Mathematics Problem Archive

Remove the imaginary-hyperelliptic and $p\ne2$ restrictions from the complexity bound stated in the slides....

Test and analyze whether deleting extra points and fixing the lift $x\mapsto x^p$ improves Frobenius lifts for nondegenerate curves....

Develop the higher-dimensional analogue of the nondegenerate-curve Frobenius-lift method....

Find useful deformations of nondegenerate curves together with easy starting matrices for Frobenius computation....

Work out effective zeta-function computations for nondegenerate surfaces in toric threefolds....

Factor $N=p^rq^s$ with $r\approx s$ using fewer known bits of the factors than existing methods require....

Develop methods to factor an integer when the known bits of its factors are nonconsecutive....

For large $N$, efficiently find the required short five-term geometric progressions modulo $N$ that avoid first- and second-order recurrence, thereby ...

Study the distribution of group structures $E(\mathbb{F}_q)$ as elliptic curves $E/\mathbb{F}_q$ vary; in particular, determine the correct nonuniform...

Is the exponent $e_q(E)$ of $E(\mathbb{F}_q)$ typically close to $q$?...

How often is the group of a random elliptic curve over $\mathbb{F}_q$ cyclic?...

Characterize the typical arithmetic structure of $\#E(\mathbb{F}_q)$ for elliptic curves over finite fields....

Prove that there are sufficiently many prime-order elliptic curves over every finite field $\mathbb{F}_q$....

Prove that sufficiently many elliptic curves $E/\mathbb{F}_p$ have smooth group order $\#E(\mathbb{F}_p)$....

Quantify elliptic curves over finite fields whose group order has a large prime divisor....

Prove the conjecture that the EC-LCG, EC-PG, and EC-NRG sequences defined in the slides are very well distributed....

Given $n$, efficiently choose a prime power $q$ and construct an elliptic curve $E/\mathbb{F}_q$ with $\#E(\mathbb{F}_q)=n$....

Given the Weil polynomial of an abelian-threefold isogeny class over a finite field, determine whether the class contains a Jacobian....

Decide whether a given principally polarized abelian threefold over a field $k$ is the Jacobian of a curve over $k$....

Find more MNT curves, including usable larger embedding degrees, more curve families, and smaller cofactors....

Construct pairing-friendly hyperelliptic curves suitable for cryptography....

Determine the computational hardness of the Pairing Inversion Problem....

Prove an analogous small-polynomial-factor worst-case hardness result for SVP and SIVP on ideal lattices....

Is the shortest vector problem on ideal or cyclic lattices NP-hard, either exactly or under approximation?...

Reduce computational problems on arbitrary lattices to corresponding problems on cyclic or ideal lattices....

Does SVP reduce to CVP while remaining inside the class of cyclic or ideal lattices?...

Exhibit cyclic or ideal lattices on which LLL achieves its worst-case approximation factor....

Develop an algebraic analogue of the LLL lattice-reduction algorithm that exploits ideal-lattice structure....

Construct efficient pseudorandom functions from ideal-lattice problems....

Use algebraic tools to solve computational problems on ideal lattices efficiently....

Develop an efficient quantum algorithm for the shortest vector problem on ideal lattices....

Use a baby-step/giant-step infrastructure framework to speed infrastructure discrete logarithms or point counting by a polynomial factor....

Give efficient reductions in both directions between the degree-zero divisor-class-group discrete logarithm problem and the infrastructure discrete lo...

For a compact convex domain $\Omega$, the values of $F_\Omega$ at the vertices of its corner locus $C_\Omega$ give complete coordinates. How are these...

Give another proof of the paper's identities (Ж) and (ж) using the methods for identity (1). Can $f(a,b,c,d)$ be interpreted as a residue at $(a+b)+(c...

Prove that every bounded-degree graph roughly isometric to a nonamenable Cayley graph is non-Liouville....