Secret sharing using the Chinese remainder theorem
Alright kiddo, imagine you have a big treasure chest and you want to keep it safe. But you don't want to carry the key to the chest with you all the time or give it to one person, because what if that person loses it or runs away with all your treasure? That would be terrible, right?
So, you come up with a clever plan to share the key to the chest among your friends. But you don't want just one friend to have the whole key either, because that would be risky. Instead, you want to split the key into parts and give each part to a different friend. But you don't want any friend to be able to open the chest on their own - you need all the parts put together to unlock it.
This is where the Chinese remainder theorem comes in. It's a special way of splitting up a number into smaller pieces that are easier to keep secret. The idea is to divide the key into different parts, and give each part to a different friend. But the way you divide the key makes it so that none of your friends can figure out what the full key is just by looking at their part.
Here's how it works:
Let's say your key is the number 56. You want to split it into two parts, and give each part to a different friend.
First, you pick two other numbers - let's call them 3 and 5. These numbers are called "moduli". You divide your key by each of these moduli (which means you figure out the remainder when you divide 56 by 3, and the remainder when you divide 56 by 5).
The remainder when you divide 56 by 3 is 2. So, you give one part of the key to your first friend - let's call this part "A" - and it's the number 2.
The remainder when you divide 56 by 5 is 1. So, you give the other part of the key to your second friend - let's call this part "B" - and it's the number 1.
Now, your friends can't figure out the full key just by looking at their parts. But when you want to open the chest, you can ask your friends to come together and combine their parts using the Chinese remainder theorem.
The Chinese remainder theorem allows you to take different numbers that have been split into different parts using different moduli, and combine them back into the original number. In our example, your friends would use the moduli 3 and 5 to combine their parts A and B.
They would follow a special formula to find the full key:
Full key = A * 5 * (5^-1 mod 3) + B * 3 * (3^-1 mod 5)
This might look a bit confusing, but it just means that your friends would use their parts A and B, along with the moduli 3 and 5, to calculate the full key.
When they do the math, they get:
Full key = 2 * 5 * (5^-1 mod 3) + 1 * 3 * (3^-1 mod 5)
Full key = 56
Ta-da! Now your friends can unlock the treasure chest together, but none of them knows the full key on their own. And if one of your friends loses their part or runs away with it, the other friend still can't unlock the chest without both parts and the moduli.
That's secret sharing using the Chinese remainder theorem - a clever way to keep your treasure safe by dividing it up into pieces that don't give away the full secret.
So, you come up with a clever plan to share the key to the chest among your friends. But you don't want just one friend to have the whole key either, because that would be risky. Instead, you want to split the key into parts and give each part to a different friend. But you don't want any friend to be able to open the chest on their own - you need all the parts put together to unlock it.
This is where the Chinese remainder theorem comes in. It's a special way of splitting up a number into smaller pieces that are easier to keep secret. The idea is to divide the key into different parts, and give each part to a different friend. But the way you divide the key makes it so that none of your friends can figure out what the full key is just by looking at their part.
Here's how it works:
Let's say your key is the number 56. You want to split it into two parts, and give each part to a different friend.
First, you pick two other numbers - let's call them 3 and 5. These numbers are called "moduli". You divide your key by each of these moduli (which means you figure out the remainder when you divide 56 by 3, and the remainder when you divide 56 by 5).
The remainder when you divide 56 by 3 is 2. So, you give one part of the key to your first friend - let's call this part "A" - and it's the number 2.
The remainder when you divide 56 by 5 is 1. So, you give the other part of the key to your second friend - let's call this part "B" - and it's the number 1.
Now, your friends can't figure out the full key just by looking at their parts. But when you want to open the chest, you can ask your friends to come together and combine their parts using the Chinese remainder theorem.
The Chinese remainder theorem allows you to take different numbers that have been split into different parts using different moduli, and combine them back into the original number. In our example, your friends would use the moduli 3 and 5 to combine their parts A and B.
They would follow a special formula to find the full key:
Full key = A * 5 * (5^-1 mod 3) + B * 3 * (3^-1 mod 5)
This might look a bit confusing, but it just means that your friends would use their parts A and B, along with the moduli 3 and 5, to calculate the full key.
When they do the math, they get:
Full key = 2 * 5 * (5^-1 mod 3) + 1 * 3 * (3^-1 mod 5)
Full key = 56
Ta-da! Now your friends can unlock the treasure chest together, but none of them knows the full key on their own. And if one of your friends loses their part or runs away with it, the other friend still can't unlock the chest without both parts and the moduli.
That's secret sharing using the Chinese remainder theorem - a clever way to keep your treasure safe by dividing it up into pieces that don't give away the full secret.