As it is mentioned above, I am confused about “Gen takes as input $1^n$ and outputs a key“. Does it mean that the input of Gen is 1111....111 with length of n?
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In my lecture, the lecturer said:      Let $K$ be the key generation algorithm. Given a security parameter represented in unary, $1^k$, $K(1^k)$ will output a keypair $(pk; sk)$, known as the public key and the private (or secret) key, respectively.    My question is: When I want to implement the cryptosystem, how should I represent this $1^k$?  When I implement the key generation procedure, do I literally require callers to pass me a string of $k$ one-bits that the procedure will never use?  And, if so, why is it there in the first place?
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Sorry about asking such an elementary question, but I have been wondering about this exact definition for a while. What power of $n$ is $\log(n)$. I know that it is $n^\epsilon$ for a very small $\epsilon$, but what value is $\epsilon$ exactly?
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I believe I see that $a_n = 2^n(a_0+1) - 1$ but am somewhat uncertain where to proceed afterwards. I am a complete beginner at number theory and would appreciate it if someone could point me in the right direction--surely there is some obvious argument I am missing.
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If I perform RSA on my input 5 times, I get the original input back out. Why is it that often, iterating RSA encryption just a few times cycles yields back the original value, when the public modulus N is small?
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My question is, for the question below, in the inductive step, where does $\dfrac{1}{2^{(k+1)}}$ come from?And where does $2^k$ come from in the third last step?
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"" (number $5$).  I'm looking at this, and i'm not sure what the $\equiv$ means in this instance? I know the rest of the values, but what does $\equiv$ mean right now?  Thanks in advance!
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I have been a witness to many a discussion about numbers to the power of zero, but I have never really been sold on any claims or explanations.  This is a three part question, the parts are as follows...    1) Why does $n^{0}=1$ when $n\neq 0$?  How does that get defined?  2) What is $0^{0}$?  Is it undefined?  If so, why does it not equal 1?  3) What is the equation that defines exponents?  I can easily write a small program to do it (see below), but what about in equation format?    I just want a little discussion about numbers to the power of zero, for some clarification.    Code for Exponents: (pseudo-code/Ruby)  def int find_exp (int x, int n){     int total = 1;     n.times{total*=x}     return total; }
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Why does:   $$ \sum_{k=0}^{n} k \binom nk p^k (1-p)^{n-k} = np $$ ? Taking the derivative of:   $$ \sum_{k=0}^{n}  \binom nk p^k (1-p)^{n-k} = (1 + [1-P])^n = 1 $$   does not seem useful, since you would get zero. And induction hasn't yet worked for me, since – during the inductive step – I am unable to prove that:   $$ \sum_{k=0}^{n+1} k \binom {n+1}{k} p^k (1-p)^{n+1-k} = (n+1)p  $$   assuming that:   $$ \sum_{k=0}^{n} k \binom {n}{k} p^k (1-p)^{n-k} = np  $$
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Prove by induction that for all $n \ge 0$:  $${n \choose 0} + {n \choose 1} + ... + {n \choose n} = 2^n.$$  In the inductive step, use Pascal’s identity, which is:  $${n+1 \choose k} = {n \choose k-1} + {n \choose k}.$$  I can only prove it using the binomial theorem, not induction.
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What's the best way to write subsequences?  $x_n_i$ gives an error, while $x_{n_{i}}$ works, but it just looks like $x_{ni}$ (the i isn't lowered enough below the n)...
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For example say, $n = 2$. So our set is $\{1, 2, 3, 4\}$ in base $10$ and  $\{1, 10, 11, 100\}$ in base $2$. So Output $1$, because only one number i.e. $3$ is there such that it has $``11"$ in it.  For $n = 3$, we get $\{3, 6, 7\}$ as having $``11"$, so output $3$.
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Prove that $S_n$ is generated by elements of the form $(1k)$.   I know that all elements in $S_n$ can be expressed as a multiplication of transpositions. I tried to take the direction of proving that every transposition can be written in the form of a product of elements of the form $(1k)$.  I started with a simple example of $(23)\in S_3$, but no matter what I tried I could not write it in the form of a multiplication of elements of the form $(1k)$. Am I in the right direction?
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In a , the setup function is mentioned with the $setup(1^n)$ or $setup(1^l)$. I want to know what is meant here.   Is it multiples of ones or representing something else? Also, why is this used here?
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What determines the number of possible RSA key pairs for a given $n$? Is it because $\varphi(n)$ produces the number of values that are less than $n$ and coprime with $n$? Is it as simple as $\varphi(n)$?
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The function p(n) counts the number of ways a number can be made up of smaller numbers. For example, the p(5) = 7 because you can make 5 in the following ways:  1-1-1-1-1  1-1-1-2  3-2  3-1-1  2-2-1  1-4  5  According to Wikipedia, the generating function for this, counting the possible partitioning of a number, is as follows:    After much effort, I have been unable to make any sense of this, or contort it in such a way that it actually answer the question: what is p(n) when n = 1000?. There is a great deal about this expression that I do not understand. Please answer any of the following which, given your expertise, you can.   n is the argument of this function, the number for which we are counting the partitions. What is x here? Where does it come from? Same for k? What is the significance of k? Why does it go from 1 to infinity?  What is the significance of the small dot on the right side? We see the sign on the left "Sigma" so often, but the sign on the right (Big Pi) appears, at least for me, in very few places. Why is this? On the right we see that we get the product of the expression for every value of k from = 1 to infinity. How does this make sense? If we go from 1 to infinity, isn't the resulting value necessarily infinite? It is known that actually getting a series of numbers from a generating function can be non-trivial. Is it even possible to solve for p(n) using this equation? Or does it simply explain the pattern with no means of getting the numbers? How can this be implemented in a computer language, one that is general, and not math-based, for example in Python, or in JavaScript? Is it possible? Can you provide an implementation of this generating function in computer code? (See PS) What is p(1000)? How do you know? Generally, in solving generating functions, using a "Taylor series" is proposed as a possible solution. How would you explain this to someone who does not know what a Taylor series is? Is this problem even solvable without a deep knowledge of that subject? Is it correct to call this an "equation"? Or is a generating function fundamentally a different thing?   PS I am asking this question because it is my hobby. I am NOT in a university and asking you to do my work, NOR am I employed, either as a mathematician or otherwise. This is fun for me, and I hope answering it is fun for you!
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I have implemented a key pair generation scheme for RSA algorithm. I have taken the length of private key exponent as RSA key size, but then I've got to know that RSA key size is the size of the modulus $n$. Is it wrong to take RSA key size as the size of the private key exponent?  To generate a private key of 1024 bits, I increase the size of private key exponent regardless the modulus $n$.
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Disclaimer: This is exercise 2.1 in Katz-Lindell book.  Given a (symmetric) encryption scheme $\Pi=(Gen,Enc,Dec)$ where $Gen$ takes the security parameter $1^n$ as input and generates a key $k$ of length $n$.  Usually we simply assume that $Gen$ chooses key uniformly from $\{0,1\}^n$ but this is not always the case.  Given $Gen$, how would you construct a new generation algorithm $Gen'$ that indeed outputs $k$ uniformly at random?