 Let's consider another example of arranging distinct objects in this case using an ideal block cipher so what we have is we have some plaintext as input and We want to encrypt that plaintext We take a key which tells us how to take the input plaintext and Applying the encryption algorithm proofs and produce some ciphertext So for an example, let's say we have a plaintext which is two bits in length very simple example and With the block cipher it produces ciphertext, which is also two bits We want to know with an ideal block cipher how many possible keys are there? Let's consider that with a two-bit plaintext. There are four possible plaintext inputs We could have input of zero zero zero one one zero or one one and When we encrypt we take a key and it takes a plaintext input and produces a ciphertext output and the output should be a reversible mapping in that the ciphertext if we have zero zero input and Or one of the other three inputs then the ciphertext should be unique So one possible arrangement is that if we have zero zero's input Then the output with a particular key is also zero zero that's not so very not very good encryption, but let's consider this case and another rate and If we have zero one as input plaintext it produces zero one as ciphertext if it takes one zero as plaintext One zero is ciphertext one one produces one one So that's one possible arrangement if we use a key. We in fact don't do anything not good encryption, but it's possible Another case would be that if we have zero zero's plaintext and we use a different key We'd get a different arrangement of ciphertext for example zero one here zero zero one zero one one that's using a second key or If we used a third key we could get a different arrangement of the ciphertext for example if zero one is the input plaintext using the third key the ciphertext may be one zero Using zero one as input plaintext with a third key maybe we get one one as output ciphertext and One zero becomes zero zero and one one Become zero one if we use the third key The question is how many possible keys are there in this case? Well, all we're doing really is rearranging these distinct four values So when we have four values the number of arrangements that we have we will not write them all is four factorial or 24 That is there are 24 possible keys For this ideal block cipher we take two bits input This output ciphertext depends upon which key we use if the two bits input is zero one and We use the third key then the output ciphertext is one one so we have 24 Possible keys with this ideal block cipher or four factorial. Let's try that but with Different values for example more realistic and I do a block cipher that takes 64 bits input and Produces a 64-bit ciphertext how many possible plaintext inputs are there? well Two to the power of 64 We will not of course try to list them in our previous example We had two bits input. There are two to the power of two possible plaintext now. We have 64 bits input There are two to the power of 64 possible plaintext messages, and there are two to the power of 64 ciphertext that will come out how many arrangement of arrangements of those ciphertext and Therefore how many possible keys with an ideal block cipher? It's two to the power of 64 factorial Which is a very big number it turns out it's approximately 10 to the power of 10 to the power of 88 Maybe look up on the Wolfram alpha to calculate that one so with an ideal block cipher with such a with a typical block size of 64 bits there in fact two to the power of 64 factorial possible keys and That illustrates one of the problems with ideal block ciphers in that the key size is and the key length is not manageable