University of Nevada, Reno Permutation Steganography in Many Systems A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Mathematics by William G. Rudebusch Dr. Edward C. Keppelmann/Thesis Advisor December, 2011 THE GRADUATE SCHOOL We recommend that the thesis prepared under our supervision by William Rudebusch entitled Permutation Steganography in Many Systems be accepted in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Edward C. Keppelmann, Ph.D., Advisor Thomas Quint, Ph.D., Committee Member Sushil Louis, Ph.D., Graduate School Representative Marsha H. Read, Ph. D., Associate Dean, Graduate School December, 2011 i Abstract The newest method we wish to offer to the world of steganography and cryp- tology makes use of permutations. The initial idea of permeation steganog- raphy was developed independently by Peter Hendrickson (1). Here we extended on his idea by adding composition of permutations into the reply step. We have also included cycle notation; this allows for more freedom and saves bandwidth. ii Contents List of Tables v 1 Introduction 1 1.1 What is Steganography? . .1 1.2 Our Proposed System . .3 1.2.1 twitter . .3 1.2.2 Cards . .4 1.2.3 DNA . .4 2 Background Mathematics 5 2.1 Shannon's Confusion and Diffusion . .5 2.2 Combinatorial Compositions . .6 2.3 Symmetric Group . .6 2.3.1 Notation . .6 2.3.2 Composition . .7 2.3.3 Inverses . .7 2.4 Factorial Number System . .7 2.5 Rank of a Permutation . .9 2.5.1 Example: Rank 4 Permutation of S3 ................9 2.6 Packing Number . 10 2.6.1 Example: the Packing Number of S52 ............... 11 2.7 The 36 Character Alphabet . 12 3 twitter 13 3.1 Idea . 13 3.2 Preliminary Discussion . 13 iii 3.3 Set up . 14 3.4 Encoding . 15 3.5 Decoding . 15 3.6 Reply . 16 3.6.1 Round Two Communication . 16 3.6.2 Multiple Permutations Present . 17 3.7 Implementation . 17 3.7.1 Case 1: Number Station . 17 3.7.2 Case 2: Dialogue Between Parties . 17 3.8 Dead Drop Example . 18 3.8.1 Geocache Preliminary . 18 3.8.2 Geocache Encode . 18 3.8.3 Geocache Decode . 19 3.9 Longer Example . 21 3.10 Analysis . 23 3.10.1 Authentication . 23 3.10.2 Extra Characters . 24 3.10.3 Limitations of twitter . 24 4 Cards 26 4.1 Idea . 26 4.2 Preliminary Discussion . 26 4.3 Set Up . 27 4.4 Encoding . 27 4.5 Decoding . 28 4.6 Replying . 28 4.7 Card Example . 29 4.7.1 Card Encode . 29 4.7.2 Card Decode . 30 4.7.3 Card Reply . 30 4.8 Analysis . 33 4.8.1 Shuffling . 33 4.8.2 Message Length . 35 iv 4.8.3 False Positive . 35 4.8.4 Multiple Decks . 35 5 Genetics 37 5.1 Motivation . 37 5.2 Preliminary Genetic Information . 37 5.3 Combinatorial Compositions of DNA . 39 5.4 Possible Process . 40 5.5 Example . 41 5.5.1 Encode . 41 5.5.2 Decode . 41 5.5.3 Reply . 42 5.6 Analysis . 42 5.6.1 Ideal Length . 42 5.6.2 Maximum Length . 42 6 Future Frontiers 44 6.1 twitter . 44 6.2 Cards . 45 6.3 Genetics . 45 6.4 New Systems . 45 Bibliography 47 v List of Tables 2.1 36 Character Alphabet . 12 5.1 title of table . 38 1 1 Introduction 1.1 What is Steganography? Steganography is the art and science of hiding messages in plain sight, \thereby ob- scuring not just its meaning but its very existence." (10) Typically the application of steganography makes use of the wasted space inherent in all channels of communication. By wasted space I mean the places in communication where metadata exists. At any instant in time there is more information being shown to an observer than he or she is processing. For example, when reading a sentence the observer typically does not exactly count the white spaces or frequency of the letters. These properties of every-day communication could be used to hide information. Stenography has just as many legal uses as covert. It can be used to authenticate a copy of a piece of software while not affecting the data in any decreeable way. This is called watermarking. This method is used in the industry to assure authenticity of a product. Clandestine as well as legal activities require creativity. This is because steganog- raphy requires information to unnoticeably mix into existing forms of communication. In this paper we will be focusing on encoding and transmitting plaintext messages. Whether this is cloak-and-dagger or completely innocent is left to the reader. A historical example of technology and steganography coming together happened in the 1980s. Prime minister Margret Thatcher thought she had a leak in her cabinet. Her idea to fix it was to slightly alter the word processors of government employees. These tampered machines now encoded the information of the author into the word spacing of 2 the document. Therefore making each piece of internal correspondence secretly linked to its source. Now she could trace a leaked document to is originating word processor. (2) Due to advancing technology the applications of steganography are growing. One popular technique uses of digital photography. The least significant pixels in a picture can be slightly altered from their original state. This small change in value will not arouse suspicion in a human viewer but can be used to hide information.There are many open source computer programs that can do this operation. A cryptologist will be quick to point out that these systems are not secure. In the cryptographic sense they are absolutely correct. Steganography relies on the fact that the enemy would first need to know where to look to be able to find anything. In cryp- tology the code is usually transmitted in the open. This encrypted data is available to anyone who wants to spend the resources to try and decrypt (crack) it. Whereas it is difficult to quantify exactly how much computing time it would take to crack a steganographic system. The first problem is finding it and then checking every item for hidden data. On a any given website with frequent updates, such as a photo-centric tumblr stream, checking every new photo could become quite burden- some. Furthermore, two parties speaking in code is typically enough to arouse suspicion. On the other hand, two parties communicating with steganography does not. This ability for encrypted communication to be carried out in an open unsuspicious fashion is yet strength of steganography. For example, suppose that two parties wish to communicate in a prison. In this situation cryptography is not an option because code will obviously alert the officials to possible illicit activity. Hence, stenography in a letter would be permissible and an effective way for them to still communicate while imprisoned. Another example is if two operatives, Alice and Bob, wish to keep their identities secret but still need to communicate. If Alice were caught transmitting secret data to Bob then Bob would also be caught since he was the obvious recipient. Supposing that their relationship were kept ambiguous this avoids unnecessary risk for all parities involved. In theses situations steganography has many advantages over traditional cryptology. 3 1.2 Our Proposed System The newest method we wish to offer to the world of steganography and cryptology makes use of permutations. The initial idea was developed independently by Peter Hendrickson (1). Here we extended on his idea by adding composition of permutations into the reply step. In some cases we use cycle notation which allows for a lot more freedom. Since we are using permutation composition we are thereby obscuring the nature of the permutations being used between parties. For an eavesdropper to be successful at decoding the message she or he would need to have the entire conversation. This can be quite difficult to assertion in the situations we propose. Another added feature is that we are using cycle notation instead of tabular. This makes it so fewer characters have to be embedded into the message in order to transmit a permutation. Saving space will be important when using twitter. The math involved deals mostly with combinatorics, group and number theory. All of our computations are done in Sage; an open source mathematics software system (6). 1.2.1 twitter The idea of using steganographic techniques in twitter should not come as a shock; many rebel groups have used twitter and Facebook as their method of communication and organization (3). Social networking offers an endless sea of information where one more drop couldn't possibly cause a stir. This idea of information that can exist in an inconspicuous way is the basis of steganography. Depending on personal creativity there can be quite a few bits of hidden information encoded in the message body of a tweet. One system makes use of the white space between words and sentences. Explicitly a single space for a \0" and two for a \1". This can become a little cumbersome and not very efficient. With a 140 character per post limit, very few bits can be encoded this way. Another method makes use of special alphabets. Suppose that synonyms for a commonly used word are in fact binary values. This scheme needs some amount of non-twitter communication and a shared codebook but there is a fair amount of data that could be encoded (4). It should be noted that the messages themselves may cause 4 suspicion depending on the choice of special alphabet.