Mathematics Problem Archive

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?...

Explicitly construct test elements or principal homogeneous spaces having prescribed ramification and a prescribed large prime order $\ell$....

Work efficiently with the testing characters and principal homogeneous spaces without constructing them explicitly....

Identify and solve tractable special cases of the signature problem described in the slides....

For nearby moduli $n_i=n_1+d_i$ and maps $f_i(r)=r^{e_i}\bmod n_i$, prove the conjecture that with sufficiently many components at least one $f_i$ is ...

For a prime $p$, conjecturally $p$ does not divide the class number of the maximal real subfield $\mathbb{Q}(\zeta_p+\overline{\zeta_p})$ of the $p$th...

Let $\zeta_p:\mathbb{Z}_p\to\mathbb{Q}_p$ be the $p$-adic zeta function. Is $\zeta_p(k)\ne0$ for every even integer $k$?...

Given an integer $n$, determine whether there are rational numbers $x,y,z$ satisfying $x^2+y^2=z^2$ and $xy=2n$; equivalently, determine whether $n$ i...

Is every integer $n\equiv5,6,$ or $7\pmod 8$ a congruent number?...

For $E_n:y^2=x^3-n^2x$, is $n$ a congruent number if and only if $L(E_n,1)=0$?...

Given an elliptic curve $E:y^2=x^3+Ax+B$ over $\mathbb{Q}$, determine whether $E$ has infinitely many rational points....

For an elliptic curve $E/\mathbb{Q}$, let $N_p$ be its number of solutions modulo $p$ plus one and put $f(X)=\sum_{p\le X}\log(N_p/p)$. Is $f(X)$ boun...

For an elliptic curve $E/\mathbb{Q}$ of rank $r$, does $f(X)=\sum_{p\le X}\log(N_p/p)$ grow asymptotically like $r\log\log X$?...

Find a higher-dimensional analogue of the paper's cropping and summation argument; in dimension three the expected sum ranges over quadruples $v_1,v_2...

For $Z(s)=\sum f(a,b,c,d)^s$, which is known to converge for real $s>1/2$, extend $Z$ to complex values of $s$....

Can the function $f$ on $SL(2,\mathbb{Z})$ be extended naturally to $\mathbb{C}/SL(2,\mathbb{Z})$? Can analogous series be constructed for other latti...

For every odd prime $p$ and $k\ge1$, is $s(p^k)=k$? For $k\ge2$, is $d(p^k)=p^{k-1}d(p)$?...

For every $k\ge1$, is $s(2^k)=u_k$? Is $d(2^k)=2^k$ for $k\ne2$, with $d(4)=2$?...

Is Arnold's sequence $(u_k)_{k\ge1}$ the $f$-transform of the quadruple $(2,4,4,4)$?...