Mathematics Problem Archive

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 ...

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

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

Do there exist positive integers $a,b,c,d$ such that all six pairwise face diagonals $\sqrt{a^2+b^2}$, $\sqrt{a^2+c^2}$, $\sqrt{a^2+d^2}$, $\sqrt{b^2+...

Given a prime modulus $p$, a group generator $g$, and the public values $g^a$ and $g^b$ modulo $p$, can the shared value $g^{ab}\bmod p$ be computed e...

Can Fermat's Last Theorem be proved using only mathematical techniques that were available in the seventeenth century?...

Given a unit square, does there exist a point in its plane, inside or outside the square, whose distances from all four vertices are rational? Equival...

Does there exist a $3\times3$ semi-magic square whose nine entries are distinct positive integer cubes and whose three row sums and three column sums ...

Büchi's problem on sufficiently large sequences of square numbers with constant second difference....

Carmichael's totient function conjecture: do all values of Euler's totient function have multiplicity greater than $1$?...

Catalan–Dickson conjecture on aliquot sequences: no aliquot sequences are infinite but non-repeating....

Exponent pair conjecture: for all $\varepsilon > 0$, is the pair $(\varepsilon, 1/2 + \varepsilon)$ an exponent pair?...

Are there any pairs of betrothed numbers which have same parity?...

Are there any pairs of relatively prime amicable numbers?...

Are there infinitely many betrothed numbers?...

Do any odd noncototients exist?...

Do any (2, 5)-perfect numbers exist?...

Do any Taxicab(5, 2, n) exist for n > 1?...