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Mathematical Aspects of Modern Algebraic Cryptography

Instructor:  Professor Delaram Kahrobaei

Dr. Delaram Kahrobaei is an Associate Professor at the City University of New York. She has a joint appointment at CUNY Graduate Center in the PhD program in Computer science and at New York City College of Technology (CUNY) in the Mathematics Department. Her main research area is Information Security, Cryptography, Computational and Combinatorial Group Theory.
She was previously an Assistant Professor in Pure Mathematics at the Mathematical Institute, at Scotland’s prestigious University of St Andrews (2004-2006).
She has done substantial research in Mathematics and computer science both as individual and as a team member and publishes her work in peer reviewed journals and proceedings. Her work is attracting strong international interest and many invited talks (105). She has been awarded substantial prestigious research grants, including Office of Naval Research Office ($448K), American Association for Advances in Sciences, the National Science Foundation, PSC-CUNY Research Foundation, Faculty Fellowship Publication award, London Mathematical Society and Edinburgh Mathematical Society Grants and intends to apply for more grants from government agencies.

She has published in the areas of combinatorial group theory, geometric group theory, computational group theory, combinatorics, logic as well as algebraic cryptography and computational complexity and representation theory.
Delaram is the director of C-LAC, Center for Logic, Algebra and Computation. She enjoys involving students in her research and in the past years she has been a research supervisor to PhD (4), MSc (2) and undergraduate (30) students. Indeed two of her Ph.D. students graduated in 2012 from CUNY Graduate Center in Mathematics. Maggie Habeeb is now a tenure track assistant professor at California University in Pennsylvania and Bobby Koupparis is a vice president and quant at the Royal Bank of Canada in New York.
She is also keen to teach mathematics and computer science at both undergraduate and graduate levels and for the past fourteen years she has done so, at CUNY Graduate Center, City Tech, University of St Andrews and Hunter College, New York University Polytechnic Institute.
Kahrobaei has been co-organizing several conferences, including the ones for American Mathematical Society special sessions (in Newark, New Orleans, Las Vegas, Cornell U., San Diego, Boston, Israel). She is a co-founder of the annual Manhattan Algebra Day (December 6, 2013).
She co-founded the New York Applied Algebra Colloquium with Andrew Douglas. Kahrobaei is also running the Mathematical Aspects of Cryptography Student seminars at CUNY Graduate Center.
She is also co-organizing Algebra Cryptography Seminar at the CUNY Graduate Center.
She is a member of the Algebraic Cryptography Center.
She is the member of the editorial board of the International Journal of Open Problems in Computer Science and Mathematics.

In the Fall 2013, Professor Kahrobaei is teaching, Mathematical Aspects of Modern Algebraic Cryptography, in the PhD programs in Mathematics and Computer Science at CUNY Graduate Center.
Delaram will be traveling in 2013-2014 for conferences and research activity to Barcelona, South Korea, Scotland, Louisville, Paris, Tel Aviv, New Castle, Buenos Aires, Moscow, Baltimore.


In this course, we cover mathematical aspects of algebraic cryptography. The topics include finite fields, computational and combinatorial group theory, mathematical definitions and proof of provably secure cryptosystems and algebraic attacks.

Course Topics

  • Cryptography (Public Key, RSA, DH, Digital Signatures Algorithms, Secret Sharing, Coin Flipping, Passwords, Practical and Useful Ones)

  • Complexity of Computation (P, NP, NP-Completeness, Randomized Algorithms and Complexity Classes)

  • Groups and Finite Fields (Example, Important Facts, with some proofs, Polynomia Rings)

  • Combinatorial-Algebraic Cryptosystems

  • Combinatorial group theory (groups presented as finite presentations, Decision problems, Search Problems)

  • Computational Group Theory (Some computations with Groups, mention about the related software, GAP, Magnus)

  • Elliptic Curves, Homomorphic Encryption

  • Linear Algebraic Attacks, Length Based Attacks

  • Computability and Complexity of algorithms in group theory used in cryptography as well as generic case complexity.

  • Cryptographic problems : Commutative and non-Commutative (public key exchange problems, digital signatures, authentication)

  • Provable Security in Cryptography

  • Quantum Algorithms in group theory and applications in cryptography

  • Open interesting problems

Suggested Texts

  • Aspects of non-abelian group-based cryptography: a survey and open problems, B.Fine, M.Habeeb, D.Kahrobaei, G.Rosenberger, JP Journal of Algebra, Number Theory, Applications, 2012

  • Lecture notes will be distributed in each class, by Kahrobaei

  • All the slides will be available via Black Board.

  • Algebraic Aspects of Cryptography by Kolbitz, Springer, 2000

  • Non-commutative Cryptography and Complexity of Group Theoretic Problems, by Myasnikov,Shpilrain, Ushakov, American Mathematical Society, 2011

  • Introduction to Identity-Based Encryption, By Martin, 2008, Artech House, Boston, London

  • Introduction to Modern Cryptography, by Katz, Lindell, Chapman&Hall/CRC, 2008


  • There will be no exam, but projects and homework.

  • There will be some suggested projects and presentations.

  • Active participation in the class