Different Types of Matrices

Matrices

Zero Matrix

• Every element is zero, exemplified by $\begin{bmatrix}0 & 0 \ 0 & 0\end{bmatrix}$.

Identity Matrix

• A square matrix, having 1s on the diagonal and 0s elsewhere, for example $\begin{bmatrix}1 & 0 \ 0 & 1\end{bmatrix}$.

Diagonal Matrix

• A square matrix where all non-diagonal elements equal zero, such as $\begin{bmatrix}a & 0 \ 0 & d\end{bmatrix}$.

Transpose Matrix

• Denoted as $A^T$, formed by swapping the rows and columns of matrix A.
• For $A = \begin{bmatrix}a & b \ c & d \ e & f\end{bmatrix}$, the transpose is $A^T = \begin{bmatrix}a & c & e\ b & d & f\end{bmatrix}$.

Symmetric Matrix

• Square matrix that equals its transpose, $A = A^T$, as in $A = \begin{bmatrix}a & b \ b & d\end{bmatrix}$.

Skew-Symmetric Matrix

• Square matrix where the transpose equals its negative, $A^T = -A$.
• Diagonal elements are always zero, for example $A = \begin{bmatrix}0 & b \ -b & 0\end{bmatrix}$.

Triangular Matrix

• Upper Triangular Matrix: All elements below the main diagonal are zero, such as $\begin{bmatrix}a & b \ 0 & d\end{bmatrix}$.
• Lower Triangular Matrix: All elements above the main diagonal are zero, such as $\begin{bmatrix}a & 0 \ c & d\end{bmatrix}$.

Matrix Operations

• A and B are matrices of the same size; c is a scalar.

Addition

• Add corresponding elements: $A + B = [a_{ij} + b_{ij}]$.
• For example, $\begin{bmatrix}a & b \ c & d\end{bmatrix} + \begin{bmatrix}e & f \ g & h\end{bmatrix} = \begin{bmatrix}a+e & b+f \ c+g & d+h\end{bmatrix}$.

Subtraction

• Subtract corresponding elements: $A - B = [a_{ij} - b_{ij}]$.
• For example, $\begin{bmatrix}a & b \ c & d\end{bmatrix} - \begin{bmatrix}e & f \ g & h\end{bmatrix} = \begin{bmatrix}a-e & b-f \ c-g & d-h\end{bmatrix}$.

Scalar Multiplication

• Multiply each element by the scalar: $cA = [ca_{ij}]$.
• For example, $c\begin{bmatrix}a & b \ c & d\end{bmatrix} = \begin{bmatrix}ca & cb \ cc & cd\end{bmatrix}$.

Matrix Multiplication

• If A is an $m \times n$ matrix and B is an $n \times p$ matrix, then the product AB is an $m \times p$ matrix.
• The (i,j)-th entry of AB obtained by multiplying the entries of the i-th row of A by the corresponding entries of the j-th column of B and then adding them.
• $(AB){ij} = \sum{k=1}^{n} a_{ik}b_{kj}$
• For example, $\begin{bmatrix}a & b \ c & d\end{bmatrix} \begin{bmatrix}e & f \ g & h\end{bmatrix} = \begin{bmatrix}ae+bg & af+bh \ ce+dg & cf+dh\end{bmatrix}$.

Algorithmes de Hachage

Définition

• A function whose input of arbitrary size produces a fixed-size output, known as a hash.

Propriétés

• Deterministic: Same input yields same hash.
• Fast Computation: Hash calculation must be quick.
• Preimage Resistance: Difficulty finding an input that generates a given hash.
• Second Preimage Resistance: Difficulty finding a second, different input that generates the same hash as a given one.
• Collision Resistance: Difficulty finding two unique inputs that generate the same hash.
• One-way Function: Easy calculation in one direction, hard to reverse.
• Avalanche Effect: Minor change in input results in a significant change in hash.

Applications

• Data Integrity Verification: Compare hashes of two file versions to detect changes.
• Password Storage: Store hashed passwords (instead of the actual passwords) for security.
• Hash Tables: Use hashes as indices for storing and retrieving data.
• Digital Signatures: Hash a document before digitally signing it for efficiency.
• Blockchain: Use hashes to link blocks in a blockchain and ensure chain integrity.

Exemples d'algorithmes de hachage

• MD5: Produces a 128-bit hash (obsolete due to security weaknesses).
• SHA-1: Produces a 160-bit hash (obsolete due to security weaknesses).
• SHA-256: Produces a 256-bit hash (widely used).
• SHA-384: Produces a 384-bit hash.
• SHA-512: Produces a 512-bit hash.
• SHA-3: A family of hashing algorithms replacing SHA-2.
• bcrypt: Password hashing algorithm based on Blowfish.
• scrypt: Password hashing algorithm needing a large amount of memory.
• Argon2: Award-winning password hashing algorithm.

Attaques Courantes

• Attaque par force brute: Trying all possible inputs until finding one that produces the target hash.
• Attaque par dictionnaire: Trying inputs from a dictionary of common words/phrases until finding one that produces the target hash.
• Attaque arc-en-ciel: Using a pre-calculated table of hashes to find an input that produces the target hash.
• Attaque de collision: Finding two different inputs that produce the same hash.
• Attaque de longueur d'extension: Exploiting the internal structure of some hashing algorithms to calculate the hash of a longer input without knowing the original input.

Sélection d'un Algorithme de Hachage

• Sécurité: Choose an algorithm resistant to known attacks.
• Performance: Choose an algorithm fast enough for the application.
• Taille du haché: Choose a hash size large enough to avoid collisions.
• Disponibilité: Choose a widely available and supported algorithm.
• Standard: Choose a standardized algorithm to ensure interoperability.

Conclusion

• Hashing algorithms are essential for many security applications. It's important to understand properties, applications and limits to use them effectively.

Factorial ANOVA

Basic Concepts

• Factorial Design
  • Two or more IVs are manipulated.
  • Allows investigation of interaction effects.
• Example
  • Exercise and Diet affect Weight Loss with two IVs.
    • Exercise: yes or no.
    • Diet: yes or no.
    • Four groups of participants.
    • Exercise and diet.
    • Exercise and no diet.
    • No exercise and diet.
    • No exercise and no diet.
• Main Effect
  • The effect of one IV on the DV, ignoring the other IV(s).
• Interaction Effect
  • The effect of one IV on the DV depends on the level of the other IV(s).

Two-Way ANOVA

• Hypotheses
  • Main effect of exercise: $H_0: \mu_{exercise} = \mu_{no\ exercise}$.
  • Main effect of diet: $H_0: \mu_{diet} = \mu_{no\ diet}$.
  • Interaction effect: The effect of exercise on weight loss depends on diet.
• ANOVA Table

• Degrees of Freedom
  • $df_{ex} = a - 1$, where $a$ is the number of levels of exercise.
  • $df_{di} = b - 1$, where $b$ is the number of levels of diet.
  • $df_{ex*di} = (a - 1)(b - 1)$.
  • $df_e = N - ab$, where $N$ is the total number of participants.
  • $df_t = N - 1$.
• Sum of Squares
  • $SS_{ex} = n \cdot b \cdot \sum_{i=1}^{a} (\bar{x}{i.} - \bar{x}{..})^2$, where $n$ is the number of participants in each group, $\bar{x}{i.}$ is the mean of exercise level $i$, and $\bar{x}{..}$ is the grand mean.
  • $SS_{di} = n \cdot a \cdot \sum_{j=1}^{b} (\bar{x}{.j} - \bar{x}{..})^2$, where $\bar{x}_{.j}$ is the mean of diet level $j$.
  • $SS_{ex*di} = n \cdot \sum_{i=1}^{a} \sum_{j=1}^{b} (\bar{x}{ij} - \bar{x}{i.} - \bar{x}{.j} + \bar{x}{..})^2$, where $\bar{x}_{ij}$ is the mean of cell $ij$.
  • $SS_t = \sum_{i=1}^{a} \sum_{j=1}^{b} \sum_{k=1}^{n} (x_{ijk} - \bar{x}{..})^2$, where $x{ijk}$ is the score of participant $k$ in cell $ij$.
  • $SS_e = SS_t - SS_{ex} - SS_{di} - SS_{ex*di}$.
• Mean Squares
  • $MS_{ex} = \frac{SS_{ex}}{df_{ex}}$.
  • $MS_{di} = \frac{SS_{di}}{df_{di}}$.
  • $MS_{exdi} = \frac{SS_{exdi}}{df_{ex*di}}$.
  • $MS_e = \frac{SS_e}{df_e}$.
• F-Ratios
  • $F_{ex} = \frac{MS_{ex}}{MS_e}$.
  • $F_{di} = \frac{MS_{di}}{MS_e}$.
  • $F_{exdi} = \frac{MS_{exdi}}{MS_e}$.
• Example
  • $N = 20$.
  • $n = 5$. | | Diet | No diet | | | :---------- | :--- | :------ | :---- | | Exercise | 8 | 4 | $\mu$ = 6 | | No exercise | 2 | 6 | $\mu$ = 4 | | | $\mu$ = 5 | $\mu$ = 5 | |

$H_0$: No main effect of exercise: $(6 - 5) = (4 - 5) = 0$.

$H_0$: No main effect of diet: $(5 - 5) = (5 - 5) = 0$.

$H_0$: No interaction effect: $(8 - 4) = (2 - 6) = 0$.

• ANOVA Table

• Calculation of SS
  • $SS_{exercise} = n \cdot b \cdot \sum_{i=1}^{a} (\bar{x}{i.} - \bar{x}{..})^2 = 5 \cdot 2 \cdot [(6 - 5)^2 + (4 - 5)^2] = 20$.
  • $SS_{diet} = n \cdot a \cdot \sum_{j=1}^{b} (\bar{x}{.j} - \bar{x}{..})^2 = 5 \cdot 2 \cdot [(5 - 5)^2 + (5 - 5)^2] = 0$.
  • $SS_{exercise*diet} = n \cdot \sum_{i=1}^{a} \sum_{j=1}^{b} (\bar{x}{ij} - \bar{x}{i.} - \bar{x}{.j} + \bar{x}{..})^2 = 5 \cdot [(8 - 6 - 5 + 5)^2 + (4 - 6 - 5 + 5)^2 + (2 - 4 - 5 + 5)^2 + (6 - 4 - 5 + 5)^2] = 60$.
  • $SS_t = \sum_{i=1}^{a} \sum_{j=1}^{b} \sum_{k=1}^{n} (x_{ijk} - \bar{x}_{..})^2$.
  • $SS_e = SS_t - SS_{ex} - SS_{di} - SS_{ex*di} = 144 - 20 - 0 - 60 = 64$.
• Interpretation
  • Main effect of exercise: $F(1, 16) = 5, p