T-type Safety Instrumented Function Block

T-type Safety Instrumented Function Block

• Implements safety functions
• Includes on-line testing to ensure integrity and availability.

Icons and Pinout

• PE_SIF: Input, process enable; execution of safety function is possible only with high signal.
• SD_DEMAND: Input, safety demand; dangerous situation signal.
• TEST: Input, test signal; high signal performs diagnostic test.
• Q: Output of safety function.
• S: Output, status signal; diagnostic status indicator.

Parameters

• Test interval time: Time between automatic tests (default: 10s).
• Test pulse time: Minimal time of test pulse (default: 100ms).
• Manual test pulse time: Minimal manual test pulse time (default: 1s).
• Safe state delay time: Delay time for safe state (default: 0s).
• Fail to danger response: Can be enabled or disabled.
• Error reset type: Auto or Manual.
• Manual reset signal: Input pin for manual error reset.
• Category: Allows grouping of similar blocks.

Functional Description

• Operates in safety function mode and diagnostic test mode.
• Safety Function Mode: Activates when PE_SIF is high, monitors SD_DEMAND.
• Diagnostic Test Mode: Activated by TEST input or internal timer.

Functional Diagram Signals

• PE_SIF: Process Enable SIF; only executes safety functions with high signal.
• SD_DEMAND: Safety Demand; reflects a dangerous situation.
• TEST: Test signal; when high, diagnostic test runs instead of safety function.
• Test.ACTIVE: Internal signal; signals a diagnostic test is in progress.
• Test.ERROR: Internal signal; diagnostic error detected.
• Test.TIMEOUT: Internal signal; indicates a test timeout.
• Q: Output of safety function; de-energized when safety function is active.
• S: Status signal; reports the diagnostic status of the block.
• INIT: Block initialization; TRUE after successful initialization, FALSE on error.

Truth Table

• PE_SIF = 0: Q=1, S=0 Safety function is disabled. Q is energized i.e. inactive state
• PE_SIF = 1, SD_DEMAND = 0, TEST = 0: Q=1, S=0 Normal operation. Q is energized i.e. inactive state
• PE_SIF = 1, SD_DEMAND = 1, TEST = 0: Q=0, S=0 Safety function active. Q is de-energized
• PE_SIF = 1, TEST = 1: Q= Test., S= Test.* Diagnostic test in progress.

Diagnostic Features

• Fault detection includes input signal failures, output signal failures, internal timer failures, and logic solver failures.
• Status signal S indicates diagnostic status and fail-to-danger response can be activated.

Error Reset Options

• Auto: Error resets after successful test completion.
• Manual: Error remains until a reset signal is applied.

Category Parameter

• Used to group similar blocks and is visible in library/catalog but not on the block symbol or in the Logic Builder.

Safety Parameters

• Safety Integrity Level (SIL): Designed for SIL 3 applications (IEC 61508).
• Probability of Failure on Demand (PFD): Must be calculated (IEC 61508).
• Diagnostic Coverage (DC): Determined by on-line testing effectiveness.
• Mean Time to Repair (MTTR): Must be considered.

Example Block Functional Diagram

• T-type SIF block contains four input signals (PE_SIF, SD_DEMAND, TEST), several internal signals (Test.Active, Test.Error, Test.Timeout, INIT), and two outputs (Q, S).
• Flowchart illustrates the interplay between the SIF logic path and the diagnostic test procedure.

Why is cryptography difficult?

• Complexity: Due to interactions between different components.
• Counter-intuitive: Attackers may act unexpectedly.
• Paranoia: Assume attackers know everything except the key.
• Evolution: Techniques and tools continually change.

What do you need to know for cryptography?

• Basics: Modular arithmetic, linear algebra, probability, information theory.
• Classical: Caesar, Vigenère, historical methods.
• Modern Symmetric: AES, ChaCha20, modes of operation (CBC, CTR, GCM).
• Asymmetric (Public Key): RSA, Diffie-Hellman, ECC.
• Cryptographic Hash Functions: SHA-256, SHA-3.
• Message Authentication Codes (MAC): HMAC.
• Digital Signatures: RSA, DSA, ECDSA.
• Cryptographic Protocols: TLS, SSH, Signal.

How to approach a cryptography problem

• Problem definition: Identify known and unknown elements, creating a model of the system.
• Security assessment: Locate known vulnerabilities, considering common attacks and countermeasures.
• Research and experimentation: Find relevant resources, testing and simulations.
• Implementation and verification: Develop prototypes and test, analyzing performance and improvements.

Practical tips for cryptography problems

• Use tested cryptographic libraries (e.g., OpenSSL, PyCryptodome).
• Understand key lengths and their impact on security.
• Keep up with developments and standards in the field.
• Document assumptions, decisions, and test results.
• Seek feedback from experts.

Useful resources for cryptography problems

• Textbooks: "Cryptography Engineering" by Ferguson, Schneier, Kohno.
• Online courses: Coursera, edX, Khan Academy.
• Research Articles: IACR publications.
• Tools: Wireshark, Nmap, Burp Suite.

Resolution Methodology: AES Cryptoanalysis example

• Given ciphertext and partial plaintext, find the AES key.
• AES is a symmetric block cipher with key sizes of 128, 192, or 256 bits.
• Look for side-channel attacks or known/chosen plaintext attacks.
• Implement known plaintext attack.
• Use tools like PyCryptodome for testing, and document each step.

Additional Tips

• Visualize protocols with flow diagrams.
• Automate tasks with scripts.
• Collaborate to gain different perspectives.

Conclusion for cryptography

• Solving cryptographic problems requires theoretical knowledge, practical skills, and analytical thinking.

Inference rules

• Logical forms that permit deriving valid conclusions from presented premises.

Modus Ponens (MP)

• If P then Q. P is true. Therefore, Q it true.

Modus Tollens (MT)

• If P then Q. Q is false. Therefore, P is false.

Hypothetical Syllogism (HS)

• If P then Q. If Q then R. Therefore, if P then R.

Disjunctive Syllogism (DS)

• P or Q is true. P is false. Therefore, Q is true.

Simplification (Simp)

• P and Q are true. Therefore, P is true.

Conjunction (Adj)

• P is true. Q is true. Therefore, P and Q are true.

Addition (Ad)

• P is true. Therefore, P or Q is true.

Double Negation (DN)

• Not not P is true. Therefore, P is true.
• These rules are foundational and utilized for valid and proving theorems.

Common Functions in calculus

• Linear: f(x) = mx + b
• Quadratic = f(x) = ax^2 + bx + c
• Polynomial = f(x) = a_n x^n + a_{n-1} x^{n-1} +... + a_1 x + a_0
• Rational: f(x) = p(x) / q(x), where p(x) and q(x) are polynomials
• Exponential: f(x) = a^x, where a > 0
• Logarithmic: f(x) = log_a(x), where a > 0 and a ≠ 1
• Trigonometric: f(x) = sin(x), f(x) = cos(x), f(x) = tan(x), etc.

Definitions for calculus

• Limit: Value that a function "approaches" as the input "approaches" some value. $\lim_{x \to a} f(x) = L$ means that as x gets closer to a, f(x) gets closer to L.
• Continuity: A function is continuous at a point if: the limit exists at that point, the function is defined at that point, and the limit is equal to the function value. $f(x)$ is continuous at $x = a$ if $\lim_{x \to a} f(x) = f(a)$.
• Derivative: The instantaneous rate of change of a function with respect to one of its variables. $f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$
• Integral: Represents the area under a curve. $\int_a^b f(x) dx$ gives the area under the curve $f(x)$ from $x = a$ to $x = b$.

Theorems for calculus

• Intermediate Value Theorem (IVT): If f is continuous on the closed interval [a, b] and k is any number between f(a) and f(b), then there exists at least one number c in the interval (a, b) such that f(c) = k.
• Mean Value Theorem (MVT): If f is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then there exists at least one number c in the interval (a, b) such that $f'(c) = \frac{f(b) - f(a)}{b - a}$.
• Fundamental Theorem of Calculus (FTC): Part 1: If $f(x)$ is continuous on $[a, x]$, then $\frac{d}{dx} \int_a^x f(t) dt = f(x)$. Part 2: $\int_a^b F'(x) dx = F(b) - F(a)$, where $F'(x) = f(x)$.

Applications of Derivatives

• Optimization: Finding the maximum or minimum values of a function.
• Related Rates: Determining how the rate of change of one quantity affects the rate of change of another.
• Curve Sketching: Using derivatives to analyze the shape of a graph (increasing, decreasing, concavity, etc.).

Applications of Integrals

• Area Between Curves: Finding the area between two or more curves.
• Volume of Solids: Finding the volume of a solid of revolution.
• Average Value of a Function: Determining the average value of a function over an interval.

Techniques of Integration

• Substitution (u-substitution): Reversing the chain rule.
• Integration by Parts: Reversing the product rule.
• Partial Fractions: Decomposing rational functions into simpler fractions.

Infinite Series

• Convergence: An infinite series converges if the sequence of its partial sums approaches a finite limit.
• Divergence: An infinite series diverges if it does not converge.
• Tests for Convergence: Ratio Test, Root Test, Comparison Test, etc.

Differential Equations

• Ordinary Differential Equation (ODE): An equation that relates a function to its derivatives.
• Solution to an ODE: A function that satisfies the differential equation.
• Types of ODEs: Separable, Linear, Exact, etc.

Systems of Linear Equations

• Linear Equation: $a_1x_1 + a_2x_2 + … + a_nx_n = b$
• System of Linear Equations: Set of linear equations.
• Solution: Values for variables satisfying all equations.
• Compatible system: Has at least one solution.
• Incompatible system: Has no solution.

Matrix Representation

• $Ax = b$, where A is coefficients matrix, x is variable vector, b is constants vector.
• Augmented Matrix: Constants vector appended to coefficients matrix.

Elementary Operations

• Exchange of two rows.
• Multiplication of a row by a nonzero constant.
• Addition of a multiple of one row to another row.

Solving Linear Equations

Gauss´ Method

• Transform the augmented matrix into reduced echelon form.

Rouché-Fontené Theorem

• For a system $Ax = b$ with $m \times n$matrix A:
• If $rank(A) < rank([A|b])$, the system is incompatible.
• If $rank(A) = rank([A|b]) = n$, the system is compatible and determined.
• If $rank(A) = rank([A|b]) < n$, the system is compatible and indeterminate.

Examples

• Solving linear equation systems using Gauss method involving row operations.
• Solving linear equation systems with Rouche-Fontene involves rank analysis.

Point Estimation

• Function of data to estimate population parameter.
• Unbiasedness: expected value equals true parameter.
• Efficiency: small variance.
• Consistency: converges in probability as sample size increases.

Methods of Finding Estimators

• Method of Moments: Equate sample moments to population moments and solve for parameters.
• Sample moment:* $\frac{1}{n}\sum_{i=1}^{n}X_i^k$
• Population moment:* $E(X^k)$
• Maximum Likelihood Estimation (MLE): Maximize likelihood function.
• Likelihood function:* $L(\theta;x) = f(x|\theta)$

Confidence intervals

• Range likely containing the true parameter with certain confidence.
• General Formula: Estimator ± (Critical Value) * (Standard Error)
• Mean (σ known): $\bar{x} \pm z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}}$
• Mean (σ unknown): $\bar{x} \pm t_{n-1,\alpha/2} \cdot \frac{s}{\sqrt{n}}$
• Proportion: $\hat{p} \pm z_{\alpha/2} \cdot \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$

Hypothesis Testing

• Procedure to determine if sufficient evidence exists to reject the null.

Steps in Hypothesis Testing

• State null and alternative hypotheses.
• Choose significance level (α).
• Calculate test statistic.
• Determine p-value.
• Reject null if p-value < α.

Common Hypothesis Tests

• Mean (σ known): z-test.
• Mean (σ unknown): t-test.
• Proportion: z-test.
• Two means: t-test or z-test.
• Two proportions: z-test.
• Chi-squared: for independence between categorical variables.

Regression Analysis

• Modeling relationship between dependent and independent variables.
• Simple linear Equation: $y = \beta_0 + \beta_1x + \epsilon$. # $beta_0$: intercept $\beta_1$: slope. # $\epsilon$: error term.
• Multiple linear: $y = \beta_0 + \beta_1x_1 + \beta_2x_2 +... + \beta_kx_k + \epsilon$

Assumptions of Linear Regression

• Linearity
• Independence
• Homoscedasticity
• Normality of error term. R-squared: measures how well the model fits.

Formulas for Spatial Geometry

Prisms

• For upright prisms, $S_{tp} = S_{xq} + 2B$ and $V = B * h$.

Pyramids

• For pyramids, the volume $V= (1/3) B*h$.

Cylinders

• $S{xq} = 2 \pi r h$.
• $S_{tp} = S_{xq} + 2 \pi R^{2}$.
• The volume $ \pi r^{2} h$.

Cones

• $S_{xq} = \pi r l$.
• $S_{tp} = \pi r l + \pi r^2$.
• $ V= \frac{1}{3} \pi r^{2}H$.

Spheres

• $S= 4 \pi R^2$; $V= \frac{4}{3} r \pi^{3}$.

Heat Equation

• Describes the diffusion of heat in a given region over time,
• $\frac{\partial u}{\partial t} = k \nabla^2 u$. Where: $u(x, t)$ is the temperature at point $x$ and time $t$, $k$ is the thermal diffusivity.

Numerical Solution

1D Explicit Scheme

• $u_i^{n+1} = u_i^n + k \frac{\Delta t}{\Delta x^2} (u_{i+1}^n - 2u_i^n + u_{i-1}^n)$
• With $r = k \frac{\Delta t}{\Delta x^2}$
• $u_i^{n+1} = u_i^n + r(u_{i+1}^n - 2u_i^n + u_{i-1}^n)$
• Stability condition: $\Delta t \leq \frac{\Delta x^2}{2k}$ or $r \leq \frac{1}{2}$

2D Explicit Scheme

• $u_{i,j}^{n+1} = u_{i,j}^n + k \Delta t (\frac{u_{i+1,j}^n - 2u_{i,j}^n + u_{i-1,j}^n}{\Delta x^2} + \frac{u_{i,j+1}^n - 2u_{i,j}^n + u_{i,j-1}^n}{\Delta y^2})$
• If $\Delta x = \Delta y$, then let $r = k \frac{\Delta t}{\Delta x^2}$:
• $u_{i,j}^{n+1} = u_{i,j}^n + r(u_{i+1,j}^n + u_{i-1,j}^n + u_{i,j+1}^n + u_{i,j-1}^n - 4u_{i,j}^n)$
• Stability condition: $\Delta t \leq \frac{\Delta x^2}{4k}$ or $r \leq \frac{1}{4}$

Implicit scheme

• Using backward difference in time and central difference in space: $-r u_{i-1}^{n+1} + (1 + 2r)u_i^{n+1} - r u_{i+1}^{n+1} = u_i^n$. Solving linear equations can be achieved using Thomas algorithm (Tridiagonal matrix algorithm). Implicit schemes are unconditionally stable, solving a system of equations at each time step.