RSA Encryption
One commonly used cipher of this forms called RSA Encryption,
where RSA are the initials of the three creators: Rivest, Shamir, and Adleman. It
is based on the following idea:
It is very simply to multiply numbers together, especially with computers. But it can
be very difficult to factor numbers. For example, if I ask you to multiply together
34537 and 99991, it is a simple matter to punch those numbers into a calculator and
3453389167. But the reverse problem is much harder.
Suppose I give you the number 1459160519. I'll even tell you that I got it by multi-
plying together two integers. Can you tell me what they are? This is a very difficult
problem. A computer can factor that number fairly quickly, but (although there are
some tricks) it basically does it by trying most of the possible combinations. For any
size number, the computer has to check something that is of the order of the size of
the square-root of the number to be factored. In this case, that square-root is roughly
38000.
Now it doesn't take a computer long to try out 38000 possibilities, but what if the
number to be factored is not ten digits, but rather 400 digits? The square-root of a
number with 400 digits is a number with 200 digits. The lifetime of the universe is
approximately 10^18 seconds an 18 digit number. Assuming a computer could test
one million factorizations per second, in the lifetime of the universe it could check
10^24 possibilities. But for a 400 digit product, there are 10^200 possibilties. This means
the computer would have to run for 10^176 times the life of the universe to factor the
large number.
It is, however, not too hard to check to see if a number is prime in other words to
check to see that it cannot be factored. If it is not prime, it is difficult to factor, but if it
is prime, it is not hard to show it is prime.
So RSA encryption works like this. I will find two huge prime numbers, p and q that
have 100 or maybe 200 digits each. I will keep those two numbers secret (they are
my private key), and I will multiply them together to make a number N = pq. That
number N is basically my public key. It is relatively easy for me to get N; I just need
to multiply my two numbers. But if you know N, it is basically impossible for you
to find p and q. To get them, you need to factor N, which seems to be an incredibly
difficult problem.
But exactly how is N used to encode a message, and how are p and q used to decode
it? Below is presented a complete example, but I will use tiny prime numbers so it is
easy to follow the arithmetic. In a real RSA encryption system, keep in mind that the
prime numbers are huge.
where RSA are the initials of the three creators: Rivest, Shamir, and Adleman. It
is based on the following idea:
It is very simply to multiply numbers together, especially with computers. But it can
be very difficult to factor numbers. For example, if I ask you to multiply together
34537 and 99991, it is a simple matter to punch those numbers into a calculator and
3453389167. But the reverse problem is much harder.
Suppose I give you the number 1459160519. I'll even tell you that I got it by multi-
plying together two integers. Can you tell me what they are? This is a very difficult
problem. A computer can factor that number fairly quickly, but (although there are
some tricks) it basically does it by trying most of the possible combinations. For any
size number, the computer has to check something that is of the order of the size of
the square-root of the number to be factored. In this case, that square-root is roughly
38000.
Now it doesn't take a computer long to try out 38000 possibilities, but what if the
number to be factored is not ten digits, but rather 400 digits? The square-root of a
number with 400 digits is a number with 200 digits. The lifetime of the universe is
approximately 10^18 seconds an 18 digit number. Assuming a computer could test
one million factorizations per second, in the lifetime of the universe it could check
10^24 possibilities. But for a 400 digit product, there are 10^200 possibilties. This means
the computer would have to run for 10^176 times the life of the universe to factor the
large number.
It is, however, not too hard to check to see if a number is prime in other words to
check to see that it cannot be factored. If it is not prime, it is difficult to factor, but if it
is prime, it is not hard to show it is prime.
So RSA encryption works like this. I will find two huge prime numbers, p and q that
have 100 or maybe 200 digits each. I will keep those two numbers secret (they are
my private key), and I will multiply them together to make a number N = pq. That
number N is basically my public key. It is relatively easy for me to get N; I just need
to multiply my two numbers. But if you know N, it is basically impossible for you
to find p and q. To get them, you need to factor N, which seems to be an incredibly
difficult problem.
But exactly how is N used to encode a message, and how are p and q used to decode
it? Below is presented a complete example, but I will use tiny prime numbers so it is
easy to follow the arithmetic. In a real RSA encryption system, keep in mind that the
prime numbers are huge.
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