Blum integer
E505280
A Blum integer is a special type of composite number formed as the product of two distinct prime numbers each congruent to 3 modulo 4, widely used in cryptography and pseudorandom number generation.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Blum integer canonical | 1 |
| Blum integers | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T5245492 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Blum integer Context triple: [Blum–Blum–Shub pseudorandom number generator, basedOn, Blum integer]
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A.
Fermat number
A Fermat number is a special type of integer of the form \(F_n = 2^{2^n} + 1\), studied in number theory for its intriguing properties related to primality and constructible polygons.
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B.
Atkin
Atkin is an English surname, typically derived as a diminutive or pet form of the given name Adam.
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C.
Bateman–Horn conjecture
The Bateman–Horn conjecture is a far-reaching unproven statement in number theory that predicts how often sets of polynomial expressions simultaneously take prime values, generalizing several earlier conjectures about the distribution of prime numbers.
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D.
Adleman–Pomerance–Rumely primality test
The Adleman–Pomerance–Rumely primality test is an early deterministic algorithm in computational number theory used to determine whether a given number is prime, notable for its theoretical importance in the development of modern primality testing methods.
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E.
Ramanujan prime
A Ramanujan prime is a type of prime number that provides a bound guaranteeing the existence of a certain number of primes in intervals of the form (x/2, x], named after the mathematician Srinivasa Ramanujan.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Blum integer Target entity description: A Blum integer is a special type of composite number formed as the product of two distinct prime numbers each congruent to 3 modulo 4, widely used in cryptography and pseudorandom number generation.
-
A.
Fermat number
A Fermat number is a special type of integer of the form \(F_n = 2^{2^n} + 1\), studied in number theory for its intriguing properties related to primality and constructible polygons.
-
B.
Atkin
Atkin is an English surname, typically derived as a diminutive or pet form of the given name Adam.
-
C.
Bateman–Horn conjecture
The Bateman–Horn conjecture is a far-reaching unproven statement in number theory that predicts how often sets of polynomial expressions simultaneously take prime values, generalizing several earlier conjectures about the distribution of prime numbers.
-
D.
Adleman–Pomerance–Rumely primality test
The Adleman–Pomerance–Rumely primality test is an early deterministic algorithm in computational number theory used to determine whether a given number is prime, notable for its theoretical importance in the development of modern primality testing methods.
-
E.
Ramanujan prime
A Ramanujan prime is a type of prime number that provides a bound guaranteeing the existence of a certain number of primes in intervals of the form (x/2, x], named after the mathematician Srinivasa Ramanujan.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
composite number
ⓘ
mathematical concept ⓘ |
| appearsIn |
complexity-theoretic cryptography literature
ⓘ
theory of pseudorandom bit generators ⓘ |
| category |
computational number theory
ⓘ
cryptographic number theory ⓘ |
| definition | a composite number n that is the product of two distinct primes p and q with p ≡ 3 (mod 4) and q ≡ 3 (mod 4) ⓘ |
| example |
21 is a Blum integer because 21 = 3 × 7 and 3 ≡ 3 (mod 4), 7 ≡ 3 (mod 4)
ⓘ
33 is a Blum integer because 33 = 3 × 11 and 3 ≡ 3 (mod 4), 11 ≡ 3 (mod 4) ⓘ |
| field | number theory ⓘ |
| generalizationOf | special case of a semiprime with additional congruence conditions on the primes ⓘ |
| hasAdvantage |
provides strong hardness assumptions for cryptographic constructions
ⓘ
simplifies analysis of quadratic residues modulo n ⓘ |
| hasConstraint |
n must be large for security
ⓘ
p and q must be kept secret in cryptographic applications ⓘ |
| hasForm | n = p × q ⓘ |
| hasProperty |
for a Blum integer n, exactly half of the elements with Jacobi symbol 1 modulo n are quadratic residues
ⓘ
for a Blum integer n, the Chinese remainder theorem gives a 1–1 correspondence between square roots modulo n and pairs of square roots modulo p and q ⓘ for a Blum integer n, −1 is not a quadratic residue modulo p or q ⓘ if n is a Blum integer, then λ(n) = lcm(p − 1, q − 1) is even and typically divisible by 4 ⓘ if n is a Blum integer, then φ(n) = (p − 1)(q − 1) is divisible by 4 ⓘ n has exactly four square roots modulo n for any quadratic residue ⓘ n is square-free ⓘ p and q are distinct primes ⓘ p ≡ 3 (mod 4) ⓘ q ≡ 3 (mod 4) ⓘ the Jacobi symbol (a/n) does not uniquely determine quadratic residuosity modulo n ⓘ the factorization of a Blum integer can be recovered from an oracle that distinguishes quadratic residues from non-residues with Jacobi symbol 1 modulo n ⓘ |
| introducedInContext | construction of provably secure pseudorandom generators ⓘ |
| namedAfter | Manuel Blum ⓘ |
| nonExample |
15 is not a Blum integer because 5 ≡ 1 (mod 4)
ⓘ
49 is not a Blum integer because it is not a product of two distinct primes ⓘ |
| relatedTo |
Blum integer-based trapdoor permutations
ⓘ
linked to:
Blum–Blum–Shub pseudorandom number generator
RSA modulus ⓘ hardness of factoring Blum integers ⓘ integer factorization problem ⓘ quadratic residues modulo n ⓘ quadratic residuosity problem ⓘ |
| usedIn |
Blum–Blum–Shub pseudorandom number generator
ⓘ
Goldwasser–Micali cryptosystem ⓘ Rabin cryptosystem variants ⓘ bit commitment schemes ⓘ cryptography ⓘ digital signature schemes ⓘ probabilistic encryption schemes ⓘ pseudorandom number generation ⓘ zero-knowledge proofs ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Blum integer Description of subject: A Blum integer is a special type of composite number formed as the product of two distinct prime numbers each congruent to 3 modulo 4, widely used in cryptography and pseudorandom number generation.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Blum integer