How Chinese remainder theorem is used in cryptography?
Table of Contents
- 1 How Chinese remainder theorem is used in cryptography?
- 2 Where can we use Remainder theorem in real life?
- 3 Is Chinese remainder theorem if and only if?
- 4 Who invented Chinese remainder theorem?
- 5 Why does remainder theorem work?
- 6 Why is the remainder theorem used to determine whether a linear binomial is a factor of a polynomial?
How Chinese remainder theorem is used in cryptography?
One of the most useful results of number theory is the Chinese remainder theorem (CRT). In essence, the CRT says it is possible to reconstruct integers in a certain range from their residues modulo a set of pairwise relatively prime moduli.
Where can we use Remainder theorem in real life?
Real-life Applications
- The remainder theorem provides a more efficient avenue for testing whether certain numbers are roots of polynomials.
- This theorem can increase efficiency when applying other polynomial tests, like the rational roots test.
What can we use the remainder theorem to test for?
The Polynomial Remainder Theorem allows us to determine whether a linear expression is a factor of a polynomial expression easily.
Why is it called the Chinese remainder theorem?
There was a much greater emphasis on algorithms used to solve more or less practical problems. The Chinese remainder theorem in its original form was an algorithm devised to solve a problem often known as “an unknown quantity of things” (物不知數), so that was the name used to refer to the original method.
Is Chinese remainder theorem if and only if?
The Chinese remainder theorem (CRT) asserts that there is a unique class a + NZ so that x solves the system (2) if and only if x ∈ a + NZ, i.e. x ≡ a(mod N). Thus the system (2) is equivalent to a single congruence modulo N.
Who invented Chinese remainder theorem?
mathematician Sun Zi
Chinese remainder theorem, ancient theorem that gives the conditions necessary for multiple equations to have a simultaneous integer solution. The theorem has its origin in the work of the 3rd-century-ad Chinese mathematician Sun Zi, although the complete theorem was first given in 1247 by Qin Jiushao.
What is practical example of using remainder?
For example; when 41 is divided by 7, the quotient is 5 and the remainder is 6. Here the remainder is greater than the quotient….Examples:
| 18 ÷ 7 | Remainder 4 |
|---|---|
| 46 ÷ 9 | Remainder 1 |
| 15 ÷ 5 | Remainder 0 |
How is factoring used in the real world?
Factoring is a useful skill in real life. Common applications include: dividing something into equal pieces, exchanging money, comparing prices, understanding time and making calculations during travel.
Why does remainder theorem work?
The remainder theorem states that when a polynomial, f(x), is divided by a linear polynomial , x – a, the remainder of that division will be equivalent to f(a). It should be noted that the remainder theorem only works when a function is divided by a linear polynomial, which is of the form x + number or x – number.
Why is the remainder theorem used to determine whether a linear binomial is a factor of a polynomial?
The Remainder Theorem is useful for evaluating polynomials at a given value of x, though it might not seem so, at least at first blush. In polynomial terms, since we’re dividing by a linear factor (that is, a factor in which the degree on x is just an understood “1”), then the remainder must be a constant value.
How do you find the inverse of the Chinese remainder theorem?
Modular multiplicative inverses are used to obtain a solution of a system of linear congruences that is guaranteed by the Chinese Remainder Theorem. t3 = 6 is the modular multiplicative inverse of 5 × 7 (mod 11). Thus, X = 3 × (7 × 11) × 4 + 6 × (5 × 11) × 4 + 6 × (5 × 7) × 6 = 3504.
What do you mean by remainder theorem?
The Remainder Theorem Definition states that when a polynomial is p ( a ) is divided by another binomial ( a – x ), then the remainder of the end result that is obtained is p ( x ).