Courses

July – November 2026

·  Algebra MTL105 and MTL2005

Main reference: Contemporary Abstract Algebra (A first course)

by Joseph A. Gallian

Group Theory: Equivalence relations and partitions, maximal elements, minimal elements, least upper bound, greatest lower bound, lattices, with examples.

Permutations, cyclic permutation, transpositions, and the sign of a permutation. Binary operations, formal definition of group, subgroup, basic examples of groups,

symmetric group (permutation group), matrix group, group of rigid motions of the plane, and finite group of motions (Dihedral group). Finite group, cyclic group,

generators and relations, cosets and Lagrange's theorem, group homomorphisms, Cayley's theorem, normal subgroup, quotient group and isomorphism theorems.

Group action, Class equation, Cauchy's theorems, Sylow's theorems, direct products and structure theorem for finite abelian groups (without proof) and their

applications. Ring Theory: Definition and examples, ring homomorphism, ideals and quotient rings, Chinese Remainder Theorem, integral domain and quotient field,

ring of Gaussian integers, Unique factorisation domain, Principal ideal domain, Euclidean domain, Gauss' lemma (statement only), polynomial ring, irreducibility

of polynomials and Eisenstein's criterion. Field Theory: Fields, extension of a field, degree of an extension, finite extension, algebraic extension, Splitting field,

Finite field and their structures.

 

On successful completion of the course, a student will be able to:

  1. Basics of Group Theory: Introduction to the fundamental concepts of group theory, including the definition and properties of groups, types of groups, and basic

examples, in order to build a foundational understanding of group structures.

  1. Some advance topics in Group Theory: These topics are foundational for the development of group theory and have various applications in physics, cryptography, and geometry.
  2. Ring Theory: This introductory module provides the basics of ring theory which has applications in many areas of mathematics such as Number Theory and Algebraic Geometry.
  3. Field Theory: In this module, we study the basics of field theory, focusing on the fundamental properties and structure of fields. Special attention is given to finite fields which
  4. plays a crucial role in modern applications such as cryptography, coding theory, and error correction.

 

Grading policy: Quizzes and assignements: 30, Midterm: 30, Final: 40.

Attendance policy: Attendance is compulsory. Attendance record will be maintained! The institute rule will be applied.

Tutorial 1, Tutorial 2 , Tutorial 3

 

Quiz 1: 4th September (Friday).

Note: Students will be given a chance to make up only one quiz (if he or she misses it)

 

For MTL105/MTL2005: Audit Pass is 50%

 

January – May 2026

·  Mathematical Theory of Coding (MTL744)

Main reference: Coding Theory – A First Course by Xing and Ling.

 

There will be a midterm and a final test. At least three quizzes are expected. There will be two types of

assignments (1) problem solving and submitting solutions, (2) understanding certain results (theorems,

propositions, lemmas, concepts, or examples) and making a presentation.

The final grading will have the following approximate proportions: Quizzes: 21%, Assignments: 15%,

Midterm: 24%, Major (final): 40%.

 

Audit pass: minimum 60% marks.

Quizzes are expected on: 20th -24th Jan, 10th -14th Feb, 17th -21th March, 10th -12th April.

Midterm: 21st Feb-27th Feb.

 

Class room: LH413.4, Time: 5 - 6:30 (Monday and Thursday). 

Courses Jan-May 2025

·         MTL 101  (Linear Algebra and Differential Equations)

See the course information sheet here.

PDF of lectures 1 to 5, 6, 7&8, 9&10, 11&12, 13&14, 15 & 16, 17 & 18.

Tutorial 1, Tutorial 2, Tutorial 3, Tutorial 4, Tutorial 5, Tutorial 6.

Solution of every question of the tutorial sheet is not provided. However, the following questions with answers may be useful for the students to understand the

concepts and the methods and to solve the problems in the tutorial sheet. Students are advised to understand the problems solved in the lecture classes too (see the

lecture notes provided above):

QWA1, QWA2, QWA3, QWA4, QWA5, QWA6, QWA7, QWA8, QWA9, QWA10A,10B, QWA11, QWA12, QWA13, QWA14, QWA15, QWA16, QWA17, 

QWA18, QWA19, QWA20, QWA21, QWA22, QWA23, QWA 24 (existence-uniqueness, diff eqn).

Solution of Minor test 1 (2015): Q1, Q2, Q3, Q4, Q5.

Solution of Minor test 1 (2014): Answer to  Q1, Q2, Q3, Q4, Q5.

Solution of Minor test 2 (2014): Answer to Q1, Q2, Q3, Q4, Q5, Q6.

Solution of Major test (2014): Answer to Q1, Q2, Q3, Q4, Q5, Q6, Q6,

Q7, Q8, Q9, Q10.

 

 

 

Course July – November 2025:

 ·  Cryptography MTL730 (or MTL7230)

office No.- 611R, Department of Mathematics, Academic Complex West, IIT Delhi).

Tutor: Anuj Kumar Bhagat (506, Department of Mathematics, Academic Complex West, IIT Delhi).

Student time: Tuesday, Friday- 6 PM to 7 PM.

Lecture time: K slot, i.e., Tuesday-5 PM to 6 PM, Wednesday-12 PM to 1 PM, and Friday-5 PM- 6 PM.

Venue: LH 410

Course Evaluation: Best of two quizzes out of 3 quizzes (10 marks each), Midterm (25 marks),

Assignments (five assignments of 3 marks each to be submitted to Gradescope), End-term (40 marks).

The date and the time for a quiz will be announced in class. One week's time will be given to submit each assignment.

Attendance Policy: 75% is mandatory, including all medical leaves. Poor attendance may result in a grade down.

Reference Books:

1.     An introduction to mathematical cryptography, Hoffstein, Pipher and Silverman, Second edition, Springer, 2014.

2.     A course in number theory and cryptography, Neal Koblitz, Second edition, Springer-Verlag, 1994.

Syllabus:

Classical cryptosystems (substitution ciphers, Vigenère ciphers, matrix ciphers, transposition ciphers, ADFGX cipher, ADFGVX cipher, permutation cipher, Playfair cipher and Enigma machine cipher) and their cryptanalysis. Preview from number theory, Congruences and residue class rings, quadratic residue, and Legendre symbol. Public Key Cryptosystems of RSA, Rabin, etc., their security and cryptanalysis. Primality tests (Fermat PT, Miller-Rabin PT, AKS PT), factorization and quadratic sieve, efficiency of other factoring algorithms (Fermat’s, Pollard’s (𝑝 − 1), Monte Carlo factorization, Factor base, quadratic sieve, and continued fraction). Finite fields: Construction and examples, polynomials on finite fields and their factorization/ irreducibility. Key-Exchange protocols: Diffie-Hellman key exchange, Massey Omura cryptosystems and ElGamal cryptosystem. Discrete logarithm problems in general and on finite fields. Algorithms for finding discrete logarithms (Shank Baby step Giant step algorithm, Pollard’s 𝜌 algorithm, index calculus algorithm in ℤ𝑝 and Pohlig-Hellman algorithm) and use them for cryptanalysis. Elliptic curves, public key cryptosystems, particularly on elliptic curves. Problems of key exchange, discrete logarithms, and the elliptic curve logarithm problem. Implementation of elliptic curve cryptosystems. Counting of points on Elliptic Curves over Galois Fields of order 2𝑚. Cryptographic hash functions. Authentication, Digital Signatures, Identification, certification infrastructure and other applied aspects. DES- security and generalizations.

Applying the corresponding algorithm programs. (Laboratory/ design activities could also be included).

 

 

 

 

 

Courses Jan-May 2025

·         MTL 101  (Linear Algebra and Differential Equations)

See the course information sheet here.

PDF of lectures 1 to 5, 6, 7&8, 9&10, 11&12, 13&14, 15 & 16, 17 & 18.

Tutorial 1, Tutorial 2, Tutorial 3, Tutorial 4, Tutorial 5, Tutorial 6.

Solution of every question of the tutorial sheet is not provided. However, the following questions with answers may be useful for the students to understand the

concepts and the methods and to solve the problems in the tutorial sheet. Students are advised to understand the problems solved in the lecture classes too (see the

lecture notes provided above):

QWA1, QWA2, QWA3, QWA4, QWA5, QWA6, QWA7, QWA8, QWA9, QWA10A,10B, QWA11, QWA12, QWA13, QWA14, QWA15, QWA16, QWA17, 

QWA18, QWA19, QWA20, QWA21, QWA22, QWA23, QWA 24 (existence-uniqueness, diff eqn).

Solution of Minor test 1 (2015): Q1, Q2, Q3, Q4, Q5.

Solution of Minor test 1 (2014): Answer to  Q1, Q2, Q3, Q4, Q5.

Solution of Minor test 2 (2014): Answer to Q1, Q2, Q3, Q4, Q5, Q6.

Solution of Major test (2014): Answer to Q1, Q2, Q3, Q4, Q5, Q6, Q6,

Q7, Q8, Q9, Q10.

 

·  Mathematical Theory of Coding (MTL744)

Main reference: Coding Theory – A First Course by Xing and Ling.

 

There will be a midterm and a final test. At least three quizzes are expected. There will be two types of

assignments (1) problem solving and submitting solutions, (2) understanding certain results (theorems,

propositions, lemmas, concepts, or examples) and making a presentation.

The final grading will have the following approximate proportions: Quizzes: 21%, Assignments: 15%,

Midterm: 24%, Major (final): 40%.

 

Audit pass: minimum 60% marks.

Quizzes are expected on: 20th -24th Jan, 10th -14th Feb, 17th -21th March, 10th -12th April.

Midterm: 21st Feb-27th Feb.

 

Class room: LH413.4, Time: 5 - 6:30 (Monday and Thursday). 

Course July – November 2024:

MTL100 (Calculus for B.Tech. 1st year). Visit the course page here.

Courses Jan-May 2024

·  Algebra (MTL105)

Main reference: Contemporary Abstract Algebra by Joseph Gallian

 

There will be a midterm and a final test. At least three quizzes are expected. There will be assignments that everyone is expected to submit.

The final grading will have the following approximate proportions: Quizzes: 25%, Assignments: 5%, Midterm: 30%, Major (final): 40%.

For attending 75% or more a bonus of up to 5 may be awarded (e.g., 5 for attending 95% or more lectures). Anyone with false attendance

may not be awarded any bonus. Moreover, if the attendance of a student falls below 75%, he/she may not be allowed to award I grade. For

an I grade, a student must have at least 75% attendance and a valid reason for not able to appear in the major test (e.g., poor medical condition).

 

Audit pass: minimum 50% marks and not lesser than 50% attendance (as recorded without any proxy)

Quizzes are expected on: 17th -19th Jan, 7th -9th Feb, 13th -15th March, 10th -12th April.

Major exam with marking scheme.

Major exam with marking scheme.

 

Courses Taught in IIT Delhi:

                   MTL101 (Linear Algebra and Differential Equations) - A course for all the first year BTech in IITD

       MAL 601 Topology (Point set topology and introduction to Algebraic topology) for MSc.

       MAL110 (Calculus and Differential Equations) for UG

       MAL 120 for BTech first year (second sem). It consists of Vector Calculus, Complex Analysis and Fouries Series (2009).

       MAL111 (Analysis and ODE) for BTech Computer Science and Electrical Engineering.

       MAL124 (Algebra and Matrix Analysis) for BTech Computer Science and Electrical Engineering.

       MAL255 (Linear Algebra, first course) for UG

       MAL256 (Algebra) for UG

       MTL502/MAL503 (Linear Algebra) for PG

        MTL509/MAL514 (Complex Analysis) for PG

       MTL503/MAL516 (algebra) for PG

       MAL734/755 (Algebraic Geometry) for PG

       MAL860 (Advance course in Linear Algebra) for PhD

       MAL863 (Algebraic Number Theory)

                    MTL 856 (Lie Algebra)

James E. Humphreys, Introduction to Lie Algebras and Representation Theory.

Courses Taught outside IIT Delhi:

v  MA 101 Real Analysis for the first year BTech in IIT Patna (2008).

v  MA 102 Linear Algebra and Ordinary Differential Equations in IIT Guwahati for BTech first year (2008).

v  MA 521 Modern Algebra for MSc in IIT Guwahati (2007).

Taught short courses in a few workshops including MTTS, AFS, IST.