Mathematics Problem Archive

There exists a positive absolute constant $C$ with the following property. Let $\alpha_1,\ldots,\alpha_n$ be nonzero algebraic numbers and $\log\alpha...

Let $x_1,\ldots,x_n$ be $\mathbb{Q}$-linearly independent complex numbers. Assume that for any $\varepsilon>0$, there exists a positive number $H_0$ s...

Let $A$ be a simple abelian variety over $\mathbb{Q}$, $\exp_A:\mathbb{R}^g\rightarrow A(\mathbb{R})^0$ the exponential map of the Lie group $A(\mathb...

Let $m$, $n$, $k$ be positive integers and $a_{ij\kappa}$ rational integers ($1\le i\le n$, $1\le j\le m$, $1\le\kappa\le k$). For $\underline{x}=(x_1...

For any $\varepsilon>0$ there exists $S_0>0$ (depending on $\varepsilon$, $\gamma_1,\ldots,\gamma_m$ and $\mathcal{K}$) such that, for any $S\ge S_0$ ...

Let $n$ be a positive integer. For almost all $n$-tuples $(x_1,\ldots,x_n)$, there are positive constants $c$ and $D_0$ (depending on $n$, $x_1,\ldots...

Turn $H^2(G_K,K_s^*)$ into an explicitly computational group....

Explicitly describe the cocycle $c_u$, equivalently fast-compute invariants of local algebras split by the generalized-dihedral extensions specified i...

Adapt Schoof's method to compute zeta functions of curves without unfavorable dependence on the genus....

Is computation of a curve's zeta function polynomial simultaneously in the genus $g$ and in $\log q$?...

Reduce the number of guesses required by lattice attacks for factoring with partially known bits....

Extract useful information from incorrect guesses in factoring attacks based on partially known bits....

Efficiently solve for, or characterize all relevant roots of, $x^2-1$ modulo a composite integer $N$ in the setting of the slides....

Improve the running time of Coppersmith-type methods for finding small modular or integer roots....

Understand and control how the shape of a polynomial affects Coppersmith-type small-root algorithms....

Give conditions or constructions that ensure algebraic independence in multivariate elimination for small-root attacks....

Find an optimal collection of polynomials for multivariate lattice-based small-root attacks....

Determine whether the lattice dimension in the stated small-root constructions can be reduced....

Resolve the zero-constant-term case in the Newton-polytope formulation of multivariate small-root methods....

Construct additional cryptographic primitives whose security follows from the hardness of finding small roots....

Analyze how effective rotation-augmented lattice reduction is on cyclic or NTRU lattices....

Speed up rotation-augmented reduction algorithms for cyclic or NTRU lattices....

For a fixed $E/\mathbb{F}_q$, prove that $\#E(\mathbb{F}_{q^n})/\#E(\mathbb{F}_q)$ is prime for infinitely many $n$....

For a torsion-free elliptic curve $E/\mathbb{Q}$, prove that $\#E(\mathbb{F}_p)$ is prime for infinitely many primes $p$....

Give an effective input representation for a principally polarized abelian threefold suitable for deciding whether it is a Jacobian....

Combine the Meagher–Ritzenthaler criteria with Deligne modules to detect Jacobians in an ordinary absolutely simple abelian-threefold isogeny class....

For fixed $q$, is $N_q(g)=\max_C\#C(\mathbb{F}_q)$ increasing as a function of the genus $g$?...

Find a sharp upper bound for the shortest-vector length in an $n$-dimensional positive-definite Hermitian space of determinant $d$ over an imaginary q...

Determine the exact computational-speed and security tradeoff for genus-four curves used in pairing cryptography....

Find an attack that breaks the pairing-based cryptographic system discussed in the slides, or establish its resistance to known attacks....

Break, or determine the true hardness of, the weaker security assumptions used in pairing-based cryptography....

Update Joux's 2002 work by developing a systematic taxonomy of pairing-related computational assumptions....

Is the Decision Linear problem strictly harder than the decisional Diffie–Hellman problem in the relevant pairing groups?...

Prove that general SVP and SIVP are hard in the worst case to approximate within small polynomial factors....

Is the minimum-distance problem for cyclic codes NP-hard?...

Prove that quasi-cyclic lattice or code public-key constructions are secure based on worst-case hardness for quasi-cyclic structures....

Use lattice reduction together with average-case problems to solve computational problems in algebraic number theory....

Develop an efficient quantum algorithm for the Smallest Conjugate problem....

Use number theory to prove non-malleability properties for real-world RSA key-generation algorithms....

Construct a malleable RSA generator producing publicly related moduli $n,n'$ such that factoring $n'$ makes $n$ easy to factor....

Construct practical groups in which discrete logarithms have an effective trapdoor....

Construct groups in which inversion is infeasible under reasonable cryptographic assumptions....

Construct improved practical trapdoor pairings....

Prove the simple construction from trapdoor groups with infeasible inversion to directed transitive signatures secure, or repair the construction....

Prove finiteness of the Shafarevich–Tate group of the elliptic-curve lift required by the Huang–Raskind method, in the general cases where it is not k...

Is the odd-class-number condition in the stated lower bound for Heegner points necessary?...

Is the no-complex-multiplication condition in the stated lower bound for Heegner points necessary?...

Prove analogues of the stated Heegner-point results for points arising from nonmaximal orders....

Find an analogue of the Deuring Lifting Theorem for Darmon–Heegner points....

Can the lifting attack be improved by allowing the number-field degree $[K:\mathbb{Q}]$ to grow?...