Transience of Simple Random Walks With Linear Entropy Growth

Using the technique of evolving sets, we explore the connection between entropy growth and transience for simple random walks on connected infinite graphs with bounded degree. In particular we show that for a simple random walk starting at a vertex $x_0$, if the entropy after $n$ steps, $E_n$ is at least $Cn$ where the $C$ is independent of $x_0$, then the random walk is transient. We also give an example which demonstrates that the condition of $C$ being independent of $x_0$ is necessary.Comment: 8 pages, 1 figur

Problems in Discrete Probability Theory and Cryptography

This thesis discusses two distinct problems, the first of which is concerning format preservingencryption in cryptography and the second is concerning transience of simple random walks on infinite graphs. In Chapter 1, we discuss format preserving encryption (FPE) under the presence of an adversary who can leak parts of the secret key. The main aim of FPE is to encrypt a plaintext into a ciphertext of the same format. For example, under FPE, an encrypted credit card number will still look like a credit card number. One way to achieve this is by generating a uniformly random permutation on the space of all plaintexts and then applying this permutation on the plaintexts to get ciphertexts. Storing such a permutation is infeasible. For example in the case of credit cards, it would take 61, 391 terabytes to store such a permutation. So instead, in cryptography what we usually look for is a random permutation which is an easy to compute function of a typically short random string called the key. Such a random permutation is never going to be equal to a uniformly random permutation in distribution as long as the key size is small. All one needs is that it is practically very hard to distinguish between the two. Such a construction only works as long as the key is secret, because knowing the key is same as knowing the permutation. In [7], Bellare, Kane and Rogaway provide an encryption scheme that is secure even in the presence of an adversary who has partial knowledge of the key. They thwart such an adversary by making the key large and putting an upper limit on the amount of information about the key that the adversary can steal. The rationale behind this is that for a threat residing in one’s network, it is hard for it to transfer huge amounts of data to an external location without being detected. Their encryption scheme isn’t format preserving. In this chapter, we discuss format preserving encryption in the same setting as [7]. In particular we provide a format preserving encryption scheme and prove that it is secure under an appropriately modified notion of security. In chapter 2 we explore the connection between the entropy growth of a simple random walk on a connected infinite graph with bounded degree and its transience. In particular, using the technique of evolving sets we show that for a simple random walk starting at any vertex of an infinite connected graph with bounded degree, if the entropy of its nth step grows at least linearly in n, with the constant of linearity being independent of the starting position, then the random walk is transient. For irreducible transitive Markov chains, it is already known that linear entropy growth implies transience. So, we end up enlarging the class of chains for which this result holds. We also give an example to show that this uniformity of the constant of linearity is an essential condition. That is, if there is no constant of linearity which works for every starting position, then the random walk is not necessarily transient