On primitive integer solutions of the Diophantine equation x3±y3=ak±bkx^3\pm y^3=a^k\pm b^k

In this note we consider the title Diophantine equation from both theoretical as well as experimental point of view. In particular, we prove that for $k=4, 6$ and each choice of the signs our equation has infinitely many co-prime positive integer solutions. For $k=5, 7$ and all choices of the signs we computed all co-prime positive integer solutions $(x, y, a, b)$ satisfying the condition \op{max}\{a, b\}\leq 50000.Comment: 15 page

Bimonotone enumeration

Solutions of a diophantine equation $f(a,b) = g(c,d)$, with $a,b,c,d$ in some finite range, can be efficiently enumerated by sorting the values of $f$ and $g$ in ascending order and searching for collisions. This article considers functions that are bimonotone in the sense that $f(a,b) \le f(a',b')$ whenever $a \le a'$ and $b \le b'$. A two-variable polynomial with non-negative coefficients is a typical example. The problem is to efficiently enumerate all pairs $(a,b)$ such that the values $f(a,b)$ appear in increasing order. We present an algorithm that is memory-efficient and highly parallelizable. In order to enumerate the first $n$ values of $f$, the algorithm only builds up a priority queue of length at most $\sqrt{2n}+1$. In terms of bit-complexity this ensures that the algorithm takes time $O(n \log^2 n)$ and requires memory $O(\sqrt{n} \log n)$, which considerably improves on the memory bound $\Theta(n \log n)$ provided by a naive approach, and extends the semimonotone enumeration algorithm previously considered by R.L. Ekl and D.J. Bernstein.Comment: 22 pages, 7 figures. The algorithms presented here have been implemented as class templates in C++ and are available on the author's homepag

Enumerating solutions to p(a)+q(b)=r(c)+s(d)

Let p; q; r; s be polynomials with integer coefficients. This paper presents a fast method, using very little temporary storage, to find all small integers (a; b; c; d) satisfying p(a)+q(b) = r(c)+s(d). Numerical results include all small solutions to a4+b4+c4 = d4; all small solutions to a4+b4 = c4+d4; and the smallest positive integer that can be written in 5 ways as a sum of two coprime cubes

Enumerating solutions to p(a)+q(b)=r(c)+s(d)

Let p; q; r; s be polynomials with integer coefficients. This paper presents a fast method, using very little temporary storage, to find all small integers (a; b; c; d) satisfying p(a)+q(b) = r(c)+s(d). Numerical results include all small solutions to a4+b4+c4 = d4; all small solutions to a4+b4 = c4+d4; and the smallest positive integer that can be written in 5 ways as a sum of two coprime cubes

Refinements of the k-tree Algorithm for the Generalized Birthday Problem

We study two open problems proposed by Wagner in his seminal work on the generalized birthday problem. First, with the use of multicollisions, we improve Wagner\u27s $3$-tree algorithm. The new 3-tree only slightly outperforms Wagner\u27s 3-tree, however, in some applications this suffices, and as a proof of concept, we apply the new algorithm to slightly reduce the security of two CAESAR proposals. Next, with the use of multiple collisions based on Hellman\u27s table, we give improvements to the best known time-memory tradeoffs for the k-tree. As a result, we obtain the a new tradeoff curve T^2 \cdot M^{\lg k -1} = k \cdot N. For instance, when k=4, the tradeoff has the form T^2 M = 4 \cdot N

FINDING MINIMUM GAPS AND DESIGNING KEY DERIVATION FUNCTIONS

"While the priest climbs a post, the devil climbs ten". The problem size gets larger as computers become faster. Using naive algorithms, even equipped with fast CPUs and large memories, computers still cannot handle many problems of certain size. Some searching tasks, however, can be answered with the help of the algorithmic technique, such as time and space trade-off.Let n and k be positive integers, n>k. Define r(n,k) to be the minimum value of the absolute value of sqrt(a_1)+sqrt(a_2)+...+sqrt(a_k)-sqrt(b_1)-sqrt(b_2)-...-sqrt(b_k)-where a_1,a_2,...,a_k,b_1,b_2,...,b_k are positive integers no larger than n. It is important to find a tight bound for r(n,k), in connection to the sum-of-square-roots problem, a famous open problem in computational geometry. The current best lower bound and upper bound are far apart. For exact values of r(n,k), only a few simple cases have been reported so far, and they can be found easily using exhaustive search. The new algorithm is developed to find r(n,k) exactly in nk+o(k) time and in nk/2+o(k) space. Space usage is decreased dramatically along with little increase in time, compared to an intuitive trade-off method. Our algorithm reduces time for swap-in and swap-out, minimizing the total running time. The problem is solved in size that was infeasible for a naive trade-off scheme. We also present lots of numerical data.The time and space trade-off technique has its limitation. For some problems, when space is reduced to a certain extent, time will be increased exponentially. The trade-off technique does not apply to this situation. We want to explore such a property to discourage trade-off attacks.Key generation is an important part of symmetric-key encryption algorithms, such as AES. A key derivation function can be used to generate symmetric cipher session keys. As CPU technology advances, key derivation functions are more vulnerable to off-line brute force attacks. Based on the Memory Wall problem, we propose a simple number theoretic way to mitigate exhaustive search attacks. We also present a formal definition of memory bounded functions. On one hand, if attackers try to reduce memory usage, they are forced to spend dramatically more time. On the other hand, a memory-bound security scheme will minimize the difference between high-end and low-end computers. Trade-off attacks will hence be deterred