Advances in Efficient Algorithms and Computational Number Theory

The field of computational number theory and efficient algorithms is witnessing significant developments, with a focus on improving the complexity of various mathematical operations. Researchers are exploring new strategies to overcome existing barriers, such as the 3/2 exponent in polynomial factorization, and are proposing innovative approaches to achieve better performance. The use of algebraic packing, graded projection recursion, and indefinite lattice reduction are some of the techniques being investigated to enhance computational efficiency. Noteworthy papers in this area include: Faster Algorithms for Structured Matrix Multiplication via Flip Graph Search, which presents improved schemes for multiplying structured matrices, and A number-theoretic conjecture implying faster algorithms for polynomial factorization and integer factorization, which proposes a new strategy to overcome the 3/2 barrier. Additionally, Indefiniteness makes lattice reduction easier presents a revised approach to lattice reduction that leads to better reduced representations.

Sources

Faster Algorithms for Structured Matrix Multiplication via Flip Graph Search

A number-theoretic conjecture implying faster algorithms for polynomial factorization and integer factorization

Fast polynomial computations with space constraints

Graded Projection Recursion (GPR): A Framework for Controlling Bit-Complexity of Algebraic Packing

Compact Schemes for $A^+B$, $A^+AB$ and $AA^+B$

A Deterministic Dimension Property of Twisted Goppa Codes

Monimial Matrix Analogue of Yoshida's theorem

LinApart2: efficient parallel partial fraction decomposition algorithm for denominators with polynomials of general degree

Indefiniteness makes lattice reduction easier

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