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Computational problem used in cryptography
Short integer solution (SIS) and ring-SIS problems are two average-case problems that are used in lattice-based cryptography constructions. Lattice-based
Short integer solution problem
Short_integer_solution_problem
Mathematical optimization problem restricted to integers
An integer programming, also known as integer optimization, problem is a mathematical optimization or feasibility program in which some or all of the variables
Integer_programming
Method to solve optimization problems
variables are required to be integers, then the problem is called an integer programming (IP) or integer linear programming (ILP) problem. In contrast to linear
Linear_programming
Optimization problem in computer science
Learning with errors Short integer solution problem Khot, Subhash (2005). "Hardness of approximating the shortest vector problem in lattices". J. ACM
Lattice_problem
Computer software bug occurring in 2038
systems. Modern systems and software updates address this problem by using signed 64-bit integers, which will take 292 billion years to overflow—approximately
Year_2038_problem
Complexity class
the integers that have the same total. This problem is contained in PPP, but it is not known if it is PPP-complete. The constrained-SIS (short integer solution)
PPP_(complexity)
Decision problem in computer science
sum problem (SSP) is a decision problem in computer science. In its most general formulation, there is a multiset S {\displaystyle S} of integers and
Subset_sum_problem
On solvability of Diophantine equations
principal contributors to its solution). When all coefficients and variables are restricted to be positive integers, the related problem of polynomial identity
Hilbert's_tenth_problem
In mathematics, when is n!+1 a square
Unsolved problem in mathematics Does n ! + 1 = m 2 {\displaystyle n!+1=m^{2}} have integer solutions other than n = 4 , 5 , 7 {\displaystyle n=4,5,7}
Brocard's_problem
NP-hard problem in combinatorial optimization
Corporation, who expressed the problem as an integer linear program and developed the cutting plane method for its solution. They wrote what is considered
Travelling_salesman_problem
Problem in combinatorial optimization
Knapsack Problem Archived 14 February 2015 at the Wayback Machine Optimizing Three-Dimensional Bin Packing Knapsack Integer Programming Solution in Python
Knapsack_problem
Unsolved problem in computer science
Unsolved problem in computer science If the solution to a problem can be checked in polynomial time, must the problem be solvable in polynomial time? More
P_versus_NP_problem
Mathematical problem in number theory
cattle problem (or the problema bovinum or problema Archimedis) is a problem in Diophantine analysis, the study of polynomial equations with integer solutions
Archimedes's_cattle_problem
Problem a computer might be able to solve
science, a problem is one that asks for a solution in terms of an algorithm. For example, the problem of factoring "Given a positive integer n, find a
Computational_problem
Polynomial equation whose integer solutions are sought
Diophantine equation is a polynomial equation with integer coefficients, for which only integer solutions are of interest. A linear Diophantine equation equates
Diophantine_equation
Sum of inverse squares of natural numbers
1741. The solution to this problem can be used to estimate the probability that two large random numbers are coprime. Two random integers in the range
Basel_problem
{\displaystyle A,B,C} must share some prime factor. Brocard's problem: are there any integer solutions to n ! + 1 = m 2 {\displaystyle n!+1=m^{2}} other than
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Topics referred to by the same term
state SIS (file format), Symbian OS filename extension Short integer solution problem, a problem in lattice-based cryptography Single-instance storage
Sis
Solving an optimization problem with a quadratic objective function
x will need to take on integer values. This leads to the formulation of a mixed-integer quadratic programming (MIQP) problem. Applications of MIQP include
Quadratic_programming
How many integer lattice points there are in a circle
In mathematics, the Gauss circle problem is the problem of determining how many integer lattice points there are in a circle centered at the origin and
Gauss_circle_problem
On the distribution of prime numbers
sometime be in a position to attempt the rigorous solution of Goldbach's problem, viz., whether every integer is expressible as the sum of two positive prime
Hilbert's_eighth_problem
Mathematical problem in number theory
In number theory, Waring's problem asks whether each natural number k has an associated positive integer s such that every natural number is the sum of
Waring's_problem
Classical problem in combinatorics
to form an integer solution. The primal-dual algorithm for the set cover problem is an iterative method that constructs feasible solutions to both the
Set_cover_problem
Optimization problem
The vehicle routing problem (VRP) is a combinatorial optimization and integer programming problem which asks "What is the optimal set of routes for a
Vehicle_routing_problem
Mathematical and computational problem
NP-complete. Despite its worst-case hardness, optimal solutions to very large instances of the problem can be produced with sophisticated algorithms. In addition
Bin_packing_problem
Problem of finding the best feasible solution
economics, an optimization problem is the problem of finding the best solution from all feasible solutions. Optimization problems can be divided into two
Optimization_problem
On divisibility among sets of integers
Znám's problem asks which sets of integers have the property that each integer in the set is a proper divisor of the product of the other integers in the
Znám's_problem
Theorem in geometric topology
the Betti numbers, which associate to any manifold a list of nonnegative integers. Riemann showed that a closed connected two-dimensional manifold is fully
Poincaré_conjecture
Complexity class
known as the travelling salesman problem—is NP-hard. The subset sum problem is another example: given a set of integers, does any non-empty subset of them
NP-hardness
Initial set of valid possible values
including inequalities, equalities, and integer constraints. This is the initial set of candidate solutions to the problem, before the set of candidates has
Feasible_region
Computer arithmetic error
8-bit integer addition of 127 + 1 results in −128, a two's complement of 128). (A solution for this particular problem is to use unsigned integer types
Integer_overflow
This is a list of notable integer sequences with links to their entries in the On-Line Encyclopedia of Integer Sequences. OEIS core sequences Index to
List_of_integer_sequences
Triangle with integer side lengths
positive integers can serve as the side lengths of an integer triangle as long as it satisfies the triangle inequality: the longest side is shorter than the
Integer_triangle
Mathematical counting-out question
used to solve this problem in the general case by performing the first step and then using the solution of the remaining problem. When the index starts
Josephus_problem
Mathematical puzzle
measure any integer amount up to the sum of the volumes. As shown in the previous section, we can construct the solution to the problem from the desired
Water_pouring_puzzle
Mathematical problem in operations research
problem reducible to the knapsack problem. The problem can be formulated as an integer linear programming problem. A paper machine can produce an unlimited
Cutting_stock_problem
Computational problem of graph theory
In graph theory, the shortest path problem is the problem of finding a path between two vertices (or nodes) in a graph such that the sum of the weights
Shortest_path_problem
Mathematical problem
be obtained using only coins of 3 and 5 units is 7 units. The solution to this problem for a given set of coin denominations is called the Frobenius number
Coin_problem
Concept in integral mathematics
optimization problem (integer programming) into a related problem that is solvable in polynomial time (linear programming); the solution to the relaxed
Linear_programming_relaxation
17th-century conjecture proved by Andrew Wiles in 1994
developed methods for the solution of some kinds of Diophantine equations. A typical Diophantine problem is to find two integers x and y such that their
Fermat's_Last_Theorem
Matrix form in linear algebra
matrices over the integers Z {\displaystyle \mathbb {Z} } . Just as reduced echelon form can be used to solve problems about the solution to the linear system
Hermite_normal_form
Branch of numerical optimization
on finding the global solutions of an optimization problem whilst providing theoretical guarantees that the reported solution is indeed the global one
Deterministic global optimization
Deterministic_global_optimization
Combinatorial optimization problem
polynomial. If the weights are integers, and all weights are at most C (where C>1 is some integer), then the problem can be solved in O ( m n log (
Assignment_problem
Complex number whose real and imaginary parts are both integers
number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers. The Gaussian integers, with ordinary addition and
Gaussian_integer
Cryptographic algorithm created by Adi Shamir
demonstrated above, which uses integer arithmetic rather than finite field arithmetic, works, there is a security problem: Eve gains information about S
Shamir's_secret_sharing
NP-complete problem in computer science
science, the partition problem, or number partitioning, is the task of deciding whether a given multiset S of positive integers can be partitioned into
Partition_problem
Optimization technique for solving (mixed) integer linear programs
Such procedures are commonly used to find integer solutions to mixed integer linear programming (MILP) problems, as well as to solve general, not necessarily
Cutting-plane_method
Mathematical algorithm
Kuṭṭaka is an algorithm for finding integer solutions of linear Diophantine equations. A linear Diophantine equation is an equation of the form ax + by
Kuṭṭaka
About polynomials in several variables
cracks yet another century-old problem". Fortune. Retrieved July 23, 2026. Matthew Sparkes (July 20, 2026). "AI's solution to 87-year-old riddle takes mathematicians
Jacobian_conjecture
Natural number
of Integer Sequences. OEIS Foundation. Sloane, N. J. A. (ed.). "Sequence A005277 (Nontotients: even numbers k such that phi(m) = k has no solution)".
300_(number)
Complexity class
contains the classes PPAD and PWPP. Notable problems in this class include the short integer solution problem. PPAD (standing for "Polynomial time Parity
TFNP
23 mathematical problems stated in 1900
Fields Medal in 1966 for his work on the first problem, and the negative solution of the tenth problem in 1970 by Yuri Matiyasevich (completing work by
Hilbert's_problems
On unit fractions adding to 4/n
{1}{y}}+{\tfrac {1}{z}}} have a positive integer solution for every integer n ≥ 2 {\displaystyle n\geq 2} ? More unsolved problems in mathematics The Erdős–Straus
Erdős–Straus_conjecture
Mathematical puzzle
studied problems requiring integer solutions in the 3rd century CE. The Euclidean algorithm for greatest common divisor which underlies the solution of such
The_monkey_and_the_coconuts
Mathematical problem set on a chessboard
queens puzzle is the problem of placing eight chess queens on an 8×8 chessboard so that no two queens threaten each other; thus, a solution requires that no
Eight_queens_puzzle
River crossing puzzle
London: Routledge & Kegan Paul. pp. 4–5. Alcuin's Transportation Problems and Integer Programming Archived 2011-07-19 at the Wayback Machine, Ralf Borndörfer
Wolf, goat and cabbage problem
Wolf,_goat_and_cabbage_problem
Arithmetic operation
the Greatest Unsolved Problem in Mathematics. New York City: Penguin Books. ISBN 978-0-452-28525-5. Weisstein, Eric W. "Integer Division". MathWorld.
Division_(mathematics)
Subfield of mathematical optimization
feasible solutions is discrete or can be reduced to a discrete set. Typical combinatorial optimization problems are the travelling salesman problem ("TSP")
Combinatorial_optimization
Family of solutions to related differential equations
when solving problems (like Laplace's equation) in cylindrical coordinates. When α {\displaystyle \alpha } is a half-integer, the solutions are called spherical
Bessel_function
Inherent difficulty of computational problems
of a solution. If the answer is yes, many important problems can be shown to have more efficient solutions. These include various types of integer programming
Computational complexity theory
Computational_complexity_theory
Mathematical problem
for n {\displaystyle n} being any positive integer. The exercise of working through this problem may be used to explain and demonstrate exponents
Wheat_and_chessboard_problem
Set of objects whose state must satisfy limits
these kinds of problems. Additionally, the Boolean satisfiability problem (SAT), satisfiability modulo theories (SMT), mixed integer programming (MIP)
Constraint satisfaction problem
Constraint_satisfaction_problem
Mathematical puzzle
various lengths and heights, or requesting unusual solutions such as cases where all values are integers. Its charm has been attributed to a seeming simplicity
Crossed_ladders_problem
If G is a finitely generated group with exponent n, is G necessarily finite?
exponent, there exists a largest one. This provides a solution for the restricted Burnside problem for the case of prime exponent. (Later, in 1989, Efim
Burnside_problem
Root of a quadratic polynomial with a unit leading coefficient
are integers, i.e. quadratic integers are algebraic integers of degree two. Thus quadratic integers are those complex numbers that are solutions of equations
Quadratic_integer
Geometry problem on grid points
no-three-in-line problem and then scaling down the integer grid to fit within a unit square produces solutions to the Heilbronn triangle problem where the smallest
No-three-in-line_problem
Even integers as sums of two primes
proved that every positive integer is the sum of four squares. See Waring's problem and the related Waring–Goldbach problem on sums of powers of primes
Goldbach's_conjecture
List of unsolved computational problems
list of notable unsolved problems in computer science. A problem in computer science is considered unsolved when no solution is known or when experts
List of unsolved problems in computer science
List_of_unsolved_problems_in_computer_science
Mathematical problem of square numbers which are also square-pyramidal
Encyclopedia of Integer Sequences. OEIS Foundation. Weisstein, Eric W. "Square Pyramidal Number". MathWorld. Weisstein, Eric W. "Cannonball Problem". MathWorld
Cannonball_problem
Mathematical puzzle
although there are actually many more correct solutions. The entries in blue are those that use four integers 4 (rather than four digits 4) and the basic
Four_fours
Does the plane contains a dense set of points whose distances are all rational
its vertices, and then scaled to make the distances integers. However, like the Erdős–Ulam problem, Harborth's conjecture remains unproven. Anning, Norman
Erdős–Ulam_problem
Probability of shared birthdays
Encyclopedia of Integer Sequences. OEIS. Retrieved 17 February 2020. DasGupta, Anirban. "The matching, birthday and the strong birthday problem: a contemporary
Birthday_problem
18 mathematical problems stated in 1998
Pardo, Luis Miguel (2009). "Smale's 17th Problem: Average Polynomial Time to compute affine and projective solutions" (PDF). Journal of the American Mathematical
Smale's_problems
Type of Diophantine equation
nonsquare integer, and integer solutions are sought for x and y. In Cartesian coordinates, the equation is represented by a hyperbola; solutions occur wherever
Pell's_equation
Problem of inverting exponentiation in groups
logarithm problem. Other base-10 logarithms in the real numbers are not instances of the discrete logarithm problem, because they involve non-integer exponents
Discrete_logarithm
Mathematical riddle
Mathematically, the problem can be formulated as follows: Given an integer m and a set V of positive integers, find the smallest integer z that cannot be
Postage_stamp_problem
Cryptographic primitives that involve lattices
well-studied lattice problems, and Cynthia Dwork showed that a certain average-case lattice problem, known as short integer solutions (SIS), is at least
Lattice-based_cryptography
Natural number
not semiperfect. 70 is also part of the only nontrivial solution pair to the cannonball problem, along with 24. In Jewish tradition, Ptolemy II Philadelphus
70_(number)
Seven mathematical problems with a US$1 million prize for each solution
for the first correct solution to each problem. The Clay Mathematics Institute officially designated the title Millennium Problem for the seven unsolved
Millennium_Prize_Problems
Type of permutation of a set of elements
Dn equals the nearest integer to n!/e, where n! denotes the factorial of n and e ≈ 2.718281828... is Euler's number. The problem of counting derangements
Derangement
Type of computational problem
include the travelling salesman problem, which asks for the route taken by the salesman, and the integer factorization problem, which asks for the list of
Function_problem
on the traveling salesman problem. The problem for graphs is NP-complete if the edge lengths are assumed integers. The problem for points on the plane is
List_of_NP-complete_problems
Unsolved problem in mathematics
Unsolved problem in mathematics Is it possible to construct a three-by-three magic square from nine distinct integer squares? More unsolved problems in mathematics
Magic_square_of_squares
Category of mathematical proof
ratio of two integers. Another consequential proof of impossibility was Ferdinand von Lindemann's proof in 1882, which showed that the problem of squaring
Proof_of_impossibility
Transportation and logistics optimization problem
The vehicle rescheduling problem (VRSP) is a combinatorial optimization and integer programming problem seeking to service customers on a trip after change
Vehicle_rescheduling_problem
Family of computing problems
producer-consumer problem (also known as the bounded-buffer problem) is a family of problems described by Edsger W. Dijkstra since 1965. Dijkstra found the solution for
Producer–consumer_problem
Solution of some Diophantine equation
hard open problem. The MRDP theorem (so named for the initials of the four principal contributors to its solution) states that a set of integers is Diophantine
Diophantine_set
3rd-century Greek mathematician
technique to solve problems in arithmetic. The book considers finding integer solutions to equations with integer coefficients, a class of problem presently called
Diophantus
Unsolved problem about sums of powers
In mathematics, the Prouhet–Tarry–Escott problem asks for two disjoint multisets A and B of n integers each, whose first k power sum symmetric polynomials
Prouhet–Tarry–Escott_problem
Computational problems no algorithm can solve
Hilbert's tenth problem: the problem of deciding whether a Diophantine equation (multivariable polynomial equation) has a solution in integers. For functions
List_of_undecidable_problems
Choosing the fewest coins to make a given amount of money
of the integer knapsack problem, and has applications wider than just currency. It is also the most common variation of the coin change problem, a general
Change-making_problem
Mathematical proof technique
most often used for problems in which a relation between two integers is given, along with a statement to prove about its solutions. In particular, it
Vieta_jumping
Problem in geometry
A006533". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Honsberger, Ross (1973). "9. A Problem in Combinatorics". Mathematical Gems.
Moser's_circle_problem
Mathematical concept
Mixed-Integer Linear Program to solve the optimization problem for a weighted sum of the two objectives to calculate a set of Pareto optimal solutions. Applying
Multi-objective_optimization
Computer science challenge
structures. A problem related to the order-maintenance problem is the list-labeling problem in which instead of the order(X, Y) operation the solution must maintain
Order-maintenance_problem
Problem of stacking blocks to maximize overhang
acceleration (value of g decreases with increase in height), the solution to the single-wide problem is that the maximum overhang is given by ∑ i = 1 N 1 2 i
Block-stacking_problem
Study of mathematical algorithms for optimization problems
optimization, in which an object such as an integer, permutation or graph must be found from a countable set. A problem with continuous variables is known as
Mathematical_optimization
Complexity class used to classify decision problems
is a solution to the problem. The complexity class P (all problems solvable, deterministically, in polynomial time) is contained in NP (problems where
NP_(complexity)
Mathematics of number-placement puzzle
class of Sudoku. The general problem of determining whether a Sudoku puzzle on n2×n2 grids of n×n blocks has a solution is known to be NP-complete. A
Mathematics_of_Sudoku
Problem-solving technique and algorithmic paradigm
satisfies the problem's statement. A brute-force algorithm that finds the divisors of a natural number n would enumerate all integers from 1 to n, and
Brute-force_search
SHORT INTEGER-SOLUTION-PROBLEM
SHORT INTEGER-SOLUTION-PROBLEM
Surname or Lastname
English
English : topographic name for someone who lived by a projecting piece of land, from Old English scēat, or a steep slope, from an unattested Old English scēot.
Boy/Male
Bengali, Indian
Resolution
Surname or Lastname
South German and Austrian
South German and Austrian : variant of Hardt 1.English : variant of Hart 1.
Girl/Female
Arabic, Muslim
Determination; Resolution
Boy/Male
Arabic
Prudence; Resolution
Boy/Male
Hindu
Female
Scandinavian
Scandinavian form of Old Norse Ingigerðr, INGEGERD means "Ing's enclosure."
Surname or Lastname
English and German
English and German : unexplained.
Girl/Female
British, English
Tiny; Small
Surname or Lastname
English
English : nickname from Middle English schort ‘short’.Scottish and northern Irish : reduced Anglicized form of Gaelic Mac an Gheairr, Mac an Ghirr ‘son of the short man’ (see McGirr).
Girl/Female
Tamil
Good or Happy condition, Solution
Surname or Lastname
English and Scottish (now mainly found in Ireland)
English and Scottish (now mainly found in Ireland) : variant spelling of Short.
Female
Swedish
Swedish contracted form of Scandinavian Ingegerd, INGER means "Ing's enclosure."
Girl/Female
Muslim
Determination, Resolution
Girl/Female
Arabic, Muslim
Determination; Resolution
Girl/Female
Hindu
Good or Happy condition, Solution
Boy/Male
Hindu, Indian
Listener
Boy/Male
Hindu, Indian, Marathi
Famous
Boy/Male
Indian, Sanskrit
Evolution; Progress
Surname or Lastname
English
English : topographic name for someone who lived by the seashore, Middle English schore.English : topographic name for someone who lived on or by a bank or steep slope, Old English scora. There are minor places named with this word in Lancashire and West Yorkshire, and the surname may also be a habitational name from these.Americanized spelling of Ashkenazic Jewish S(c)hor(r) or Szor, variants of Schauer.
SHORT INTEGER-SOLUTION-PROBLEM
SHORT INTEGER-SOLUTION-PROBLEM
Girl/Female
Hindu, Indian
Love; Beauty; Speak Less; Brilliant
Boy/Male
Indian, Tamil
Prosperous; Resourceful
Girl/Female
Hindu, Indian
Pure
Biblical
joy; rejoicing
Girl/Female
Latin American
or Mirabel: Worthy of admiration; wonderful; marvelous.
Female
Spanish
Variant spelling of Spanish Maite, MAYTE means "rebellious-harvester."
Boy/Male
Irish
Pure fire.
Girl/Female
American, Anglo, British, English
Daybreak; Sunrise; The First Appearance of Daylight
Girl/Female
Muslim/Islamic
One who speaks
Boy/Male
Indian
Writer
SHORT INTEGER-SOLUTION-PROBLEM
SHORT INTEGER-SOLUTION-PROBLEM
SHORT INTEGER-SOLUTION-PROBLEM
SHORT INTEGER-SOLUTION-PROBLEM
SHORT INTEGER-SOLUTION-PROBLEM
p. pr. & vb. n.
of Short-circuit
a.
Not living or lasting long; being of short continuance; as, a short-lived race of beings; short-lived pleasure; short-lived passion.
superl.
Breaking or crumbling readily in the mouth; crisp; as, short pastry.
n.
A short sound, syllable, or vowel.
n.
See Exsolution.
superl.
Engaging or engaged to deliver what is not possessed; as, short contracts; to be short of stock. See The shorts, under Short, n., and To sell short, under Short, adv.
n.
The act of shooting; the discharge of a missile; a shot; as, the shoot of a shuttle.
a.
Having short life.
imp. & p. p.
of Short-circuit
n.
The state of being dissolved or disintegrated; resolution; disintegration.
n.
The act or process of solving; solution; as, the resolution of an equation or problem.
adv.
In a short manner; briefly; limitedly; abruptly; quickly; as, to stop short in one's course; to turn short.
n.
Short, inferior hemp.
superl.
Not long; having brief length or linear extension; as, a short distance; a short piece of timber; a short flight.
superl.
Not extended in time; having very limited duration; not protracted; as, short breath.
n.
To analyse, or determine the strength of, by means of standard solutions. Cf. Standardized solution, under Solution.
n.
The termination of a disease; resolution.
superl.
Abrupt; brief; pointed; petulant; as, he gave a short answer to the question.