Mathematics Problem Archive

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?...

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

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 generators from ideal-lattice problems....

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

Construct efficient digital-signature schemes from ideal-lattice problems....

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

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

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

Base cryptographic constructions directly on worst-case hardness assumptions from algebraic number theory....

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

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

Construct an efficient ideal-lattice version of Regev's quantum-SVP-based cryptosystem....

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...

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...

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 $r\log\log X$?...

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...

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 $s$....

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...

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 $d(4)=2$?...