Higher residuosity problem, the Glossary

Table of Contents

1. 22 relations: Benaloh cryptosystem, Chinese remainder theorem, Cryptography, Cyclic group, Divisor, Group (mathematics), Group isomorphism, If and only if, Index of a subgroup, Integer, Integer factorization, Modular arithmetic, Naccache–Stern cryptosystem, Prime number, Public-key cryptography, Quadratic residue, Quadratic residuosity problem, Ring (mathematics), Ring homomorphism, Semantic security, Subgroup, Unit (ring theory).

2. Computational hardness assumptions
3. Computational number theory

Benaloh cryptosystem

The Benaloh Cryptosystem is an extension of the Goldwasser-Micali cryptosystem (GM) created in 1985 by Josh (Cohen) Benaloh.

See Higher residuosity problem and Benaloh cryptosystem

Chinese remainder theorem

In mathematics, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer n by several integers, then one can determine uniquely the remainder of the division of n by the product of these integers, under the condition that the divisors are pairwise coprime (no two divisors share a common factor other than 1).

See Higher residuosity problem and Chinese remainder theorem

Cryptography

Cryptography, or cryptology (from κρυπτός|translit.

See Higher residuosity problem and Cryptography

Cyclic group

In abstract algebra, a cyclic group or monogenous group is a group, denoted Cn (also frequently \Zn or Zn, not to be confused with the commutative ring of p-adic numbers), that is generated by a single element.

See Higher residuosity problem and Cyclic group

Divisor

In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by another integer m if m is a divisor of n; this implies dividing n by m leaves no remainder.

See Higher residuosity problem and Divisor

Group (mathematics)

In mathematics, a group is a set with an operation that associates an element of the set to every pair of elements of the set (as does every binary operation) and satisfies the following constraints: the operation is associative, it has an identity element, and every element of the set has an inverse element.

See Higher residuosity problem and Group (mathematics)

Group isomorphism

In abstract algebra, a group isomorphism is a function between two groups that sets up a bijection between the elements of the groups in a way that respects the given group operations.

See Higher residuosity problem and Group isomorphism

If and only if

In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements.

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Index of a subgroup

In mathematics, specifically group theory, the index of a subgroup H in a group G is the number of left cosets of H in G, or equivalently, the number of right cosets of H in G. The index is denoted |G:H| or or (G:H).

See Higher residuosity problem and Index of a subgroup

Integer

An integer is the number zero (0), a positive natural number (1, 2, 3,...), or the negation of a positive natural number (−1, −2, −3,...). The negations or additive inverses of the positive natural numbers are referred to as negative integers.

See Higher residuosity problem and Integer

Integer factorization

In number theory, integer factorization is the decomposition of a positive integer into a product of integers. Higher residuosity problem and integer factorization are computational hardness assumptions.

See Higher residuosity problem and Integer factorization

Modular arithmetic

In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus.

See Higher residuosity problem and Modular arithmetic

Naccache–Stern cryptosystem

The Naccache–Stern cryptosystem is a homomorphic public-key cryptosystem whose security rests on the higher residuosity problem.

See Higher residuosity problem and Naccache–Stern cryptosystem

Prime number

A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers.

See Higher residuosity problem and Prime number

Public-key cryptography

Public-key cryptography, or asymmetric cryptography, is the field of cryptographic systems that use pairs of related keys.

See Higher residuosity problem and Public-key cryptography

Quadratic residue

In number theory, an integer q is called a quadratic residue modulo n if it is congruent to a perfect square modulo n; i.e., if there exists an integer x such that: Otherwise, q is called a quadratic nonresidue modulo n. Originally an abstract mathematical concept from the branch of number theory known as modular arithmetic, quadratic residues are now used in applications ranging from acoustical engineering to cryptography and the factoring of large numbers.

See Higher residuosity problem and Quadratic residue

Quadratic residuosity problem

The quadratic residuosity problem (QRP) in computational number theory is to decide, given integers a and N, whether a is a quadratic residue modulo N or not. Higher residuosity problem and quadratic residuosity problem are computational hardness assumptions and computational number theory.

See Higher residuosity problem and Quadratic residuosity problem

Ring (mathematics)

In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist.

See Higher residuosity problem and Ring (mathematics)

Ring homomorphism

In mathematics, a ring homomorphism is a structure-preserving function between two rings.

See Higher residuosity problem and Ring homomorphism

Semantic security

In cryptography, a semantically secure cryptosystem is one where only negligible information about the plaintext can be feasibly extracted from the ciphertext.

See Higher residuosity problem and Semantic security

Subgroup

In group theory, a branch of mathematics, given a group under a binary operation ∗, a subset of is called a subgroup of if also forms a group under the operation ∗.

See Higher residuosity problem and Subgroup

Unit (ring theory)

In algebra, a unit or invertible element of a ring is an invertible element for the multiplication of the ring.

See Higher residuosity problem and Unit (ring theory)

See also

Computational hardness assumptions

• Computational Diffie–Hellman assumption
• Computational hardness assumption
• Decision Linear assumption
• Decisional Diffie–Hellman assumption
• Decisional composite residuosity assumption
• Diffie–Hellman problem
• Discrete logarithm
• Discrete logarithm records
• Exponential time hypothesis
• Higher residuosity problem
• Integer factorization
• Lattice problem
• Logjam (computer security)
• Phi-hiding assumption
• Planted clique
• Quadratic residuosity problem
• RSA problem
• Ring learning with errors
• Security level
• Short integer solution problem
• Small set expansion hypothesis
• Strong RSA assumption
• Sub-group hiding
• Unique games conjecture
• XDH assumption

Computational number theory

• ABC@Home
• Algorithmic Number Theory Symposium
• Computational hardness assumption
• Computational number theory
• Evdokimov's algorithm
• Factorization of polynomials over finite fields
• Fast Library for Number Theory
• Higher residuosity problem
• Itoh–Tsujii inversion algorithm
• Korkine–Zolotarev lattice basis reduction algorithm
• Lattice reduction
• Lenstra–Lenstra–Lovász lattice basis reduction algorithm
• Odlyzko–Schönhage algorithm
• Phi-hiding assumption
• Quadratic residuosity problem
• Supersingular isogeny graph
• Table of costs of operations in elliptic curves

References

[1] https://en.wikipedia.org/wiki/Higher_residuosity_problem