Modular arithmetic, sometimes referred to as modulus arithmetic or clock arithmetic, in its most elementary form,
arithmetic done with a count that resets itself to zero every time a certain whole number N greater than one
, known as the modulus (mod), has been reached.
Why is modular arithmetic?
Modular arithmetic is used extensively in pure mathematics, where it is
a cornerstone of number theory
. But it also has many practical applications. It is used to calculate checksums for international standard book numbers (ISBNs) and bank identifiers (Iban numbers) and to spot errors in them.
What is the meaning of modular arithmetic?
:
arithmetic that deals with whole numbers where the numbers are replaced by their remainders after division by a fixed number in a
modular arithmetic with modulus 5, 3 multiplied by 4 is 2.
How do you do modular arithmetic?
- Divide a by n.
- Subtract the whole part of the resulting quantity.
- Multiply by n to obtain the modulus.
What is the difference between modular arithmetic and regular arithmetic?
Modular arithmetic is almost the same as the usual arithmetic of whole numbers. The main difference is that
operations involve remainders after division by a specified number
(the modulus) rather than the integers themselves.
How do you reduce modular arithmetic?
In modular arithmetic, when we say “reduced modulo ,” we mean whatever result we obtain, we divide it by n, and report only the smallest possible nonnegative residue. The next theorem is fundamental to modular arithmetic.
Let n≥2 be a fixed integer
. If a≡b (mod n) and c≡d (mod n), then a+c≡b+d(modn),ac≡bd(modn).
What is the use of modular arithmetic in DAA?
Many complex cryptographic algorithms are actually based on fairly simple modular arithmetic. In modular arithmetic, the numbers we are dealing with are just integers and the operations used are
addition, subtraction, multiplication and division
.
What are the properties of modular arithmetic?
Properties of multiplication in modular arithmetic:
If
a ⋅ b = c a cdot b
= c a⋅b=c, then a ( m o d N ) ⋅ b ( m o d N ) ≡ c ( m o d N ) apmod Ncdot bpmod N equiv c pmod{N} a(modN)⋅b(modN)≡c(modN).
Which is use for floor division?
The real floor division operator is
“//”
. It returns floor value for both integer and floating point arguments.
How do you calculate modular?
- Start by choosing the initial number (before performing the modulo operation). …
- Choose the divisor. …
- Divide one number by the other, rounding down: 250 / 24 = 10 . …
- Multiply the divisor by the quotient. …
- Subtract this number from your initial number (dividend).
Can you divide in modular arithmetic?
Can we always do modular division? The answer
is “NO”
. … In modular arithmetic, not only 4/0 is not allowed, but 4/12 under modulo 6 is also not allowed. The reason is, 12 is congruent to 0 when modulus is 6.
What does a ≡ b mod n mean?
Definition 3.1 If a and b are integers and n > 0, we write a ≡ b mod n to mean n|(b − a). We read this as “
a is congruent to b modulo
(or mod) n. For example, 29 ≡ 8 mod 7, and 60 ≡ 0 mod 15. The notation is used because the properties of congruence “≡” are very similar to the properties of equality “=”.
How is modular arithmetic used in cryptology?
Modular Arithmetic (Clock Arithmetic)
Modular arithmetic is
a system of arithmetic for integers, where values reset to zero and begin to increase again, after reaching a certain predefined value
, called the modulus (modulo). Modular arithmetic is widely used in computer science and cryptography.
What does modulo 4 mean?
1.
An integer that leaves the same remainder when it
is the divisor of two other integers. For example, 6 modulo 4 = 2 and 14 modulo 4 = 2. In other words, 6 divided by four results in a remainder of 2, and 14 divided by 4 leaves a remainder of 2. 1.
When can you divide modular arithmetic?
We could introduce some arbitrary convention, such as choosing the smallest answer when considering the least residue as an integer, but then division will behave strangely. Instead, we require uniqueness, that is divided by modulo is only defined
when there is a unique z ∈ Z n such that x = y z .
