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3.2. Algebra and Number Theory (ANT)

This area covers the algebraic structures essential for computing: linear algebra for machine learning and graphics, abstract algebra for cryptography and coding theory, and the number-theoretic foundations of modern security protocols.

Knowledge Area (KA)	CS Core	KA Core
3.2.1 Linear Algebra	Elective
3.2.2 Group Theory	Elective
3.2.3 Rings, Fields, and Galois Theory	Elective
3.2.4 Cryptography and Coding Theory	Elective

3.2.1. ANT/Linear Algebra ↑ Back to top

Vector spaces, linear maps, matrices, determinants, eigenvalues, and inner product spaces, with applications to machine learning, computer graphics, and data science.
Topics:
Core

• Vector spaces and subspaces; bases, dimension, and linear independence
• Linear maps, kernel and image, rank-nullity theorem, and matrix representations
• Determinants: definition, properties, Cramer's rule, and geometric interpretation
• Eigenvalues and eigenvectors; diagonalization and the spectral theorem
• Inner product spaces, orthogonality, Gram-Schmidt process, and least squares
• Matrix decompositions: LU, QR, and singular value decomposition (SVD)

Learning Outcomes:
Core:

1. Identify whether a set with given operations forms a vector space and determine a basis [Familiarity]
2. Compute eigenvalues, eigenvectors, and matrix decompositions to diagonalize or factor matrices [Usage]
3. Apply the SVD and least squares methods to solve overdetermined linear systems and data approximation problems [Assessment]

3.2.2. ANT/Group Theory ↑ Back to top

Groups, subgroups, homomorphisms, quotient groups, and the classification of finite groups, with applications to symmetry in computing and cryptography.
Topics:
Core

• Groups, subgroups, cyclic groups, and order of elements
• Cosets, Lagrange's theorem, and normal subgroups
• Group homomorphisms, isomorphisms, and the isomorphism theorems
• Group actions, orbits, stabilizers, and Burnside's lemma

Learning Outcomes:
Core:

1. State Lagrange's theorem and explain the relationship between subgroups and cosets [Familiarity]
2. Apply the isomorphism theorems to analyze homomorphic images and quotient groups [Usage]
3. Analyze symmetry groups arising in computing contexts such as permutation groups over data structures [Assessment]

3.2.3. ANT/Rings, Fields, and Galois Theory ↑ Back to top

Rings, ideals, polynomial rings, field extensions, and finite fields foundational for coding theory and cryptography.
Topics:
Core

• Rings, ideals, quotient rings, and the ring isomorphism theorems
• Polynomial rings: irreducibility, factorization, and unique factorization domains
• Field extensions: algebraic and transcendental elements, degree, and splitting fields
• Finite fields: structure, existence, and applications in coding theory

Learning Outcomes:
Core:

1. Classify rings as integral domains, PIDs, or UFDs and justify the classification [Familiarity]
2. Construct and analyze finite fields used in AES and other cryptographic algorithms [Usage]
3. Apply field extension theory to analyze error-correcting codes over finite fields [Assessment]

3.2.4. ANT/Cryptography and Coding Theory ↑ Back to top

Mathematical foundations of modern cryptography and error-correcting codes, including RSA, elliptic curves, and linear codes.
Topics:
Core

• Modular arithmetic, the RSA cryptosystem, and discrete logarithm problem
• Elliptic curves over finite fields and elliptic curve cryptography (ECC)
• Linear codes: generator and parity-check matrices, Hamming distance, and error correction
• Cyclic codes: polynomial representation, BCH codes, and Reed-Solomon codes

Non Core

• Post-quantum cryptography: lattice-based schemes and the Learning With Errors (LWE) problem

Learning Outcomes:
Core:

1. Explain the mathematical hardness assumptions underlying RSA and elliptic curve cryptography [Familiarity]
2. Construct a linear code, compute its parameters, and determine its error-correction capability [Usage]

NonCore:

1. Analyze the security assumptions of lattice-based cryptographic schemes in the post-quantum context [Assessment]