Informatics and Cybersecurity

Distinguish a mathematical conjecture from a proved statement.

Construct a direct proof of a basic claim about integers.

Construct a proof by contrapositive for a conditional statement.

Construct a proof by contradiction for an irrationality or divisibility claim.

Disprove a universal statement by finding a counterexample.

Prove equality of two finite sets by double inclusion.

Use mathematical induction to prove a formula indexed by a positive integer.

Identify the invalid step in a short attempted proof.

Evaluate terms of arithmetic and geometric sequences.

Derive a closed form for a finite arithmetic sum.

Derive a closed form for a finite geometric sum.

Manipulate sigma notation using linearity and index changes.

Bound a finite sum above and below using simpler sums.

Generate a recursively defined sequence with Python.

Solve first-order linear recurrences with constant coefficients.

Verify a proposed recurrence solution using substitution and initial conditions.

Compare logarithmic, polynomial, and exponential growth rates.

Apply logarithm laws when simplifying algorithmic expressions.

Use the formal definitions of Big-O and Big-Omega to prove simple upper and lower bounds.

Establish a tight asymptotic bound using Big-Theta.

Use a loop invariant to prove the correctness of a simple iterative algorithm.

Count primitive operations in a single-loop Python algorithm.

Derive the time complexity of a basic nested-loop algorithm.

Solve a divide-and-conquer recurrence using a recursion tree or the Master Theorem.

Apply the sum and product rules to count finite outcomes.

Distinguish permutations from combinations in counting problems.

Compute the number of arrangements containing repeated objects.

Apply the pigeonhole principle to prove an existence claim.

Count outcomes using the principle of inclusion-exclusion.

Expand powers of binomial expressions using the binomial theorem.

Represent a finite graph using adjacency lists and adjacency matrices.

Use breadth-first or depth-first traversal to test reachability in a finite graph.

Model a finite random experiment by specifying its sample space, events, and probability distribution.

Compute probabilities using complementary events and set operations.

Calculate conditional probabilities from a finite sample space.

Apply Bayes' theorem to update a probability after observing evidence.

Determine whether two finite events are independent.

Compute the expectation and variance of a discrete random variable.

Calculate the Shannon entropy of a finite probability distribution.

Use Python simulations to compare empirical frequencies with exact probabilities.

Prove elementary claims using the definition of divisibility.

Compute the quotient and remainder using Euclidean division.

Compute a greatest common divisor with the Euclidean algorithm.

Express a greatest common divisor as a linear combination using the extended Euclidean algorithm.

Determine whether two integers are coprime.

Compute the least common multiple from a greatest common divisor.

Prove basic properties of greatest common divisors and coprime integers.

Implement and test the Euclidean and extended Euclidean algorithms in Python.

Determine whether two integers are congruent modulo a given modulus and identify their residue class.

Perform addition, subtraction, and multiplication in modular arithmetic.

Prove that addition and multiplication on residue classes are well defined.

Compute large modular powers using repeated squaring.

Implement binary modular exponentiation in Python.

Determine whether an integer has a multiplicative inverse modulo a given modulus.

Compute a modular inverse with the extended Euclidean algorithm.

Solve a linear congruence and state when solutions exist.

Apply the fundamental theorem of arithmetic in factorization arguments.

Generate primes up to a bound using the sieve of Eratosthenes.

Test a candidate prime by trial division up to its square root.

Compute Euler's totient function from a known prime factorization.

Apply Fermat's little theorem to simplify a modular power.

Apply Euler's theorem to simplify a modular power.

Solve simultaneous congruences using the Chinese remainder theorem.

Implement the Chinese remainder construction in Python for pairwise coprime moduli.

Verify whether a finite set and operation satisfy the group axioms.

Compute powers and inverses of elements in a finite group.

Determine the order of an element and the cyclic subgroup it generates.

Apply Lagrange's theorem to constrain possible element and subgroup orders.

Perform addition, subtraction, multiplication, and inversion in a prime field.

Perform addition and multiplication of polynomials over a prime field.

Divide polynomials with remainder and reduce products modulo an irreducible polynomial.

Compute an inverse in a small extension field using the polynomial extended Euclidean algorithm.

Represent a system of linear equations as a matrix equation.

Perform matrix addition and multiplication over a prime field.

Solve a small linear system over a prime field using Gaussian elimination.

Determine whether a square matrix over a prime field is invertible.

Compute the inverse of a small matrix over a prime field.

Determine whether a small set of vectors over a prime field is linearly independent.

Implement matrix multiplication over a prime field in Python.

Implement Gaussian elimination over a prime field in Python.

Trace RSA key generation using small prime numbers.

Justify textbook RSA decryption correctness under clearly stated assumptions.

Implement textbook RSA with small educational parameters in Python and identify why the resulting system is not suitable for production use.

Trace Diffie–Hellman key exchange in a small cyclic group.

Relate the security assumptions of RSA, finite-field Diffie–Hellman, and elliptic-curve cryptography to integer factorization and the corresponding discrete-logarithm problems.

Check that an elliptic curve over a prime field is nonsingular and determine whether a point lies on it.

Compute point addition and point doubling on a small elliptic curve over a prime field.

Perform elliptic-curve scalar multiplication using the double-and-add algorithm.