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le (5m35s)

## 2.1. Formal Definition

Welcome to the second week of this MOOC entitled Code-Based Cryptography. This week, we will talk in detail about the McEliece cryptosystem. First, in this session, we will describe formally the McEliece and the Niederreiter systems, which are the principal public-key schemes, based on error-correcting code. Let K be a security parameter. An encryption scheme is defined by the following spaces: the space of all possible messages, the space of all ciphertexts, the space of the public-keys, and the space of the secret-keys.Then, we need to define the ...
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le (4m43s)

## 2.2. Security-Reduction Proof

Welcome to the second session. We will talk about the security-reduction proof. The security of a given cryptographic algorithm is reduced to the security of a known hard problem. To prove that a cryptosystem is secure, we select a problem which we know is hard to solve, and we reduce the problem to the security of the cryptosystem. Since the problem is hard to solve, the cryptosystem is hard to break. A security reduction is a proof that an adversary able to attack the scheme is able to solve some presumably hard ...
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le (3m15s)

## 2.3. McEliece Assumptions

In this session, we will talk about McEliece assumptions. The security of the McEliece scheme is based on two assumptions as we have already seen: the hardness of decoding a random linear code and the problem of distinguishing a code with a prescribed structure from a random one. In this sequence, we will study in detail these two assumptions. The first assumption claims that decoding a random linear code is difficult.  First, notice that the general decoding problem is basically a re-writing of the Syndrome Decoding problem. And both are equivalent ...
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le (5m32s)

## 2.4. Notions of Security

In this session, we will study the notion of security of public-key scheme. A public-key scheme is one-way if the probability of success of any adversary running in polynomial time is negligible. That is, without the private key, it is computationally impossible to recover the plaintext. For the McEliece, if we assume that the general decoding problem of a linear code is on average a difficult problem and there exists no efficient distinguisher for Goppa codes, then the McEliece scheme has the One-Wayness property. However, McEliece is vulnerable to many ...
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le (5m5s)

## 2.5. Critical Attacks - Semantic Secure Conversions

In this session, we will study critical attacks against the public-key cryptosystem. The partial knowledge on the plaintext reduces drastically the computational cost of the attack to the McEliece cryptosystem. For example, suppose that the adversary knows r bits of the plaintext. Then the difficulty of recovering the remaining k - r bits in the complete McEliece with parameters [n, k] is equivalent to that of recovering the full plaintext in the McEliece with parameters [n, k - r]. This is given by this formula. You just ...
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le (3m45s)

## 2.6. Reducing the Key Size

In the next three sessions, I will explain how to reduce the key size of code-based cryptosystem. Circulant matrices are the central point in many attempts to reduce the key size of code-based cryptosystems since they provide efficient representation. A circulant matrix is a square matrix, its rows are obtained by cyclically shifting the first row. An alternative representation of an n-tuple of elements is using polynomial. Thus, this matrix can be described by a polynomial. And the i-th row of a circulant matrix can be expressed by this formula. ...
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le (4m55s)

## 2.8. Reducing the Key Size - MDPC codes

This is the last session where we will talk about reducing the key size. Here we will introduce the MDPC codes.In 2012, the MDPC codes were proposed for the McEliece schemes. An MDPC code is a code that admits a binary moderate density-parity check matrix. Typically, the Hamming weight of each row is of the order the square of the length. In this sequence, I will describe this scheme of quasi-cyclic MDPC McEliece for a binary code of rate one half. So, we use circulant matrices of blocks of size p ...
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le (3m59s)

## 2.9. Implementation

This is the last session of the second week. The cryptography community has different options for using public key cryptosystems, among others, they have RSA or DSA. But … McEliece has the same level of performance of the current protocol? eBATS is a competition to identify the most efficient public key cryptosystem. They mesure among other criteria: the key size, the time of the key generation algorithm, the encryption algorithm, and the decryption algorithm. The eBATS benchmarking includes seven public key encryption schemes. A McEliece implementation, from Biswas and Sendrier, ...
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