific way to prepare it for encryption in a specific scheme. The process of taking a cleartext and turning it into a plaintext is called encoding, and the reverse is called decoding. In homomorphic encryption, the distinction matters. Cleartexts are generally all integers, though the bit width of allowed integers can be restricted (e.

## Estimating the Security of Ring Learning with Errors (RLWE)

DevFeed: [Estimating the Security of Ring Learning with Errors (RLWE)](<https://devfeed.tech/articles/estimating-the-security-of-ring-learning-with-errors-rlwe-40461.md>)

Original publisher: [Read original article](<https://www.jeremykun.com/2022/12/28/estimating-the-security-of-ring-learning-with-errors-rlwe/>)

Published: 2022-12-28T14:52:01Z

Content type: tutorial

Language: en

Sources: [Jeremy Kun](<https://devfeed.tech/sources/jeremy-kun.md>)

Topics: [Cryptography](<https://devfeed.tech/topics/cryptography.md>), [Security](<https://devfeed.tech/topics/security.md>), [Estimator](<https://devfeed.tech/topics/estimator.md>), [FHE](<https://devfeed.tech/topics/fhe.md>)

Tags: [cod](<https://devfeed.tech/tags/cod.md>), [cryptography](<https://devfeed.tech/tags/cryptography.md>), [estimator](<https://devfeed.tech/tags/estimator.md>), [lattice-cryptography](<https://devfeed.tech/tags/lattice-cryptography.md>), [learning-with-errors](<https://devfeed.tech/tags/learning-with-errors.md>), [lwe](<https://devfeed.tech/tags/lwe.md>), [mathematics](<https://devfeed.tech/tags/mathematics.md>), [programming](<https://devfeed.tech/tags/programming.md>), [ring-learning-with-errors](<https://devfeed.tech/tags/ring-learning-with-errors.md>), [rlwe](<https://devfeed.tech/tags/rlwe.md>), [sage](<https://devfeed.tech/tags/sage.md>), [security](<https://devfeed.tech/tags/security.md>), [test](<https://devfeed.tech/tags/test.md>), [top](<https://devfeed.tech/tags/top.md>)

### AI overview

The article explains how to estimate the security of lattice-based schemes based on Learning With Errors (LWE) and Ring Learning With Errors (RLWE). It introduces LWE parameters and describes how the Lattice Estimator estimates the costs of known lattice attacks for a given instance.

### Source excerpt

This article was written by my colleague, Cathie Yun. Cathie is an applied cryptographer and security engineer, currently working with me to make fully homomorphic encryption a reality at Google. She's also done a lot of cool stuff with zero knowledge proofs. In previous articles, we've discussed techniques used in Fully Homomorphic Encryption (FHE) schemes. The basis for many FHE schemes, as well as other privacy-preserving protocols, is the Learning With Errors (LWE) problem.

## Key Switching in LWE

DevFeed: [Key Switching in LWE](<https://devfeed.tech/articles/key-switching-in-lwe-40456.md>)

Original publisher: [Read original article](<https://www.jeremykun.com/2022/08/29/key-switching-in-lwe/>)

Published: 2022-08-29T08:00:00Z

Content type: tutorial

Language: en

Sources: [Jeremy Kun](<https://devfeed.tech/sources/jeremy-kun.md>)

Topics: [Encryption](<https://devfeed.tech/topics/encryption.md>), [Cryptography](<https://devfeed.tech/topics/cryptography.md>), [scheme](<https://devfeed.tech/topics/scheme.md>)

Tags: [cryptography](<https://devfeed.tech/tags/cryptography.md>), [decrypt](<https://devfeed.tech/tags/decrypt.md>), [encryption](<https://devfeed.tech/tags/encryption.md>), [gadget-decomposition](<https://devfeed.tech/tags/gadget-decomposition.md>), [gaussian](<https://devfeed.tech/tags/gaussian.md>), [homomorphic-encryption](<https://devfeed.tech/tags/homomorphic-encryption.md>), [learning-with-errors](<https://devfeed.tech/tags/learning-with-errors.md>), [mathematics](<https://devfeed.tech/tags/mathematics.md>), [modulus-switching](<https://devfeed.tech/tags/modulus-switching.md>), [real-analysis](<https://devfeed.tech/tags/real-analysis.md>), [secret](<https://devfeed.tech/tags/secret.md>)

### AI overview

This tutorial explains key switching in the LWE encryption scheme: transforming a ciphertext encrypted under one secret key into a ciphertext under another without knowing either key. It introduces the key-switching key, shows why the naïve approach causes excessive error growth, and explains how gadget decomposition reduces that error.

### Source excerpt

Last time we covered an operation in the LWE encryption scheme called modulus switching, which allows one to switch from one modulus to another, at the cost of introducing a small amount of extra noise, roughly $\sqrt{n}$, where $n$ is the dimension of the LWE ciphertext. This time we'll cover a more sophisticated operation called key switching, which allows one to switch an LWE ciphertext from being encrypted under one secret key to another, without ever knowing either secret key.

## Modulus Switching in LWE

DevFeed: [Modulus Switching in LWE](<https://devfeed.tech/articles/modulus-switching-in-lwe-40455.md>)

Original publisher: [Read original article](<https://www.jeremykun.com/2022/07/16/modulus-switching-in-lwe/>)

Published: 2022-07-16T14:39:56Z

Content type: tutorial

Language: en

Sources: [Jeremy Kun](<https://devfeed.tech/sources/jeremy-kun.md>)

Topics: [FHE](<https://devfeed.tech/topics/fhe.md>), [homomorphic encryption](<https://devfeed.tech/topics/homomorphic-encryption.md>), [Cryptography](<https://devfeed.tech/topics/cryptography.md>), [Encryption](<https://devfeed.tech/topics/encryption.md>), [Algorithms, Complexity](<https://devfeed.tech/topics/algorithms-complexity.md>)

Tags: [algorithms](<https://devfeed.tech/tags/algorithms.md>), [cryptography](<https://devfeed.tech/tags/cryptography.md>), [encryption](<https://devfeed.tech/tags/encryption.md>), [fhe](<https://devfeed.tech/tags/fhe.md>), [homomorphic-encryption](<https://devfeed.tech/tags/homomorphic-encryption.md>), [learning-with-errors](<https://devfeed.tech/tags/learning-with-errors.md>), [mathematics](<https://devfeed.tech/tags/mathematics.md>), [modulus-switching](<https://devfeed.tech/tags/modulus-switching.md>), [programming](<https://devfeed.tech/tags/programming.md>)

### AI overview

This tutorial explains modulus switching in Learning With Errors (LWE), a technique used in some fully homomorphic encryption schemes. It introduces an LWE encryption scheme and describes how ciphertexts can be converted to a smaller modulus while addressing the role of error in encryption and decryption.

### Source excerpt

The Learning With Errors problem is the basis of a few cryptosystems, and a foundation for many fully homomorphic encryption (FHE) schemes. In this article I'll describe a technique used in some of these schemes called modulus switching. In brief, an LWE sample is a vector of values in $\mathbb{Z}/q\mathbb{Z}$ for some $q$, and in LWE cryptosystems an LWE sample can be modified so that it hides a secret message $m$.

## The Gadget Decomposition in FHE

DevFeed: [The Gadget Decomposition in FHE](<https://devfeed.tech/articles/the-gadget-decomposition-in-fhe-40450.md>)

Original publisher: [Read original article](<https://www.jeremykun.com/2021/12/11/the-gadget-decomposition-in-fhe/>)

Published: 2021-12-11T13:57:25Z

Content type: tutorial

Language: en

Sources: [Jeremy Kun](<https://devfeed.tech/sources/jeremy-kun.md>)

Topics: [FHE](<https://devfeed.tech/topics/fhe.md>), [homomorphic encryption](<https://devfeed.tech/topics/homomorphic-encryption.md>), [Cryptography](<https://devfeed.tech/topics/cryptography.md>), [Computing](<https://devfeed.tech/topics/computing.md>), [data](<https://devfeed.tech/topics/data.md>), [Code](<https://devfeed.tech/topics/code.md>), [GitHub](<https://devfeed.tech/topics/github.md>)

Tags: [bootstrapping](<https://devfeed.tech/tags/bootstrapping.md>), [core](<https://devfeed.tech/tags/core.md>), [encryption](<https://devfeed.tech/tags/encryption.md>), [fhe](<https://devfeed.tech/tags/fhe.md>), [gadget-decomposition](<https://devfeed.tech/tags/gadget-decomposition.md>), [group-theory](<https://devfeed.tech/tags/group-theory.md>), [homomorphic-encryption](<https://devfeed.tech/tags/homomorphic-encryption.md>), [learning-with-errors](<https://devfeed.tech/tags/learning-with-errors.md>), [linear-algebra](<https://devfeed.tech/tags/linear-algebra.md>), [lwe](<https://devfeed.tech/tags/lwe.md>), [mathematics](<https://devfeed.tech/tags/mathematics.md>), [matrix](<https://devfeed.tech/tags/matrix.md>), [number-theory](<https://devfeed.tech/tags/number-theory.md>), [operations](<https://devfeed.tech/tags/operations.md>), [programming](<https://devfeed.tech/tags/programming.md>), [python](<https://devfeed.tech/tags/python.md>)

### AI overview

A tutorial on gadget decomposition in fully homomorphic encryption (FHE). It explains how GSW and related schemes use random noise, how homomorphic operations increase that noise, why bootstrapping is needed, and how gadget decomposition helps limit noise growth.

### Source excerpt

Lately I've been studying Fully Homomorphic Encryption, which is the miraculous ability to perform arbitrary computations on encrypted data without learning any information about the underlying message. It's the most comprehensive private computing solution that can exist (and it does exist!). The first FHE scheme by Craig Gentry was based on ideal lattices and was considered very complex (I never took the time to learn how it worked). Some later schemes (GSW = Gentry-Sahai-Waters) are based on matrix multiplication, and are conceptually much simpler.