Modular Division

 


Modular Division

Modular division is a mathematical operation that involves dividing two integers under a specified modulus. It’s particularly useful in number theory, cryptography, and computer science. The goal is to find the remainder of the division (modulo) of one number by another.

Definition

Given three positive integers: $a, b$, and $m$, we want to compute $a/b$ under modulo $m$. In other words, we seek to find a number $c$ such that:

$(b \cdot c) \mod m = a \mod m $

When Is Modular Division Defined?
1.Inverse Existence:
  • Modular division is defined when the inverse of the divisor exists.
  • The inverse of an integer $x$ is another integer $y$ such that $(x \cdot y) \mod m = 1$, where $m$ is the modulus.
  • In other words, $x$ and $m$ must be co-prime (i.e., their greatest common divisor is 1).

2.Finding the Inverse:

  • To compute modular division, we need to find the modular multiplicative inverse of $b$ under modulus $m$.
  • If the inverse doesn’t exist (i.e., GCD(b, m) is not 1), then modular division is not defined.

How to Find Modular Division?

Check for Inverse Existence:
First, check if the inverse of  $b$ under modulus $m$ exists.
If the inverse doesn’t exist (i.e., GCD(b, m) is not 1), then print “Division not defined.”

Compute the Result:
If the inverse exists, compute the result as: 
$\text{Result of division} = ( \text{inverse} \cdot a ) \mod m $

Example:
Let $a=8 ,b=3$ and $m=5$. Compute $(\frac{8}{3} \mod 5)$

Inverse Existence:
The GCD of 3 and 5 is 1, so the inverse exists.

Compute the Result:
The inverse of 3 under modulo 5 is 2 (since $(3 \cdot 2 \mod 5 = 1)$).
The result of division is $$(8 \cdot 2) \mod 5 = 1$$.

Therefore, the solution is $\frac{8}{3} \equiv 1 \mod 5$

Remember that modular division behaves differently from ordinary division
Note: Inverse can be computed easily by using the Extended Euclid Algorithm

Example: University Question
Solve the congruence $17x \equiv 3( mod 29)$

$x=17^{-1}.3 \pmod{29}$
$x=12.3 \pmod{29}$
$x=36 \pmod{29}$
$x=7$

Python Code 
from sympy import mod_inverse
print(mod_inverse(17,29))
12

Comments

Post a Comment

Popular posts from this blog

Number Theory CST 292 KTU IV Semester Honors Course Notes Dr Binu V P 9847390760

About me Syllabus Scheme Teaching Plan Model Question Paper University Question Papers Assignment-1 Module-1  Introduction Well ordering principle Group Ring Fields Group Ring Fields in detail Divisibility Modular Arithmetic GCD- Euclidean Algorithm Bezout's Identity Extended Euclidean Algorithm LCM-Least Common Multiple Linear Diophantine Equations Modular Division Module-II Prime Numbers and Prime-Power Factorisation Fermat and Mersenne Prime Primality Testing and Factorization Miller Rabin Primality Testing Algorithm Fermat  Factorization Congruence Modular Exponentiation Linear Congruences Simultaneous Linear Congruences- Chinese Remainder Theorem Fermat's Little Theorem and Euler's Theorem Wilson's Theorem Module-III Congruence with Prime modulus Congruence with Prime power modulus Pseudoprime and Carmichael Numbers Euler's Totient Function Cryptography The Group of Units - Primitive Roots Module-IV Quadratic Residues-Legendre and Jacobi Symbols Arithmetic Func
Read more

Sum of Squares

Image
In this our aim is to determine which integers can be expressed as the sum of given number of squares ie; which have the form $x_1^2+x_2^2+\cdots+x_k^2$, where each $x_i \in \mathbb{Z}$, for a given $k$. Definition: For each integer $k \ge 1$, let $S_k=\{n|n=x_1^2+x_2^2+\cdots+x_k^2$ for some $x_1,x_2,\ldots,x_k \in \mathbb{Z} \}$ , the set of all sum of $k$ squares. Example: $S_1=\{0,1,4,9,\ldots\}$ is the set of all squares. By inspection $S_2$, the set of sum of two squares contains 0,1,2,4,5 and 8 but not 3,6, or 7. Lemma: The set $S_2$ contains, the sum of two squares is closed under multiplication ie; if $s,t \in S_2$ , then $st \in S_2$. Proof:Let $s=a_1^2+b_1^2$ and $t=a_2^2+b_2^2$ be the elements of $S_2$, where $a_1,b_1,a_2,b_2 \in \mathbb{Z}$. Then the identity $$(a_1^2+b_1^2)(a_2^2+b_2^2)=(a_1a_2-b_1b_2)^2+(a_1b_2+b_1a_2)^2$$ This shows that $st \in S_2$ Example: We have $8=2^2+2^2$ and $10=3^2+1^2$ so $8.10=80=(2.3-2.1)^2+(2.1+2.3)^2=4^2+8^2$ Note: an equivalent ident
Read more

Quadratic Residues-Legendre and Jacobi Symbols

Image
Quadratic Congruence   There is a general question whether an integer $a$ has a square root $\pmod{n}$ and if so, how many are there and how one can find them. We know that the solution of the quadratic equation can be found by  $$x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}$$ If we want to apply this to the case where $a,b,c \in Z_n$, then $2a$ to be unit in $\pmod{n}$, so that we can divide by $2a$. We can write the equation in the form $$(2ax+b)^2=b^2-4ac$$ If we can find all the square roots $s$ of $b^2-4ac$ in $\mathbb{Z}_n$, we can then find all the solutions $x \in \mathbb{Z}_n$ of the quadratic equation of the form $2ax+b=s$, or equivalently $x=(s-b)/2a$. If we look at the square roots at $\mathbb{Z}_{15}$, then  1 and 4 have four square roots each($\pm 1 ,\pm 4$ and $\pm 2,\pm 7$ respectively, while the other units have none. Example: Find all the solutions of $x^2-3x+2 \equiv 0 \pmod{15}$ given $a=1, b=−3$ and $c=2,$ we have: $b^2−4ac=(−3)^2−4(1)(2)=9−8=1$ So, $b^2−4ac=1.$ Now, we ne
Read more