Questions tagged [coding-theory]

Coding theory studies the properties of codes and their fitness for specific applications, and typically involves the removal of redundancy and the detection and/or correction of errors in transmitted data.

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ROLLO: ideal codes

In ROLLO - Rank-Ouroboros, LAKE & LOCKER, Ideal codes are definited in Definition 2.1.4. Thanks the Lemma 1, It is proved that every block of ideal matrix is non- singular. Now, why do they say &...
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What is the code when looking at the LPN problem?

We are given the problem in this question. We know that we have to use the algorithm $A_D$ in order to get $e_i$. Our idea is that we construct a vector $l$ of $l_i$'s by getting $n$ samples from the $...
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Solving LPN using algorithm for syndrome decoding

Given an algorithm $A_D$ which solves an instance of the decoding problem $e \in \mathbb{F}_2^n$ in time $T_D$ given a parity check matrix $H \in \mathbb{F}_2^{(n-k)\times n}$ and a syndrome $s \in \...
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Error correcting codes that are indistinguishable from random

Suppose you have a public program P(n) which takes message n and generates an encrypted output (utilizing asymmetric cryptography) for some entity which has the private key to decrypt it. Using a ...
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Error-Correction capabilities of encryption?

This is motivated by this question about the recovery of (corrupted) encrypted files, and in particular the statement (in the answer): But If I know the algorithm and the key and I make a custom ...
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1answer
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Decoding in Reed solomon codes

I have code encoded in GF(7) with primitive 5 Сf(4,1,0,4,5,5). (last four symbols is redundancy) While decoding using DFT we use formula $$ С_k=N^{-1}*c(z^{-kj}) $$ example: $$ C_1 = c(5^{-1*j})/6 ...
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Proof that Niederreiter Cryptosystem is correct

I read about Niederreiter Cryptosystem. I understood the Key Generator, encryption and decryption of the cryptosystem, but if I want to prove that is a correct cryptosystem, what should I do?
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Choosing the Bernoulli distribution for LPN encryption scheme

The symmetric-key encryption scheme from [1] is based on the LPN (learning parity with noise) problem. The definition of the problem is, informally, that the adversary cannot recover $\mathbf{s}$ from ...
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How to solve the system of linear equations to recover original file using “Erasure Coding”?

Following article explains a simplified version of Erasure Coding: Link to the article here is the recipe: Take a file of size M. Split the file into k chunks, each of the same size M/k. ...
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Can error correction and detection be done without adding extra bits?

I have gone through error detection and correction techniques like Hamming codes, and BCH codes require extra parity bits for detection and correction. While sending data, we always seem to introduce ...
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Constructing Low-Density Parity-Check Codes of length $n$ and minimum distance = $\delta n$ over $GF(q)$? [closed]

I am looking for a way to construct an LDPC (Low-Density Parity-Check) Code $C$ of length $n$ and minimum distance $d_C$ that scales linearly to $n$, meaning $d_C = \delta n$ for $\delta \in (0,1)$. ...
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Is McEliece secure with non-binary Goppa codes?

A binary Goppa code with codewords of length $n$ bits that can fix $t$ errors with a polynomial over $GF(2^m)$ can encode $k = n - mt$ bits long data. That is one needs to add $mt$ check bits to fix $...
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Can Reed Solomon parity blocks be used as an all-or-nothing transform?

Consider the following scheme: Perform an (N,N) Reed-Solomon encoding (i.e. N data blocks, N parity blocks) Drop the N data blocks and keep only the N parity blocks. Are these N parity blocks an all-...
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Number of words in binary linear code starting with zero and unit

Is it true that in a binary linear code number of words, which start with zero, is always at least as much as words, which start with 1? I have no idea how to prove or disprove it. Can you give me a ...
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Why doesn't “Classic McEliece” need scrambling?

The original McEliece scheme uses two random matrices S and P to scramble the generator matrix and uses $\mathsf S·\mathsf G·\mathsf P$ as the public key. The Niederreiter variant also does about the ...
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146 views

Ciphertext indistinguishability under a noisy channel w/ error correction

The objective is to broadcast information over a noisy digital channel (corruption rate of say, 10-20 %, where bits flip). Assume the broadcasting software is public and there is no possibility of ...
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2answers
212 views

Reed Solomon secret sharing and as a one-time symmetric key?

Shamir's Secret Sharing is just a special case of Reed-Solomon where only one coefficient is used to store the secret instead of the entire polynomial. Sarwate however, suggests that the latter can ...
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Hardness of LPN problem with small secret

The Learning Parity with Noise (LPN) assumption states that, for a fixed secret $s$ chosen uniformly from $\{0,1\}^n$, then the distribution that outputs $(a,a\cdot s+e)$, where $a$ is uniform in $\{0,...
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1answer
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What is the link between the parity check matrix, double-error-correcting codes and APN permutations?

I am currently reading a research paper (linked below) which mentions that a map $f:V:=GF(2^{m}) \rightarrow V$ which vanishes at 0 is APN if it satisfies the condition that it is a binary code $C_{f}$...
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1answer
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What does 2-flat mean when discussing APN permutations?

I am currently reading a research paper that uses the term 2-flat. However, I don't understand what this term means. For example, the paper mentions in its definition of APN that for all distinct $a$,...
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Unique decoding Radius in Reed Solomon Codes

In one of the coding theory books I read the unique decoding radius for Reed Solomon codes is $\frac{1-\rho}{2}$. Precisely, if the relative distance be less than these amount so the receiver is able ...
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Finding Nonlinear boolean functions

Let $\mathbb{F}_2=\{0,1\}$ be the field with two elements. I wonder if there is any known algorithm/construction that, given any $n\geq 1$, returns a boolean function $f:\mathbb{F}^n_2\rightarrow \...
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Suggestion for proof of retrievability (via coding theory)

I want to build a fully open-source open-everything protocol/service for massively-distributed shared storage (P2P). I came up with a suggestion for a proof-of-retrievability scheme, but I would like ...
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2answers
149 views

Difference between $F_2^n$ and $\Bbb F_2^n$ for a field

I am confused between the notation $F_2^n$ and $\Bbb F_2^n$ for a field in regards to codes. I thought that $F_2^n$ and $\Bbb F_2^n$ were both fields composed by codes of length n and entries in mod ...
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Entropy of the union of two sources [closed]

I am given that $S_1=(S_1,P_1)$ and $S_2=(S_2,P_2)$ are sources, where $S_1=\{s_1,...,s_n\}$, $P_1(x_i)=p_i$ and $S_2=\{y_1,...,s_m\}$, $P_2(y_j)=q_j$. I have to find the entropy of $S_{\lambda}=(S_i ...
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1answer
195 views

Code families in McEliece cryptosytem

What are the families of codes frequently used in McEliece cryptosystem or its variants? I know that binary Goppa codes were used in the original system but many codes with efficient decoding ...
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1answer
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Value of $t$ in Fuzzy Exractor

Is there a defined method to choose value of $t$ when using fuzzy extractor to reconcile two close secrets? I did try with multiple values ranging from 5 to close to half of the sequence. I ...
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1answer
124 views

Traitor tracing - codeword properties

Let $c ≥ 2$. A code $C ⊂ Q^n$ is a $c$-frameproof code if for any set $X ⊂ C$ with $|X| ≤ c$ we have desc$(X) ∩ C = X$. Thus the only codewords that a coalition of up to $c$ pirates is capable of ...
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78 views

Traitor tracing - determining a set of codewords

I am reading some notes and having trouble understanding the following example: Let $F$ be a finite set of size $q$, where $q ≥ 2$. Let $n$ be an integer, where $n ≥ 2$. For a subset $X ⊆ F^n$ of ...
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285 views

Need an example to write permutation in cycle notation [closed]

Actually, I don't quite understand the question. What does $a_{i,j}$ means? Is it the element in the matrix from row i and column j? Can anyone give me an example? Better use another matrix because I ...
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711 views

What does this notation stand for when describing a code?

This code has appeared in some online course material. I understand the $(5, 4, 3)$ refers to (length, num codewords, distance) but no explanation of the $Z_2^5$ notation is given: One $(5,4,3)$ ...
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How to hash similar strings to the same hash value?

Suppose that $s_1$ and $s_2$ are two stings that have a small hamming distance. Is there a preimage resistant "hash" function ($H$) that can map them to the same value i.e., $H(s_1) = H(s_2)$?
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Words having weight near to minimum distance

I am studying the NP-Problem of the codes Syndrome Decoding. The formulation is show below. Input: a binary matrix $H$ of dimension $r \times n$ and a bit string $S$ of length $r$. Property: there ...
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Convolution and catastrophic codes

i'm reading the article of Massey and Sain (here) and i cannot unserstand - what is "foreforward inversion"? I mean There is a description of circles in convolutional codes and a little bit ...
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1answer
352 views

Implementing the Mceliece Encryption - making the Generator Matrix

I am working on an implementation of the Mceliece Encryption system (MCE) and the Niederreiter encryption system. I have been through the basics of finite fields, polynomial arithmetic and some coding ...
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1answer
81 views

Question about block erasure codes

I have a question about linear block erasure codes that are described in this paper. I briefly describe the idea behind the linear erasure codes and then I ask my question. Given a set $d=\langle x_i ...
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331 views

About using mistakes as part of a code

Could a code be developed where one uses intentional errors say in english, as a text to encode? For example someone might have a message 'Agent X must report to station 5'. This could be distorted ...
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Does there exist a proof-of-retrievability scheme that is publicly-verifiable, limited-use, and does not use homomorphic encryption?

I find myself wanting to test out a practical implementation of a proof-of-retrievability scheme, simply out of curiosity. These schemes seem to be divided into two variations, publicly-verified and ...
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3answers
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RS Erasure Coding and Shamir's Secret Sharing

So I was trying to understand the basic difference between erasure coding and secret sharing, and I found this paper (that you can find here or here). For what I understand, it states that Shamir's ...
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3answers
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Where can I find useful data for cryptography/coding theory?

When implementing cryptographic/coding theory algorithms one need to use data like big prime numbers, numbers in $Z_n$ and their inverses, irreducible polynomials in $Z_n[x]$ and so on... While ...
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1answer
346 views

Hermitian curves introductory references

Could you give me some reference to start on Hermitian Curves. Some papers or textbooks would be perfect, and please mention if it's math inclined or comp.sci. inclined. I've only seen hermitian ...
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7answers
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Current mathematics theory used in cryptography/coding theory

What are the mainstream techniques borrowed from algebraic geometry (or some other branch of mathematics) which are currently used in cryptography/coding theory? I've only heard about a small subset ...