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Product of two distinct primes ≡ 3 (mod 4)
In mathematics, a natural number n is a Blum integer if n = p × q is a semiprime for which p and q are distinct prime numbers congruent to 3 mod 4. That
Blum_integer
Pseudorandom number generator
Blum Blum Shub (B.B.S.) is a pseudorandom number generator proposed in 1986 by Lenore Blum, Manuel Blum and Michael Shub that is derived from Michael
Blum_Blum_Shub
Venezuelan computer scientist
Manuel Blum (born 26 April 1938) is a Venezuelan-born American computer scientist who received the 1995 ACM Turing Award "In recognition of his contributions
Manuel_Blum
Natural number
congruent to 3 modulo 4, 57 = 3 ⋅ 19 {\displaystyle 57=3\cdot 19} is a Blum integer. It is a Leyland number, because 57 = 2 5 + 5 2 {\displaystyle 57=2^{5}+5^{2}}
57_(number)
Natural number
000,696,445,844,586,496) have no common digits. 713 = 23 × 31. It is a Blum integer. In Judaism there are 713 letters on a Mezuzah scroll. 714 = 2 × 3 ×
700_(number)
Natural number
77 is the second composite member of the 19-aliquot tree with 65 a Blum integer since both 7 and 11 are Gaussian primes. the sum of three consecutive
77_(number)
Topics referred to by the same term
in Washington Blum Lakes, six lakes in Washington Blum axioms, in computational complexity theory Blum integer, in mathematics Blum's speedup theorem
Blum
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
Natural number
are a total of 21 prime numbers between 100 and 200. 21 is the first Blum integer, since it is a semiprime with both its prime factors being Gaussian primes
21_(number)
Natural number
it a square-free integer. 69 is a Blum integer since the two factors of 69 are both Gaussian primes, and an Ulam number, an integer that is the sum of
69_(number)
Natural number
536 1's in all partitions of 23 into odd parts. 537 = 3 × 179. It is a Blum integer, a D-number, and a zero of the Mertens function. 538 = 2 × 269. It is
500_(number)
Natural number
is a Blum integer. 129 is a repdigit in base 6 (333). 129 is a happy number. 129 is a centered octahedral number. "Sloane's A016105 : Blum integers". The
129_(number)
Natural number
is a Blum integer. 418 = 2 × 11 × 19. It is a sphenic number, a balanced number. and the fourth 71-gonal number. It is the sum of the integers between
400_(number)
Natural number
Since those prime factors are Gaussian primes, this means that 133 is a Blum integer. 133 is the number of compositions of 13 into distinct parts. 133 is
133_(number)
Natural number
prime numbers congruent to 3 mod 4, 177 is the eleventh Blum integer, where the first such integer 21 divides the aliquot part of 177 thrice over. The first
177_(number)
Natural number
There are 632 13-bead necklaces with 2 colors 633 = 3 × 211. It is a Blum integer and the sum of three consecutive primes (199 + 211 + 223). 634 = 2 ×
600_(number)
Natural number
Since those prime factors are Gaussian primes, this means that 141 is a Blum integer. a Hilbert prime Sometimes used as an acronym [1 representing A and 4
141_(number)
Number used for counting
2, 3, and so on, possibly excluding 0. The terms positive integers, non-negative integers, whole numbers, and counting numbers are also used. The set
Natural_number
Natural number
Since its prime factors 7 and 23 are both Gaussian primes, 161 is a Blum integer. 161 is a palindromic number. 161/72 is a commonly used rational approximation
161_(number)
Natural number
proper factors of 201, 3 and 67, are both Gaussian primes, 201 is a Blum integer. 201 is the course number of basic or entry-level courses at some Canadian
201_(number)
Natural number
35,13,1,0) of three numbers to the Prime 13 in the 13-Aliquot tree. a Blum integer, since its two prime factors, 3 and 31 are both Gaussian primes. a repdigit
93_(number)
Proving validity without revealing other data
system by Oded Goldreich verifying that a two-prime modulus is not a Blum integer. Oded Goldreich, Silvio Micali, and Avi Wigderson took this one step
Zero-knowledge_proof
Natural number
natural number following 248 and preceding 250. Additionally, 249 is: a Blum integer. a semiprime. palindromic in base 82 (3382). a Harshad number in bases
249_(number)
Natural number
(151 + 157). It is the totient sum of the first 41 integers. 309 = 3 × 103. It is a Blum integer and a centered icosahedral number. 310 = 2 × 5 × 31
300_(number)
Natural number
number, a centered 36-gonal number, a Fermat pseudoprime to base 5, and a Blum integer. It is both the sum of two positive cubes and the difference of two positive
217_(number)
Natural number
balanced number, and a zero of Mertens function. 813 = 3 × 271. It is a Blum integer. 814 = 2 × 11 × 37. It is a sphenic number, a nontotient, and a zero
800_(number)
Natural number
a Blum integer. a member of the 13-aliquot tree. Sloane, N. J. A. (ed.). "Sequence A078972 (brilliant numbers)". The On-Line Encyclopedia of Integer Sequences
253_(number)
Asymmetric key encryption algorithm
and testing the two Legendre symbols. If p, q = 3 mod 4 (i.e., N is a Blum integer), then the value N − 1 is guaranteed to have the required property. The
Goldwasser–Micali cryptosystem
Goldwasser–Micali_cryptosystem
USA computer scientist and mathematician
computational hardness assumption that integer factorization is infeasible.[BBS] Blum is also known for the Blum–Shub–Smale machine, a theoretical model
Lenore_Blum
Complexity class used in circuit complexity
have been explicitly constructed under the assumption that factoring Blum integers is hard (i.e. requires circuits of size 2 p o l y ( n ) {\displaystyle
TC0
Asymmetric key encryption algorithm
The Blum–Goldwasser (BG) cryptosystem is an asymmetric key encryption algorithm proposed by Manuel Blum and Shafi Goldwasser in 1984. Blum–Goldwasser is
Blum–Goldwasser_cryptosystem
Model of computation over real numbers
In computation theory, the Blum–Shub–Smale machine, or BSS machine, is a model of computation introduced by Lenore Blum, Michael Shub and Stephen Smale
Blum–Shub–Smale_machine
Mathematical model of computer
as well as comparisons, but not modulus or rounding to integers. The reason for avoiding integer rounding and modulus operations is that allowing these
Real_RAM
Number that permute or shift cyclically when multiplied by another number
mathematics, the transposable integers are integers that permute or shift cyclically when they are multiplied by another integer n {\displaystyle n} . Examples
Transposable_integer
Approach to public-key cryptography
agreement with a symmetric encryption scheme. They are also used in several integer factorization algorithms that have applications in cryptography, such as
Elliptic-curve_cryptography
theorem of arithmetic Square-free Square-free integer Square-free polynomial Square number Power of two Integer-valued polynomial Rational number Unit fraction
List_of_number_theory_topics
Algorithm for public-key cryptography
it is practical to find three very large positive integers e, d, and n, such that for all integers x (0 ≤ x < n), both (xe)d and x have the same remainder
RSA_cryptosystem
Key agreement protocol
consisting of a private key d {\displaystyle d} (a randomly selected integer in the interval [ 1 , n − 1 ] {\displaystyle [1,n-1]} ) and a public key
Elliptic-curve_Diffie–Hellman
Product of an integer with itself
number or perfect square is an integer that is the square of an integer; in other words, it is the product of some integer with itself. For example, 9 is
Square_number
Australian computer scientist (born 1944)
most efficient general string searching algorithm known today. Along with Blum, Floyd, Rivest, and Tarjan, he described median of medians, the first worst-case
Vaughan_Pratt
Two raised to an integer power
of the form 2n where n is an integer, that is, the result of exponentiation with the number two as the base and integer n as the exponent. In the fast-growing
Power_of_two
Mechanism for authenticating cryptographic keys
v t e Public-key cryptography Algorithms Integer factorization Benaloh Blum–Goldwasser Cayley–Purser Damgård–Jurik GMR Goldwasser–Micali Naccache–Stern
Web_of_trust
Arithmetic operation
numbers: the base, b, and the exponent or power, n. When n is a positive integer, exponentiation corresponds to repeated multiplication of the base: that
Exponentiation
Inherent difficulty of computational problems
or no. Notable examples include the traveling salesman problem and the integer factorization problem. It is tempting to think that the notion of function
Computational complexity theory
Computational_complexity_theory
Non-federated cryptographic protocol
v t e Public-key cryptography Algorithms Integer factorization Benaloh Blum–Goldwasser Cayley–Purser Damgård–Jurik GMR Goldwasser–Micali Naccache–Stern
Signal_Protocol
System that can issue, distribute and verify digital certificates
v t e Public-key cryptography Algorithms Integer factorization Benaloh Blum–Goldwasser Cayley–Purser Damgård–Jurik GMR Goldwasser–Micali Naccache–Stern
Public_key_infrastructure
Asymmetric encryption algorithm developed by Robert McEliece
v t e Public-key cryptography Algorithms Integer factorization Benaloh Blum–Goldwasser Cayley–Purser Damgård–Jurik GMR Goldwasser–Micali Naccache–Stern
McEliece_cryptosystem
Number divisible only by 1 and itself
trial division, tests whether n {\displaystyle n} is a multiple of any integer between 2 and n {\displaystyle {\sqrt {n}}} . Faster algorithms include
Prime_number
Product of two prime numbers
where they are used by RSA and pseudorandom number generators such as Blum Blum Shub. These methods rely on the fact that finding two large primes and
Semiprime
Integer having only small prime factors
In number theory, an n-smooth (or n-friable) number is an integer whose prime factors are all less than or equal to n. For example, a 7-smooth number
Smooth_number
Public-key encryption scheme
function whose security, like that of RSA, is related to the difficulty of integer factorization. The Rabin trapdoor function has the advantage that inverting
Rabin_cryptosystem
Numbers with many divisors
a positive integer that has more divisors than all smaller positive integers. If d(n) denotes the number of divisors of a positive integer n, then a positive
Highly_composite_number
Short sequence of bytes used to authenticate or look up a longer public key
v t e Public-key cryptography Algorithms Integer factorization Benaloh Blum–Goldwasser Cayley–Purser Damgård–Jurik GMR Goldwasser–Micali Naccache–Stern
Public_key_fingerprint
Public-key cryptosystem
over any cyclic group G {\displaystyle G} , like multiplicative group of integers modulo n if and only if n is 1, 2, 4, pk or 2pk, where p is an odd prime
ElGamal_encryption
IEEE standardization project for public-key cryptography
signature, and encryption schemes using several mathematical approaches: integer factorization, discrete logarithm, and elliptic curve discrete logarithm
IEEE_P1363
Figurate number
The triangular numbers or triangle numbers are the sequence of positive integers that can be represented as a lattice of points arranged in an equilateral
Triangular_number
Indian mathematician
regularity lemma in Graph Theory 2020. Foundations of Data Science. (with Avrim Blum and John Hopcroft). 2009. Spectral Algorithms.(with Santosh Vempala) "Clustering
Ravindran_Kannan
Method of exchanging cryptographic keys
base g = 5 (which is a primitive root modulo 23). Alice chooses a secret integer a = 4, then sends Bob A = ga mod p A = 54 mod 23 = 4 (in this example both
Diffie–Hellman_key_exchange
Digital verification standard
1 {\displaystyle p-1} is a multiple of q {\displaystyle q} . Choose an integer h {\displaystyle h} randomly from { 2 … p − 2 } {\displaystyle \{2\ldots
Digital_Signature_Algorithm
RSA and the Blum Blum Shub pseudorandom number generator, rests in the difficulty of factorizing large integers. If factorizing large integers becomes easier
TWIRL
Cryptographic algorithm for digital signatures
Bézout's identity). Alice creates a key pair, consisting of a private key integer d A {\displaystyle d_{A}} , randomly selected in the interval [ 1 , n −
Elliptic Curve Digital Signature Algorithm
Elliptic_Curve_Digital_Signature_Algorithm
Digital signature scheme
keys are elements of G 2 {\displaystyle G_{2}} , and the secret key is an integer in [ 0 , q − 1 ] {\displaystyle [0,q-1]} . Working in an elliptic curve
BLS_digital_signature
Unsolved problem in cryptography
modulus N, a task believed to be impractical if N is sufficiently large (see integer factorization). The RSA key setup routine already turns the public exponent
RSA_problem
Quantum-safe key encapsulation mechanism
v t e Public-key cryptography Algorithms Integer factorization Benaloh Blum–Goldwasser Cayley–Purser Damgård–Jurik GMR Goldwasser–Micali Naccache–Stern
ML-KEM
Type of cryptosystem
v t e Public-key cryptography Algorithms Integer factorization Benaloh Blum–Goldwasser Cayley–Purser Damgård–Jurik GMR Goldwasser–Micali Naccache–Stern
Threshold_cryptosystem
Positive integer of the form (2^(2^n))+1
them, is a positive integer of the form: F n = 2 2 n + 1 , {\displaystyle F_{n}=2^{2^{n}}+1,} where n is a non-negative integer. The first few Fermat
Fermat_number
Infinite integer series where the next number is the sum of the two preceding it
The Lucas sequence is an integer sequence named after the mathematician François Édouard Anatole Lucas (1842–1891), who studied both that sequence and
Lucas_number
Integer whose multiples are digit rotations
A cyclic number is an integer for which cyclic permutations of the digits are successive integer multiples of the number. The most widely-known cyclic
Cyclic_number
Number that is less than the sum of its proper divisors
excessive number is a positive integer for which the sum of its proper divisors is greater than the number. The integer 12 is the first abundant number:
Abundant_number
Hybrid encryption in cryptography
v t e Public-key cryptography Algorithms Integer factorization Benaloh Blum–Goldwasser Cayley–Purser Damgård–Jurik GMR Goldwasser–Micali Naccache–Stern
Integrated_Encryption_Scheme
Class of natural numbers with many divisors
number of divisors an integer has and that integer raised to some positive power. For any possible exponent, whichever integer has the greatest ratio
Superior highly composite number
Superior_highly_composite_number
Numbers k where x - phi(x) = k has many solutions
theory, a branch of mathematics, a highly cototient number is a positive integer k {\displaystyle k} which is above 1 and has more solutions to the equation
Highly_cototient_number
Ten raised to an integer power
of the integer powers of the number ten; in other words, ten multiplied by itself a certain number of times (when the power is a positive integer). By definition
Power_of_10
Integer having a non-trivial divisor
number is a positive integer that can be formed by multiplying two smaller positive integers. Accordingly, it is a positive integer that has at least one
Composite_number
Mathematical scheme for verifying the authenticity of digital documents
is the product of two random secret distinct large primes, along with integers, e and d, such that e d ≡ 1 (mod φ(N)), where φ is Euler's totient function
Digital_signature
Cryptographic key management algorithm
v t e Public-key cryptography Algorithms Integer factorization Benaloh Blum–Goldwasser Cayley–Purser Damgård–Jurik GMR Goldwasser–Micali Naccache–Stern
Double_Ratchet_Algorithm
Numbers obtained by adding the two previous ones
Fibonacci numbers Fn are: The Fibonacci sequence can be extended to negative integer indices by following the same recurrence relation in the negative direction
Fibonacci_sequence
Prime number of the form 2^n – 1
of two. That is, it is a prime number of the form Mn = 2n − 1 for some integer n. They are named after Marin Mersenne, a French Minim friar, who studied
Mersenne_prime
Digital signature scheme
v t e Public-key cryptography Algorithms Integer factorization Benaloh Blum–Goldwasser Cayley–Purser Damgård–Jurik GMR Goldwasser–Micali Naccache–Stern
Schnorr_signature
Number that when multiplied by another number moves its last digit to its front
general, an n-parasitic number can be found as follows. Pick a one digit integer k such that k ≥ n, and take the period of the repeating decimal k/(10n−1)
Parasitic_number
Form of public key cryptography
problem). The problem is as follows: given a set of integers A {\displaystyle A} and an integer c {\displaystyle c} , find a subset of A {\displaystyle
Merkle–Hellman knapsack cryptosystem
Merkle–Hellman_knapsack_cryptosystem
Cryptographic key agreement scheme
v t e Public-key cryptography Algorithms Integer factorization Benaloh Blum–Goldwasser Cayley–Purser Damgård–Jurik GMR Goldwasser–Micali Naccache–Stern
Station-to-Station_protocol
Multiparty cryptographic process
v t e Public-key cryptography Algorithms Integer factorization Benaloh Blum–Goldwasser Cayley–Purser Damgård–Jurik GMR Goldwasser–Micali Naccache–Stern
Distributed_key_generation
Integer filtered out using a sieve similar to that of Eratosthenes
(Numbers that are both lucky and prime)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Guy, Richard K. (2004). Unsolved problems in
Lucky_number
Algorithm for public key cryptography
=\operatorname {lcm} (p-1,q-1)} . lcm means Least Common Multiple. Select random integer g {\displaystyle g} where g ∈ Z n 2 ∗ {\displaystyle g\in \mathbb {Z} _{n^{2}}^{*}}
Paillier_cryptosystem
Algorithm that generates an approximation of a random number sequence
pseudo-random number generator, or CBPRNG) is a kind of PRNG that uses only an integer counter as its internal state: output = f ( n , key ) {\displaystyle
Pseudorandom_number_generator
Result of multiplying four instances of a number together
fourth power is always 1. Every positive integer can be expressed as the sum of at most 19 fourth powers; every integer larger than 13792 can be expressed as
Fourth_power
Number equal to the sum of its proper divisors
In number theory, a perfect number is a positive integer that is equal to the sum of its positive proper divisors, that is, divisors excluding the number
Perfect_number
Positive integer that is an integer power of another positive integer
factors, or, in other words, an integer that can be expressed as a square or a higher integer power of another integer greater than one. More formally
Perfect_power
Cryptographic method
for Falcon-1024. The signature verification can be done entirely using integers modulo q {\displaystyle q} . Contrarily, the signature generation uses
Falcon_(signature_scheme)
Scheme often used with RSA encryption
v t e Public-key cryptography Algorithms Integer factorization Benaloh Blum–Goldwasser Cayley–Purser Damgård–Jurik GMR Goldwasser–Micali Naccache–Stern
Optimal asymmetric encryption padding
Optimal_asymmetric_encryption_padding
Lovász number of the complement and then rounding the approximation to an integer would not necessarily produce a monotone function, however. To make the
Tardos_function
Measure of algorithmic complexity
Levin (1974). An axiomatic approach to Kolmogorov complexity based on Blum axioms (Blum 1967) was introduced by Mark Burgin in the paper presented for publication
Kolmogorov_complexity
Digital signature scheme
parameters, the second phase computes the key pair for a single user: Choose an integer x {\displaystyle x} randomly from { 1 … p − 2 } {\displaystyle \{1\ldots
ElGamal_signature_scheme
Cryptographic signature scheme
arbitrarily long message being signed. Let k {\displaystyle k} be a positive integer and let P = { 0 , 1 } k {\displaystyle P=\{0,1\}^{k}} be the set of messages
Lamport_signature
Integer divisible by sum of its digits
mathematics, a Harshad number (or Niven number) in a given number base is an integer that is divisible by the sum of its digits when written in that base. Harshad
Harshad_number
Augmented password-authenticated key exchange protocol
logarithms modulo N is infeasible. All arithmetic is performed in the ring of integers modulo N, Z N {\displaystyle \scriptstyle \mathbb {Z} _{N}} . This means
Secure Remote Password protocol
Secure_Remote_Password_protocol
Number that cannot be written as an aliquot sum
untouchable number is a positive integer that cannot be expressed as the sum of all the proper divisors of any positive integer. That is, these numbers are
Untouchable_number
Sum of a number's digits
sum of the base 10 digits of the integers 0, 1, 2, ... is given by OEIS: A007953 in the On-Line Encyclopedia of Integer Sequences. Borwein & Borwein (1992)
Digit_sum
Number raised to the third power
cube of an integer. The non-negative perfect cubes up to 603 are (sequence A000578 in the OEIS): Geometrically speaking, a positive integer m is a perfect
Cube_(algebra)
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