Public Key Cryptography

Chapter 37: Public Key Cryptography Presented By Sean Szumlanski January 29, 2003

Presentation Agenda 1. Cameo Appearance by Symmetric Encryption 2. Purpose and Definition of Asymmetric Encryption 3. An Interlude by Math Theory 4. Example of Public Key Encryption 5. Problems Associated with Public Key Encryption 6. A Few Historical Asides

Symmetric Encryption

Symmetric Encryption Problems • Susceptibility to Linguistic Patterns: · Most common letter: E · Second most common letter: T · Frequency of single, double, and triple letter words • The Catch-22 of Symmetric Encryption: Establishing a secure key over insecure channels.

Asymmetric Encryption

Asymmetric Encryption Keys • Public Keys derived from Private Keys: · One-way mathematical functions · Also called “trap-door” functions • Main idea: · Bob generates Public Key from his Private Key · Alice encrypts a file with Bob’s Public Key · The Public Key will not decrypt the file · Only Bob’s Private Key will decrypt the file

Subset-Sum Problem Given some set of integers {a1, a2, … an} and B, find a subset of the ai that sums to B. Example: Our set of integers is {5, 8, 4, 11, 6}, and B = 20. What subset of these five integers will add up to 20? {5, 4, 11} is a solution because 5 + 4 + 11 = 20 This is an NP-Complete problem Any known algorithm guaranteed to solve the problem is O(2n)

Super-Increasing Sequences Each number in a super-increasing sequence is greater than the sum of all the preceding numbers. Example: {1, 2, 5, 11, 32, 87, 141} The subset-sum problem is very easy to solve on super-increasing sequences in linear time!

Public Key Encryption Example Whitfield Diffie, Ralph Merkle, Martin Hellman. First commercial public key algorithm. Developed at Stanford, late 1970’s. Encryption Procedure: 1. You have a public key consisting of n integers: (a1, a2, … an) 2. Convert your text string to a binary string. 3. Partition the binary string into blocks of length n. 4. Map each block (x1, x2, …, xn) into the integer:

Public Key Encryption Example Creation of the Public Key: 1. You have a private key consisting of n integers: (a’1, a’2, … a’n) · The private key is a super-increasing set. 2. You have two special integers, w and w-1 · w and w’ are modular inverses of each other. 3. The public key set is derived by the equation: ai = w * a’i mod m and the public key is then denoted by (a1, a2, … an)

Public Key Encryption Example Given some encrypted character Bx that can be decrypted with the public key using the subset-sum problem, we can find some corresponding B’x that can be decrypted using the subset-sum problem on the private key. Private Key = super-increasing sequence => easy problem to solve Bx is calculated using the aforementioned w-1 and m as follows: B’x = Bx * w-1 mod m This is because w, w-1, and m have a special relationship such that: n * w * w-1 mod m = n (for any integer n)

Usefulness of Public Key Crypto • Signing, Non-Repudiation, Authentication · If Alice can decrypt a message with Bob’s Public Key, it must be from Bob, unless someone else has his Private Key • Algorithm speed concerns • Man-in-the-Middle attacks · Public Key Infrastructure (PKI) · Certificate Authorities (CA’s)

Historically Significant Advances The real start of Public Key Encryption: GCHQ, NSA · James Ellis in 1969, Clifford Cocks in 1973 · Jerome Weisner Memorandum, 1962 Adi Shamir (MIT) found hole in subset-sum problem used in Diffie-Merkle-Hellman algorithm Rivet, Shamir, Adleman develop RSA algorithm · Stronger, more secure algorithm still used today

Public Key Cryptography