- Questions & Answers
- Accounting
- Computer Science
- Automata or Computationing
- Computer Architecture
- Computer Graphics and Multimedia Applications
- Computer Network Security
- Data Structures
- Database Management System
- Design and Analysis of Algorithms
- Information Technology
- Linux Environment
- Networking
- Operating System
- Software Engineering
- Big Data
- Android
- iOS
- Matlab

- Economics
- Engineering
- Finance
- Thesis
- Management
- Science/Math
- Statistics
- Writing
- Dissertations
- Essays
- Programming
- Healthcare
- Law

- Log in | Sign up

Assignment #2

Due date: Wednesday, April 28, 2021, 11:59pm

Course: Introduction to Cryptography

Quarter: Spring 2021

Instructor: Prabhanjan Ananth

Instructions

• This assignment is worth 100 points.

• Please read all the instructions very carefully.

• Submissions must be made on Gradescope. The enrollment code is on Slack.

• Make sure that the handwriting in your solutions is legible (using LATEX to typeset

your solutions is appreciated a lot!). Be precise and concise in your solutions. Explain

your solutions clearly and provide mathematical proofs wherever necessary.

• Write the solution to each problem on a separate page.

• If you don’t know how to solve a problem, write “I don’t know” and you get 20% of

the points.

• You are allowed to collaborate with at most two more students on these assignments.

However, everyone is expected is write up the solutions on their own and submit it

individually. Clearly state the names of all your collaborators.

• Do not use any theorem or lemma from the internet that was not taught in the class.

Use only the material covered in the class to solve these problems.

• More importantly, have fun solving these problems!

1

Free Hand

Problem 1 [10 points]

Are the following functions negligible or non-negligible? Justify your answer.

1. 51(=) = 5 (=) · =2 , for some constant 2 > 0 and some negligible function 5 (=)

2. 52(=) = =− log log =

Problem 2 [15 points]

Recall that two ensembles {-=}, {.=} where -= , .= ∈ {0, 1}?>;H(=) are computationally indistin-

guishable if for any efficient adversaryA, there exists a negligible function &(=) such that

|Pr[A(1= , G) = 1; G ← -=] − Pr[A(1= , H) = 1; H ← .=]| ≤ &(=)

Let 5 : {0, 1}∗ → {0, 1}∗ be a polynomial-time computable function. Show that if {-=}, {.=} are

computationally indistinguishable, then { 5 (-=)}, { 5 (.=)} are also computationally indistinguish-

able.

[Hint: Use reductions.]

Problem 3 [15 points]

If {-=}, {.=} are computationally indistinguishable, and {.=}, {/=} are computationally indis-

tinguishable, show that {-=}, {/=} are computationally indistinguishable.

Problem 4 [15 points]

Give examples of non-negligible functions 51(=) and 52(=) such that 51(=) · 52(=) is actually a

negligible function.

Problem 5 [25 points]

Let � : {0, 1}= → {0, 1}3= be a secure pseudorandom generator, prove that the following pseu-

dorandom generators are secure.

B, A ∈ {0, 1}= below are perfectly random strings. You can think of them as the seeds of the

PRG.

1. �1(B) : {0, 1}= → {0, 1}2= , �1(B) = �(B)[2=]. That is, �1(B) is obtained by dropping the last

= bits of �(B).

2. �2(A, B) : {0, 1}2= → {0, 1}6= , �2(A, B) = (�(A), �(B)).

Problem 6 [20 points]

1. Is the following function 5 : {0, 1}= → {0, 1}= a one-way function? ∀G ∈ {0, 1}= ,

5 (G) = G + 10 (mod 2=)1.

1Here G ∈ {0, 1}= is interpreted as a number between 0 and 2= − 1. For instance, 10 ≡ XXXXXXXXXX

2

2. Suppose 5 : {0, 1}= → {0, 1}= is a one-way function. Using 5 , can you construct another

function 5 ′ : {0, 1}= → {0, 1}=+1 such that 5 ′ is also a one-way function but not a secure

pseudorandom generator.

3

Due date: Wednesday, April 28, 2021, 11:59pm

Course: Introduction to Cryptography

Quarter: Spring 2021

Instructor: Prabhanjan Ananth

Instructions

• This assignment is worth 100 points.

• Please read all the instructions very carefully.

• Submissions must be made on Gradescope. The enrollment code is on Slack.

• Make sure that the handwriting in your solutions is legible (using LATEX to typeset

your solutions is appreciated a lot!). Be precise and concise in your solutions. Explain

your solutions clearly and provide mathematical proofs wherever necessary.

• Write the solution to each problem on a separate page.

• If you don’t know how to solve a problem, write “I don’t know” and you get 20% of

the points.

• You are allowed to collaborate with at most two more students on these assignments.

However, everyone is expected is write up the solutions on their own and submit it

individually. Clearly state the names of all your collaborators.

• Do not use any theorem or lemma from the internet that was not taught in the class.

Use only the material covered in the class to solve these problems.

• More importantly, have fun solving these problems!

1

Free Hand

Problem 1 [10 points]

Are the following functions negligible or non-negligible? Justify your answer.

1. 51(=) = 5 (=) · =2 , for some constant 2 > 0 and some negligible function 5 (=)

2. 52(=) = =− log log =

Problem 2 [15 points]

Recall that two ensembles {-=}, {.=} where -= , .= ∈ {0, 1}?>;H(=) are computationally indistin-

guishable if for any efficient adversaryA, there exists a negligible function &(=) such that

|Pr[A(1= , G) = 1; G ← -=] − Pr[A(1= , H) = 1; H ← .=]| ≤ &(=)

Let 5 : {0, 1}∗ → {0, 1}∗ be a polynomial-time computable function. Show that if {-=}, {.=} are

computationally indistinguishable, then { 5 (-=)}, { 5 (.=)} are also computationally indistinguish-

able.

[Hint: Use reductions.]

Problem 3 [15 points]

If {-=}, {.=} are computationally indistinguishable, and {.=}, {/=} are computationally indis-

tinguishable, show that {-=}, {/=} are computationally indistinguishable.

Problem 4 [15 points]

Give examples of non-negligible functions 51(=) and 52(=) such that 51(=) · 52(=) is actually a

negligible function.

Problem 5 [25 points]

Let � : {0, 1}= → {0, 1}3= be a secure pseudorandom generator, prove that the following pseu-

dorandom generators are secure.

B, A ∈ {0, 1}= below are perfectly random strings. You can think of them as the seeds of the

PRG.

1. �1(B) : {0, 1}= → {0, 1}2= , �1(B) = �(B)[2=]. That is, �1(B) is obtained by dropping the last

= bits of �(B).

2. �2(A, B) : {0, 1}2= → {0, 1}6= , �2(A, B) = (�(A), �(B)).

Problem 6 [20 points]

1. Is the following function 5 : {0, 1}= → {0, 1}= a one-way function? ∀G ∈ {0, 1}= ,

5 (G) = G + 10 (mod 2=)1.

1Here G ∈ {0, 1}= is interpreted as a number between 0 and 2= − 1. For instance, 10 ≡ XXXXXXXXXX

2

2. Suppose 5 : {0, 1}= → {0, 1}= is a one-way function. Using 5 , can you construct another

function 5 ′ : {0, 1}= → {0, 1}=+1 such that 5 ′ is also a one-way function but not a secure

pseudorandom generator.

3

May 02, 2021

- 1 CYB 220 Project Three Guidelines and Rubric Overview Sound guidance in selecting technologies that ensure compliance with organizational security requirements is a key contribution that a...Oct 21, 2021
- CSE 531_Project_gRPC Overview and Rubric Document CSE 531: Distributed and Multiprocessor Operating Systems gRPC Project Purpose The goal of this project is to build a distributed banking system that...Oct 20, 2021
- Assessment task description: You are required to complete six activities. • Activity 1: Selecting and preparing REST APIs • Activity 2: Creating a new project and configure RESTful API • Activity 3:...Oct 19, 2021
- NET 413 Lab #3 – Per-VLAN STP and DHCP ASSIGNMENT DOCUMENT Due Sunday, October 26th 2021, by 11:59 pm Introduction Students can choose to do this lab on an equipment pod in CDM 348 or can do this lab...Oct 18, 2021
- Project 2: Incident Response Start Here Incident Response You've recently been promoted to the role of a cybersecurity incident manager as part of a new contract with a major media and entertainment...Oct 14, 2021

- You are a delegate to the New York Ratification Convention. Write a statement to your constituents explaining your reasons for supporting ratification of the Constitution. Include specific details...Oct 21, 2021
- CPSC 231: Introduction to Computer Science for Computer Science Majors I Assignment 2: Graphing Calculator Weight: 7% Collaboration Discussing the assignment requirements with others is a reasonable...Oct 21, 2021
- CS 202 – Assignment #6 Purpose: Learn operator overloading both as member and non-member functions, friend functions, this pointer, copy constructor. Points: 100 Why we need Operator Overloading in...Oct 21, 2021
- public class LinkedList { private class Node { public Object value; public Node next; public char[] data; } public Node head = null; public Node tail = null; public String About() { return "About me";...Oct 21, 2021
- Self-Service Course 4 Project - Kiva Robot Remote Control (Amazon_Technical_Academy.Self_Service_Courses.Course4.JavaProgrammingPostDuke.project.KivaRobotRemoteControl.WebHome) - XWiki Self-Service...Oct 21, 2021

Copy and Paste Your Assignment Here

Copyright © 2021. All rights reserved.