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Lossy Compression Algorithm
Modulo-N code is a lossy compression algorithm used to compress correlated data sources using modular arithmetic. When applied to two nodes in a network
Modulo-N_code
Computational operation
In mathematics and computing, modulo is the binary operation of which the value is the remainder or signed remainder of a division of one number by another
Modulo
Computation modulo a fixed integer
exponentiation Modulo (mathematics) Multiplicative group of integers modulo n Pisano period (Fibonacci sequences modulo n) Primitive root modulo n Quadratic
Modular_arithmetic
Type of two-dimensional barcode
A QR code, short for quick-response code, is a type of two-dimensional matrix barcode invented in 1994 by Masahiro Hara of the Japanese company Denso
QR_code
Ordering of binary values, used for positioning and error correction
summation operation in the prefix sum is performed modulo two. To construct the binary-reflected Gray code iteratively, at step 0 start with the code0=𝟶
Gray_code
Integer that is a perfect square modulo some integer
quadratic residue modulo n if it is congruent to a perfect square modulo n; that is, if there exists an integer x such that x 2 ≡ q ( mod n ) . {\displaystyle
Quadratic_residue
Compact encoding of digital data
Minimum description length Modulo-N code Motion coding Range coding Set redundancy compression Sub-band coding Universal code (data compression) Vector
Data_compression
Barcode format
zone The check symbol is calculated from a weighted sum (modulo 103) of all the symbols. Code 128 includes 108 symbols: 103 data symbols, 3 start symbols
Code_128
Input method
characters in the current code page). In the IBM PC BIOS, typing an Alt code greater than 255 produced the same as that number modulo 256. Some applications
Alt_code
Natural number
reduction modulo 2 records the parity of an integer: even integers are congruent to 0 modulo 2, and odd integers are congruent to 1 modulo 2. In algebra
2
Family of linear error-correcting codes
≥ 2 there is a code-word with block length n = 2r − 1 and message length k = 2r − r − 1. Hence the rate of Hamming codes is R = k / n = 1 − r / (2r −
Hamming_code
Mathematical sequences
as b(n) = a(q ⋅ n), where q = 2N/2+1. Modified sequences are then formed by adding a(n) and cyclically time shifted versions of b(n) using modulo-two arithmetic
Kasami_code
Linear error correcting code
(LDPC) codes, also known as Gallager codes, are a class of error-correction codes first proposed in 1960. Together with the closely related turbo codes, they
Low-density_parity-check_code
Logical problem studied in computer science
In computer science and mathematical logic, satisfiability modulo theories (SMT) is the problem of determining whether a mathematical formula is satisfiable
Satisfiability modulo theories
Satisfiability_modulo_theories
Mapping arbitrary data to fixed-size values
Algebraic coding is a variant of the division method of hashing which uses division by a polynomial modulo 2 instead of an integer to map n bits to m
Hash_function
Unique numeric book identifier since 1970
Bookland "country code", is prefixed to the ISBN in the barcode data, and the check digit is recalculated according to the EAN-13 formula (modulo 10, 1× and
ISBN
Computer arithmetic error
a result larger than the maximum above for an N-bit integer, an overflow reduces the result to modulo N-th power of 2, retaining only the least significant
Integer_overflow
Concept in modular arithmetic
positive integer n and an integer a coprime to n, the multiplicative order of a modulo n is the smallest positive integer k such that ak ≡ 1 (mod n). In other
Multiplicative_order
Barcode system for tracking trade items
The Universal Product Code (UPC or UPC code) is a barcode symbology that is used worldwide for tracking trade items in stores. The chosen symbology has
Universal_Product_Code
Barcode symbology
A typical code 93 barcode has the following structure: A start character * Encoded message First modulo-47 check character "C" Second modulo-47 check character
Code_93
Extension of the Luhn algorithm
associated code-point and then performing the computations in mod N (where N is the number of valid input characters). Finally, the resulting check code-point
Luhn_mod_N_algorithm
Kei car produced by Honda
"コンプリートカーブランド「Modulo X」の展開を開始" [Launch of the complete car brand "Modulo X"]. Honda. 24 December 2012. Retrieved 22 July 2026. "新型軽乗用車「N-BOX SLASH（エヌボックス
Honda_N-Box
Variable length, discrete barcode symbology
Code 39 is sometimes used with an optional modulo 10 or 43 check digit. Using it requires this feature to be enabled in the barcode reader. The code with
Code_39
Signals broadcast by GPS satellites
modulo operations correspond to resets. Note that both are reset each millisecond (synchronized with C/A code epochs). In addition, the extra modulo operation
GPS_signals
Subroutine call performed as final action of a procedure
argument (product in the above example) to the function. Tail recursion modulo cons is a generalization of tail-recursion optimization introduced by David
Tail_call
Algorithm for generating pseudo-randomized numbers
the parameters m and a. For example, a = 1 and c = 1 produces a simple modulo-m counter, which has a long period, but is obviously non-random. Other values
Linear_congruential_generator
Technique in digital communications
then the symbol to be transmitted for xi is where ⊕ indicates binary or modulo-2 addition. On the decoding side, xi is recovered as That is, xi depends
Differential_coding
Serial number used to identify a periodical publication
0⋅8+3⋅7+7⋅6+8⋅5+5⋅4+9⋅3+5⋅2=0+21+42+40+20+27+10=160. The remainder of this sum modulo 11 is then calculated: 16011=14 remainder 6=14+611 If there is no remainder
ISSN
Class of quantum error correcting codes
addition modulo 2. Then CSS ( C 1 , C 2 ) {\displaystyle {\text{CSS}}(C_{1},C_{2})} as quantum correcting code [ [ n , k 1 − k 2 , d ] ] {\displaystyle [[n,k_{1}-k_{2}
CSS_code
Lossless data compression method
For N, the number to be encoded, find quotient = q = floor(N/M) remainder = r = N modulo M Generate codeword The code format : <Quotient code><Remainder
Golomb_coding
Type of cyclic code
{\displaystyle l} is a quadratic residue modulo p {\displaystyle p} . Its generator polynomial as a cyclic code is given by f ( x ) = ∏ j ∈ Q ( x − ζ j
Quadratic_residue_code
Algorithm for shuffling a finite sequence
low-order bits of a linear congruential PRNG with modulo 2e are less random than the high-order ones: the low n bits of the generator themselves have a period
Fisher–Yates_shuffle
Encoding of 1D cellular automaton rules
Rule 90 is also interesting because it creates Pascal's triangle modulo 2. A code of this type suffixed by an R, such as "Rule 37R", indicates a second-order
Wolfram_code
Type of block code
{\displaystyle l} is a quadratic residue modulo the prime p {\displaystyle p} there is a quadratic residue code which is a cyclic code of length p {\displaystyle p}
Cyclic_code
Concept in modular arithmetic
modulo p form a cyclic group of order p − 1. For any integer n > 1 {\displaystyle n>1} , it's always the case that n 2 − n + 1 {\displaystyle n^{2}-n+1}
Modular multiplicative inverse
Modular_multiplicative_inverse
Algorithm for public-key cryptography
d, and n, such that for all integers x (0 ≤ x < n), both (xe)d and x have the same remainder when divided by n (they are congruent modulo n):(xe)d≡x(modn)
RSA_cryptosystem
Probabilistic primality test
Certainly 1 and −1, when squared modulo n, always yield 1. It remains to show that there are no other square roots of 1 modulo n. This is a special case, here
Miller–Rabin_primality_test
Algebraic structure
of X n − 1 {\displaystyle X^{n}-1} , whose discriminant n n {\displaystyle n^{n}} is nonzero modulo p {\displaystyle p} . It follows that the n {\displaystyle
Finite_field
Type of error-correcting code using convolution
n modulo-2 adders (a modulo 2 adder can be implemented with a single Boolean XOR gate, where the logic is: 0+0 = 0, 0+1 = 1, 1+0 = 1, 1+1 = 0), and n
Convolutional_code
Retro-styled kei car produced by Honda
Premium (JG1) N-One RS (JG1) N-One RS (JG1) N-One Modulo X (JG1) N-One Modulo X interior (JG1) The second-generation N-One went on sale in November 2020
Honda_N-One
ID number
26; the remainder (Modulo) will give the last character, according to the following table: Should two people yield the same code, the fiscal administration
Italian_fiscal_code
Pair of related error-correcting codes
are evaluated modulo 3. Linear combinations, or vector addition, of the rows of the matrix produces all possible words contained in the code. This is referred
Ternary_Golay_code
Type of linear congruential generator with no additive constant
congruential generator (LCG) that operates in multiplicative group of integers modulo n. The general formula is X k + 1 = a ⋅ X k mod m , {\displaystyle X_{k+1}=a\cdot
Lehmer random number generator
Lehmer_random_number_generator
Sequence of n-tuples of integers
all zeros. The linear map in modulo 2 can further be identified as the cellular automata denoted as rule 102 in Wolfram code and related to rule 90 through
Ducci_sequence
{\displaystyle p} be an odd prime, and let g {\displaystyle g} be a primitive root modulo p {\displaystyle p} . Let x 0 {\displaystyle x_{0}} be a seed, and let x
Blum–Micali_algorithm
International shipping standard
languages have a modulo operator for step 3; some languages (also) return the decimal portion which must be multiplied by 11 to get a proper modulo. 11 is used
ISO_6346
Mathematics concept
an n by n matrix with entries in GF ( p ) {\displaystyle \operatorname {GF} (p)} satisfying H H T = n I {\displaystyle HH^{\textsf {T}}=nI} modulo p
Hadamard_matrix
Problem of inverting exponentiation in groups
n , {\displaystyle \log _{b}\colon H\to \mathbf {Z} _{n},} where Z n {\displaystyle \mathbf {Z} _{n}} denotes the additive group of integers modulo n
Discrete_logarithm
Arithmetic in a field with a finite number of elements
degree strictly less than n over GF(p). Operations are then performed modulo m(x) where m(x) is an irreducible polynomial of degree n over GF(p), for instance
Finite_field_arithmetic
Simple and widely known encryption technique
is performed similarly: D n ( x ) = ( x − n ) mod 26. {\displaystyle D_{n}(x)=(x-n)\mod {26}.} The modulo operation (mod) may need to be performed twice
Caesar_cipher
Error correction code
q} modulo n {\displaystyle n} is m . {\displaystyle m.} Therefore, the simplified definition is indeed a special case of the general one. A BCH code with
BCH_code
A prime p divides a^p–a for any integer a
of y, e and n is easy if one knows φ(n). In fact, the extended Euclidean algorithm allows computing the modular inverse of e modulo φ(n), that is, the
Fermat's_little_theorem
Computer checksum algorithm
} See the zlib source code for a more efficient implementation that requires a fetch and two additions per byte, with the modulo operations deferred with
Adler-32
Binary representation for signed numbers
be 2 N − 0 = 2 N {\displaystyle 2^{N}-0=2^{N}} , but in N {\displaystyle N} bits, all values are taken modulo 2 N {\displaystyle 2^{N}} , and 2 N {\displaystyle
Two's_complement
About simultaneous modular congruences
{Z} /N\mathbb {Z} \cong \mathbb {Z} /n_{1}\mathbb {Z} \times \cdots \times \mathbb {Z} /n_{k}\mathbb {Z} } between the ring of integers modulo N and the
Chinese_remainder_theorem
Number of subsets of a given size
n 8 k ( n − k ) 2 n H ( k / n ) ≤ ( n k ) ≤ n 2 π k ( n − k ) 2 n H ( k / n ) {\displaystyle {\sqrt {\frac {n}{8k(n-k)}}}2^{nH(k/n)}\leq {\binom {n}{k}}\leq
Binomial_coefficient
Part of a machine instruction
Łukasz (2012). "7.1.4. Benchmark suite". Application of CLP to instruction modulo scheduling for VLIW processors. Gliwice, Poland: Jacek Skalmierski Computer
Opcode
Pseudorandom number generator
location // 0 <= state_index <= n-1 always // int k = k - n; // point to state n iterations before // if (k < 0) k += n; // modulo n circular indexing // the
Mersenne_Twister
Type of pseudorandom binary sequence
recursive relation: {a3[n+1]=a0[n]+a1[n]a2[n+1]=a3[n]a1[n+1]=a2[n]a0[n+1]=a1[n] where n is the time index and + represents modulo-2 addition. For bit values
Maximum-length_sequence
Attack applicable to block and stream ciphers
how the cipher operates over equivalence classes (congruence classes) modulo n. The method was first suggested in 1999 by John Kelsey, Bruce Schneier
Mod_n_cryptanalysis
Factorization algorithm
are irreducible over the rationals, and which, when interpreted modulo n {\displaystyle n} , have a common integer root m {\displaystyle m} . An optimal
General_number_field_sieve
Error-detecting code for detecting data changes
longer error bursts that it will detect is approximately (1 − 2−n). Specification of a CRC code requires definition of a so-called generator polynomial. This
Cyclic_redundancy_check
Mathematical concept and applications in software development
of the n entries in the tuple such that either σ ( i ) ≡ ( i + 1 ) {\displaystyle \sigma (i)\equiv (i+1)} modulo n, for all entries i = 1, ..., n or σ (
Circular_shift
Type of linear code
polynomial code is defined over the Binary Galois Field G F ( 2 ) = { 0 , 1 } {\displaystyle GF(2)=\{0,1\}} , polynomial elements are represented as a modulo-2
Polynomial_code
Numbers obtained by adding the two previous ones
every n. Every prime number p divides a Fibonacci number that can be determined by the value of p modulo 5. If p is congruent to 1 or 4 modulo 5, then
Fibonacci_sequence
Barcode system used by Royal Mail
being 4+2+0+0 = 6, modulo 6 of which is 0. The check symbol is computed by summing the top and bottom half values separately, modulo 6, and combining the
RM4SCC
Natural number
1. If n is greater than 2, both n − 1 and n + 1 are greater than 1 so their product is not prime. The integers modulo 3 form the finite field 𝔽3, the
3
Type of barcode
the Luhn algorithm is popular. When all 16 symbols are possible, a simple modulo-16 checksum is used. The values 10 through 19 are assigned to the symbols
Codabar
experiments on channel code partitioning using the turbo codes, Hamming codes and irregular repeat-accumulate codes. Modulo-N code is a simpler technique
DISCUS
airports by ICAO code: A - B - C - D - EB-EG - EH-EY - FA-FM - FN-FZ - G - H - I - J - KA-KH - KI-KP - KR-KZ - LA-LJ - LK-LZ - M - N - O - P - Q - R -
List of airports by ICAO code: SV-SY
List_of_airports_by_ICAO_code:_SV-SY
Type of shift register in computing
generalized to addition modulo-q (note that XOR is addition modulo 2), and the feedback bit (output bit) is multiplied (modulo-q) by a q-ary value, which
Linear-feedback shift register
Linear-feedback_shift_register
Type of matrix barcode
DotCode is a two-dimensional (2D) matrix barcode invented in 2008 by Hand Held Products as a replacement for the outdated Code 128. At this time, it is
DotCode
redundancy check is derived from the mathematics of polynomial division, modulo two. In practice, it resembles long division of the binary message string
Computation of cyclic redundancy checks
Computation_of_cyclic_redundancy_checks
Natural number
that is 4 or 5 modulo 9 cannot be represented as the sum of three cubes. There are nine Heegner numbers, or square-free positive integers n that yield an
9
Type of finite commutative rings
the rings of integers modulo a prime power. A Galois ring is constructed from the ring Z / p n Z {\displaystyle \mathbb {Z} /p^{n}\mathbb {Z} } similar
Galois_ring
Topics referred to by the same term
partitioning attack applicable to block and stream ciphers Modulo (mathematics) Modular arithmetic Modulo operation Modular exponentiation MOD., a science museum
Mod
Number divisible only by 1 and itself
functions modulo large prime numbers. Carter and Wegman generalized this method to k-independent hashing by using higher-degree polynomials, again modulo large
Prime_number
Method of exchanging cryptographic keys
element of G as a point on an elliptic curve instead of as an integer modulo n. Variants using hyperelliptic curves have also been proposed. The supersingular
Diffie–Hellman_key_exchange
Algorithm for generating prime numbers
initially be marked non prime (composite). For each entry number n in the sieve list, with modulo-sixty remainder r : If r is 1, 13, 17, 29, 37, 41, 49, or 53
Sieve_of_Atkin
Motor vehicle
S660 Concept Edition Honda S660 α Honda S660 Modulo Honda S660 Modulo Honda S660 Modulo X Honda S660 Modulo X Interior S07A Turbo engine Takahashi, Yoshio
Honda_S660
Methods of error detection and correction in communications
division in the ring of polynomials over GF(2) (the finite field of integers modulo 2). That is, the set of polynomials where each coefficient is either zero
Mathematics of cyclic redundancy checks
Mathematics_of_cyclic_redundancy_checks
In coding theory, the Lee distance is a distance between two strings x1x2…xn and y1y2…yn of equal length n over the q-ary alphabet {0, 1, …, q − 1} of
Lee_distance
Alphanumeric code that uniquely identifies a bank account in any participating country
in use does not provide the ability to handle integers of this size, the modulo operation can be performed in a piece-wise manner (as is the case with the
International Bank Account Number
International_Bank_Account_Number
Cryptographic algorithm created by Adi Shamir
Division in integers modulus p means finding the inverse of the denominator modulo p and then multiplying the numerator by this inverse (Note: inverse of A
Shamir's_secret_sharing
Numbers that contain only the digit 1
n−1 non-zero residues modulo n there exist two repunits Ri(b) and Rj(b) with 1 ≤ i < j ≤ n such that Ri(b) and Rj(b) have the same residue modulo n.
Repunit
Computer function
overflow, the result is reduced modulo 232. input: key, len, seed c1 ← 0xcc9e2d51 c2 ← 0x1b873593 r1 ← 15 r2 ← 13 m ← 5 n ← 0xe6546b64 hash ← seed for each
MurmurHash
Class of data processing algorithms
vector addition modulo the simulation space's dimension to realize toroidal topologies: I c = { j ∣ ∃ x ∈ s : j = ( ( c + x ) mod ( n 1 , … , n k ) ) } {\displaystyle
Stencil_computation
Minivan produced by Honda
(facelift) 2007–2009 Honda Stepwgn Spada (facelift) The Modulo Stepwgn Concept and Stepwgn Modulo Concept X Final Room are concept cars based on the Stepwgn
Honda_Stepwgn
Algorithm
message, dividing by 255 and keeping only the remainder. (In practice, the modulo operation is performed during the summation to control the size of the result
Fletcher's_checksum
Exponentation in modular arithmetic
exponent e to be negative by finding the multiplicative inverse d of b modulo m (for instance by using extended Euclidean algorithm). More precisely:
Modular_exponentiation
2026 release of AI-generated mathematical solutions
roots: every integer that is neither −1 nor a square is a primitive root modulo infinitely many primes. An extension of the modularity theorem to elliptic
OpenAI bulk release of mathematical papers
OpenAI_bulk_release_of_mathematical_papers
Method for computing the relation of two integers with their greatest common divisor
is the modular multiplicative inverse of a modulo b, and y is the modular multiplicative inverse of b modulo a. Similarly, the polynomial extended Euclidean
Extended_Euclidean_algorithm
General-purpose programming language
operators (+, -, *, /), the floor-division operator //, and the modulo operator %. (With the modulo operator, a remainder can be negative, e.g., 4 % -3 == -2
Python_(programming_language)
Quantum algorithm for integer factorization
group of integers modulo N, having a multiplicative inverse modulo N. Thus, a has a multiplicative order r modulo N, meaning ar≡1modN, and r is the smallest
Shor's_algorithm
Topics referred to by the same term
order type of a well-ordered set V ordn(a), the multiplicative order of a modulo n Ord Publishing, an imprint of the German group VDM Publishing devoted to
Ord
Study of analyzing information systems in order to discover their hidden aspects
with a long key using the "exclusive or" operator, which is also known as "modulo-2 addition" (symbolized by ⊕ ): Plaintext ⊕ Key = Ciphertext Deciphering
Cryptanalysis
Measure of angles
angles. Namely, the integer result of the operation is automatically reduced modulo 2n, matching the fact that angles that differ by an integer number of full
Binary_angular_measurement
Algorithm for integer factorization
the non-modular case and in the case when n is prime, because ℤ/nℤ (the integers modulo n) forms a group when n is prime. When modular numbers are used
Lenstra elliptic-curve factorization
Lenstra_elliptic-curve_factorization
Generalization of the Legendre symbol in number theory
symbol: If (a/n) = −1 then a is a quadratic nonresidue modulo n. If a is a quadratic residue modulo n and gcd(a,n) = 1, then (a/n) = +1. But, unlike
Jacobi_symbol
Sometimes, Z n {\displaystyle \mathbb {Z} _{n}} denotes the integers modulo n, where n is an integer greater than 0. The notation Z / n Z {\displaystyle
Glossary of mathematical symbols
Glossary_of_mathematical_symbols
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MODULO N-CODE
MODULO N-CODE
MODULO N-CODE
MODULO N-CODE
MODULO N-CODE
MODULO N-CODE
MODULO N-CODE
MODULO N-CODE
MODULO N-CODE
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