Trouble understanding modulo equivalence notation

[FrankJ777]

Homework Statement

I'm trying to understand RSA and modular math better. I've studied some of the relations and rules, but I keep getting stuck when i try to follow an example all the way through. I'm also getting confused by how I'm supposed to treat ≡ vs = at some times. Here is the background in this particular example:

The encryption function is y ≡ x^e (mod n), where x is the integer value of the plain text and y the integer value of the cipher text.
I'm not sure if this means: y = x^e (mod n), meaning y = the remainder of x^e/n or should i interpret it to mean:
y (mod n) ≡ x^e (mod n) where i should solve for y, so that the remainder of y/n = the remainder of x^e/n ?

Would I do that using the Chinese remainder theorem or Euclid algorithm? or the relationship y = x^e (mod n): ⇒ y-x/m = k ?

Homework Equations

In this example: e = 11, p = 13 , q = 29 and n = pq = 377. Φ(n) = (p-1)(q-1)= 336.
the values for the integer plain text is 110, 111, 119. Y is the cipher text.

y ≡ 110¹¹ ≡ 310 (mod 377)
y ≡ 111¹¹ ≡ 132 (mod 377)
y ≡ 119¹¹ ≡ 189 (mod 377)

What does y equal, and where did the numbers 310, 132, 189 (in red) come from??
According to the example the cipher text is 310, 132, 189.

Here is a hyperlink to the entire example. http://math.gcsu.edu/%7Eryan/12capstone/papers/maxey.pdf
3. The Attempt at a Solution
So now I'm confused. Considering the first line of the relevant equations...
Is it saying
y = 110 ¹¹ ( mod 377) = 310, and ⇒ y ≡ 310 (mod 377) ⇒ y (mod 377) = 310 (mod 377) ?

if I calculate 110 ¹¹ (mod 377) does not equal 310 (mod 377)...
110 ¹¹ ( mod 377) = 28531167061100000000000 (mod 377) = 372
and 310 (mod 377) = 310.

So where does the number 310 come from??
According to the example y is the cipher text, and the cipher text is 310, so how do we find y.
Are we supposed to find y through the relationship..
y ≡ 110¹¹ (mod 377)
y (mod 377) ≡ 110¹¹ (mod 377)
Then we solve for y using anther method and come up with y = 310.

So, to recap...
1) Does y ≡ x^e (mod n) mean find y for y (mod n) = x^e (mod n) ?? In other words is the cipertext y, the remainder of x^e (mod n) or is it a number that when divided by n has a remainder equal to the remainder of x^e divided by n.
2) Where did the red numbers (310, 132, and 189) come from in the example. Are these equal to the y's??

I would like to work through this by myself as much as possible, but I'm stuck and thoroughly confused.
Thanks for the help guys.

 

[fresh_42]

Homework Statement

I'm trying to understand RSA and modular math better. I've studied some of the relations and rules, but I keep getting stuck when i try to follow an example all the way through. I'm also getting confused by how I'm supposed to treat ≡ vs = at some times. Here is the background in this particular example:

The encryption function is y ≡ x^e (mod n), where x is the integer value of the plain text and y the integer value of the cipher text.
I'm not sure if this means: y = x^e (mod n), meaning y = the remainder of x^e/n or should i interpret it to mean:
y (mod n) ≡ x^e (mod n) where i should solve for y, so that the remainder of y/n = the remainder of x^e/n ?

Would I do that using the Chinese remainder theorem or Euclid algorithm? or the relationship y = x^e (mod n): ⇒ y-x/m = k ?

Homework Equations

In this example: e = 11, p = 13 , q = 29 and n = pq = 377. Φ(n) = (p-1)(q-1)= 336.
the values for the integer plain text is 110, 111, 119. Y is the cipher text.

y ≡ 110¹¹ ≡ 310 (mod 377)
y ≡ 111¹¹ ≡ 132 (mod 377)
y ≡ 119¹¹ ≡ 189 (mod 377)

What does y equal, and where did the numbers 310, 132, 189 (in red) come from??
According to the example the cipher text is 310, 132, 189.

Here is a hyperlink to the entire example. http://math.gcsu.edu/%7Eryan/12capstone/papers/maxey.pdf
3. The Attempt at a Solution
So now I'm confused. Considering the first line of the relevant equations...
Is it saying
y = 110 ¹¹ ( mod 377) = 310, and ⇒ y ≡ 310 (mod 377) ⇒ y (mod 377) = 310 (mod 377) ?

if I calculate 110 ¹¹ (mod 377) does not equal 310 (mod 377)...
110 ¹¹ ( mod 377) = 28531167061100000000000 (mod 377) = 372
and 310 (mod 377) = 310.

So where does the number 310 come from??
According to the example y is the cipher text, and the cipher text is 310, so how do we find y.
Are we supposed to find y through the relationship..
y ≡ 110¹¹ (mod 377)
y (mod 377) ≡ 110¹¹ (mod 377)
Then we solve for y using anther method and come up with y = 310.

So, to recap...
1) Does y ≡ x^e (mod n) mean find y for y (mod n) = x^e (mod n) ?? In other words is the cipertext y, the remainder of x^e (mod n) or is it a number that when divided by n has a remainder equal to the remainder of x^e divided by n.
2) Where did the red numbers (310, 132, and 189) come from in the example. Are these equal to the y's??

I would like to work through this by myself as much as possible, but I'm stuck and thoroughly confused.
Thanks for the help guys.

$a ≡ b \, (mod \, n)$ means, that by division with $n$ both $a$ and $b$ have the same remainder, i.e. $a - b$ is divisible by $n.$
Thus we can write it as an equation $a = k \cdot n + b$ for some integer (!) $k$.
$a ≡ b \, (mod \, n)$ is sometimes also written $a ≡ b \, (\, n)$

1) Does y ≡ x^e (mod n) mean find y for y (mod n) = x^e (mod n) ??

I don't know why you want to find something. It simply means division by $n$ gets the same remainder for $y$ and $x^e$.

In other words is the cipertext y, the remainder of x^e (mod n)

Yes. As long as you accept all remainders $k \cdot n + y$ for we can't know whether $y$ is already the smallest one possible. Usually it is. But $23 ≡ 1 (mod \, 2)$ is also true because the division of $23$ by $2$ gets $1$ as remainder.

or is it a number that when divided by n has a remainder equal to the remainder of x^e divided by n.

Yes.

2) Where did the red numbers (310, 132, and 189) come from in the example. Are these equal to the y's??

Calculate $110^{11}$, divide it by $377$, subtract the integer part and multiply the rest by $377$ to get the remainder, which is $310$. The other two can be found in the same way.