Communication Theory of Secrecy Systems On a paper by Shannon (and the industry it didn’t spawn) Gilad Tsur Yossi Oren December 2005 Gilad Tsur Yossi Oren December 2005

What you’ll see today Shannon – his life and work Cryptography before Shannon Definition of a cryptosystem Theoretical and practical security Product ciphers and combined cryptosystems Closing thoughts Shannon – his life and work Cryptography before Shannon Definition of a cryptosystem Theoretical and practical security Product ciphers and combined cryptosystems Closing thoughts

Shannon – his life and work

Claude E. Shannon ( )

Important facts: M. Sc. Thesis founded an industry Ph. D. finished in 1.5 years Married a computer in 1949 Wrote scientific papers on a variety of topics, including juggling Important facts: M. Sc. Thesis founded an industry Ph. D. finished in 1.5 years Married a computer in 1949 Wrote scientific papers on a variety of topics, including juggling

Shannon’s Information Theory Paper “Mathematical Theory of Communication”, published in 1948 Main claim: All sources of data have a rate All channels have a capacity If the capacity is greater than the rate, transmission with no errors is possible Introduced concept of entropy of a random variable/process “Mathematical Theory of Communication”, published in 1948 Main claim: All sources of data have a rate All channels have a capacity If the capacity is greater than the rate, transmission with no errors is possible Introduced concept of entropy of a random variable/process

Cryptography before Shannon From

Themes in cryptography From dia/sharemed/targets/images/pho/t025/ T025102A.jpg Seals were used as authentication means for signing contracts, for royal decrees and for other documents. Passwords were used by military and other organizations to identify members. Codes are semantic while ciphers are syntactic. All these methods (in the case of seals, as rubber stamps) are also in use today.

Ancient Ciphers I תשרקצפעסנמלכיטחזוהדגבא אבגדהוזחטיכלמנסעפצקרשת Atbash cipher used in old testament “ בבל = ששך “ Of course, anyone who’d ever heard of this cipher could easily crack it. This is also true for another famous cipher, the Caesar cipher. ABC…XYZ DEF…ABC

Ancient Ciphers II The Caesar cipher is just a specific case of what are generally known as Shift Ciphers. A Shift cipher is one where the code is simply a rotation of the alphabet with K steps, where the number K can be considered the key. Easier for us in CS – think of it as a constant added modulo the size of the alphabet. Obviously, finding the key for such a code is not a lengthy process.

Ancient Ciphers III Text took an active effort to understand (This was used with ROT-13 on Usenet). Probably the real reason – security through obscurity. The concept of cryptography was not that well known, and codes such as Atbash were simply assumed not to be known by people you didn’t want reading them. So what were these ciphers good for?

Ancient Ciphers IV Both Atbash and Shift ciphers are specific cases of a more general type of ciphers used in the ancient world: Monoalphabetic Substitution Ciphers. As these ciphers were used by people who wanted to remember them, keyword and keyphrase ciphers were often used. The keyword could be changed daily to make it harder to decrypt. Some of these ciphers didn’t use a 1-1 correspondence, trusting the redundancy of language or allowing multiple representations.

Ancient Ciphers V Not all ancient ciphers used substitution methods. The earliest known cryptographic device (to the best of our knowledge) is the Spartan scytale. Using this device the letters of the message weren’t changed, but their order was.

Ancient Ciphers VI The scytale was a device assisting in the creation of a Transposition Cipher. Perhaps the most notable example of a transposition cipher is the column transposition. Other geometrical transposition ciphers abound, mostly route ciphers. Transposition ciphers based on a local permutation are also common, but offer a less apparently convenient way of writing quickly.

Ancient Ciphers VII We have written records of frequency analysis dating to the 9 th century. From Using multiple options to substitute frequent letters could make frequency analysis much harder.

Cryptography during the dark ages (‘till around 14 th century) Cryptography didn’t advance in Europe much during the dark ages. Some religious and mystical sects used cryptographic techniques to encode their writings, often substitution ciphers to an arcane alphabet. However, the church considered most people using cryptography as heretics, sorcerers or witches, and was in the habit of burning them. Coupled with low levels of literacy, cryptography was only studied outside of Europe. While texts (such as the cryptanalytic one mentioned above) appeared, we are unaware of major advances.

Codes and ciphers in the renaissance In Italy, and later all over Europe, cryptography returns to fashion. Different city-states and countries begin to employ professional cryptanalysts for encoding and decoding mail. The most common codes are Polyalphabetic Substitution Ciphers. Many devices are made to aid encryption and decryption.

Polyalphabetic substitution ciphers These ciphers can simply be considered as a list of shift ciphers or monoalphabetic substitution ciphers to be used consecutively. The use of some of these ciphers was aided by a cipher disk. Other such ciphers used tables to assist encryption and decryption. Notably, some in of these cipher were polygraphic – each encoded symbol represented a combination of plaintext symbols.

Cryptanalysis of polyalphabetic substitution ciphers The major classic techniques used for this process involve two steps: 1. Discover the length of cycle. 2. Use monoalphabetic cryptanalysis techniques for each alphabet (+ information gained from previous alphabets). Step 1 can be done systematically (Brute force approach) but this may be a very hard process. A shortcut that often helps (and is published in the 19 th century, ‘though probably known before) is finding repeating sequences in the text.

Cryptography in the 19 th and 20 th Centauries I WWI sees the full use of cryptography in the battle field. Advances in radio and telegraph allow military units to communicate better than ever before. This means easy to use, generic ciphers are required. Mechanized cipher machines offer this option.

Cryptography in the 19 th and 20 th Centauries II WWII is famous for being a scientific war in general, and for cryptography in particular. German Enigma cracked by British, Japanese “Purple” by the US. Enigma, in fact, a polyalphabetic cipher system with around 20,000 alphabets.

Definitions of a Cryptosystem

Definitions of a Cryptosystem: Shannon’s version II A cryptosystem can be viewed as a distribution of possible plaintexts (P), a set of possible ciphertexts (C), a distribution of possible keys (K) and an encoding transformation (E) With its inverse (D).

Definitions of a Cryptosystem: modern variations Many things have changed in our thinking about cryptography. Different functions: Not only trying to transmit secret information. Different settings for “Alice” and “Bob” – we now have public key cryptosystems and extensive use of randomness. Different settings for “Eve” – we now have a variety of attacks such as known plain text, chosen ciphertext, chosen plain text and side channel attacks.

Shannon’s 1948 Paper Published one year after his monumental “information theory” paper Inspired by Von-Neumann’s paper on game theory “transformed cryptography from art to science” Published one year after his monumental “information theory” paper Inspired by Von-Neumann’s paper on game theory “transformed cryptography from art to science”

Main Contributions Notions of theoretical security and practical security Observation that the secret is all in the key, not in the algorithm – “the enemy knows the system” (also attributed to Auguste Kerckhoffs) Product ciphers and mixing transformations – inspiration for LUCIFER and later DES Proof that Vernam’s cipher (one-time pad) was theoretically secure Notions of theoretical security and practical security Observation that the secret is all in the key, not in the algorithm – “the enemy knows the system” (also attributed to Auguste Kerckhoffs) Product ciphers and mixing transformations – inspiration for LUCIFER and later DES Proof that Vernam’s cipher (one-time pad) was theoretically secure

Theoretical Security and Practical Security

Theoretically secure cryptosystems cannot be broken – even by an all-powerful adversary Practically secure cryptosystems “require a large amount of work to solve” Bad news: The only theoretically secure cryptosystem is the one- time pad The only practically secure cryptosystem is… the one-time pad We do have some cryptosystems which are provably [as] secure as a difficult problem Theoretically secure cryptosystems cannot be broken – even by an all-powerful adversary Practically secure cryptosystems “require a large amount of work to solve” Bad news: The only theoretically secure cryptosystem is the one- time pad The only practically secure cryptosystem is… the one-time pad We do have some cryptosystems which are provably [as] secure as a difficult problem

Review: Bayes’ Theorem Let X and Y be two random variables. Define: Theorem (Chain Rule): Theorem (Bayes): Let X and Y be two random variables. Define: Theorem (Chain Rule): Theorem (Bayes): Apriori Aposteriori

Theoretical (Perfect) Security What does it mean for a cryptosystem to be perfectly secure? Essentially, the adversary doesn’t learn anything from the ciphertext: What does it mean for a cryptosystem to be perfectly secure? Essentially, the adversary doesn’t learn anything from the ciphertext:

Perfect Security means | K | ¸ | P | Reminder: adversary “knows the system” and has unlimited power If key-space is finite, each ciphertext must map to a finite number of plaintexts If | P | > | K |, some plaintexts will be “impossible” for some ciphertexts Reminder: adversary “knows the system” and has unlimited power If key-space is finite, each ciphertext must map to a finite number of plaintexts If | P | > | K |, some plaintexts will be “impossible” for some ciphertexts

The Vernam Cipher (1) Is there a perfectly secure cryptosystem for which | K |=| P | ? Theorem (Shannon): Let ( P, K, C, E, D ) be a cryptosystem for which | K |=| P |=| C |. Then the cryptosystem provides perfect secrecy iff: Is there a perfectly secure cryptosystem for which | K |=| P | ? Theorem (Shannon): Let ( P, K, C, E, D ) be a cryptosystem for which | K |=| P |=| C |. Then the cryptosystem provides perfect secrecy iff:

The Vernam Cipher (2) Proof: Let ( P, K, C, E, D ) be a cryptosystem for which | K |=| P |=| C |. Because of perfect secrecy: | K |=| P |=| C |, so there is a unique key associated with every pair (p,c) Proof: Let ( P, K, C, E, D ) be a cryptosystem for which | K |=| P |=| C |. Because of perfect secrecy: | K |=| P |=| C |, so there is a unique key associated with every pair (p,c)

The Vernam Cipher (3) Fix c. For all possible plaintexts p i, let k i be the key satisfying e ki (p i )=c By Bayes: Fix c. For all possible plaintexts p i, let k i be the key satisfying e ki (p i )=c By Bayes:

The Vernam Cipher (4) Cipher was invented by Gilbert Vernam of Bell Labs in 1919 Idea – key is a long random sequence, C=P © K By above proof, cipher is unbreakable Disadvantage – key is huge and cannot be used twice Advantage – algorithm is so simple we can give it to the Soviets… Cipher was invented by Gilbert Vernam of Bell Labs in 1919 Idea – key is a long random sequence, C=P © K By above proof, cipher is unbreakable Disadvantage – key is huge and cannot be used twice Advantage – algorithm is so simple we can give it to the Soviets…

Towards real-world cryptography How secure are cryptosystems with a smaller key space? Rules of the game: Symmetric (deterministic) encryption | P |=| C |, all keys chosen equiprobably Ciphertext-only attack Adversary wishes to recover the key Question: How fast does the set of possible keys shrink as the amount of ciphertext grows? How secure are cryptosystems with a smaller key space? Rules of the game: Symmetric (deterministic) encryption | P |=| C |, all keys chosen equiprobably Ciphertext-only attack Adversary wishes to recover the key Question: How fast does the set of possible keys shrink as the amount of ciphertext grows?

A Brief Introduction to Information Theory Some random events are more unexpected than others Some facts are more significant than others Shannon Entropy measures the amount of uncertainty regarding a random variable, or the amount of information an event provides Entropy Rate measures the growth of information in an infinitely-long sequence Some random events are more unexpected than others Some facts are more significant than others Shannon Entropy measures the amount of uncertainty regarding a random variable, or the amount of information an event provides Entropy Rate measures the growth of information in an infinitely-long sequence

Definition of Entropy If X is a random variable taking values from finite alphabet X, then (note: lim x ! 0 xlogx=0) If X is a random variable taking values from finite alphabet X, then (note: lim x ! 0 xlogx=0)

Entropy Rate If L is a language formed of a sequence of identically distributed (possibly dependent) variables, then The redundancy of a language is defined as: If L is a language formed of a sequence of identically distributed (possibly dependent) variables, then The redundancy of a language is defined as:

Basic Properties of Entropy H(X) ¸ 0, with equality iff X is constant H(X) · log 2 | X |, with equality iff p(x=X)=1/| X | 8 x 2 X H(X,Y) · H(X)+H(Y), with equality iff X and Y are independently distributed H(X|Y) · H(X), with equality iff X and Y are independently distributed Chain Rule:H(X,Y)=H(X|Y)+H(Y) H(X) ¸ 0, with equality iff X is constant H(X) · log 2 | X |, with equality iff p(x=X)=1/| X | 8 x 2 X H(X,Y) · H(X)+H(Y), with equality iff X and Y are independently distributed H(X|Y) · H(X), with equality iff X and Y are independently distributed Chain Rule:H(X,Y)=H(X|Y)+H(Y)

Entropy of Cryptosystem Components Reminder – Cryptosystem = (P,K,C,E,D) H(C|K) =H(P) H(C|P,K)=H(P|C,K)=0 H(P,K)=H(P)+H(K) H(C) ¸ H(P) H(C,P,K)=H(C,K)=H(P,K) H(K|C)=H(K)+H(P)-H(C) H(K|C n )=H(K)+H(P n )-H(C n ) Reminder – Cryptosystem = (P,K,C,E,D) H(C|K) =H(P) H(C|P,K)=H(P|C,K)=0 H(P,K)=H(P)+H(K) H(C) ¸ H(P) H(C,P,K)=H(C,K)=H(P,K) H(K|C)=H(K)+H(P)-H(C) H(K|C n )=H(K)+H(P n )-H(C n )

Example: A strong cipher which is very weak (1) Everything Haley says is encrypted with a monoalphabetic substitution cipher Could you break it? Q: Which 2 romantic era authors had their heroes break this cipher? Everything Haley says is encrypted with a monoalphabetic substitution cipher Could you break it? Q: Which 2 romantic era authors had their heroes break this cipher? (archive.org cache)

A strong cipher which is very weak (2) Observation: There are 26! ¼ possible substitution ciphers over the lowercase English alphabet This is equivalent to 88-bit security – so why was it so easy to break? Shannon: Any monoalphabetic cipher over the English language is easily broken, given a sequence of 25 letters of unknown ciphertext Observation: There are 26! ¼ possible substitution ciphers over the lowercase English alphabet This is equivalent to 88-bit security – so why was it so easy to break? Shannon: Any monoalphabetic cipher over the English language is easily broken, given a sequence of 25 letters of unknown ciphertext

Unicity distance of a language (1) By definition of the entropy rate: Since | P |=| C |, we have: Substituting into the formula for H(K|C n ): By definition of the entropy rate: Since | P |=| C |, we have: Substituting into the formula for H(K|C n ):

Unicity distance of a language (2) The cryptosystem is broken when H(K|C n )=0: Plug in English (| P |=26, R L ¼ 0.75) and the substitution cipher (log 2 | K | ¼ 88) and we get n 0 ¼ 25 (woops). The cryptosystem is broken when H(K|C n )=0: Plug in English (| P |=26, R L ¼ 0.75) and the substitution cipher (log 2 | K | ¼ 88) and we get n 0 ¼ 25 (woops).

Tricks to raise the unicity distance The idea – raise the entropy of the language without disturbing content Adding random nulls – “hello” becomes “h ; e ;; l ; lo ;; ” Replace characters with homophonic sets – “hello” becomes “hello” Compress the data Good compression – good for encryption Good encryption – bad for compression The idea – raise the entropy of the language without disturbing content Adding random nulls – “hello” becomes “h ; e ;; l ; lo ;; ” Replace characters with homophonic sets – “hello” becomes “hello” Compress the data Good compression – good for encryption Good encryption – bad for compression

Product Ciphers and Combined Cryptosystems

“Weighted Sum” Cryptosystems Different cryptosystems can be combined to create a new cryptosystem. Given two cryptosystems with the same message space, consider a probabilistic combination of the two systems: with probability p use system A, otherwise use system B. Different cryptosystems can be combined to create a new cryptosystem. Given two cryptosystems with the same message space, consider a probabilistic combination of the two systems: with probability p use system A, otherwise use system B.

Endomorphic cryptosystems and product ciphers Another way to use two cryptosystems is to encrypt and decrypt messages consecutively. We call this a product cipher. An endomorphic cryptosystem is a system where the message space is transformed to itself. With such a system we can even create a product of the system with itself. The set of endomorphic cryptosystems with the aforementioned operations almost create a linear associative algebra. Another way to use two cryptosystems is to encrypt and decrypt messages consecutively. We call this a product cipher. An endomorphic cryptosystem is a system where the message space is transformed to itself. With such a system we can even create a product of the system with itself. The set of endomorphic cryptosystems with the aforementioned operations almost create a linear associative algebra.

Idempotent and commutative cryptosystems A cryptosystem S is called idempotent if S 2 = S. Combining two idempotent secrecy systems that commute will create another idempotent secrecy system – isn’t of any use. A cryptosystem S is called idempotent if S 2 = S. Combining two idempotent secrecy systems that commute will create another idempotent secrecy system – isn’t of any use.

Designing cryptosystems that are hard to attack I Shannon recognizes that aside of brute force attacks, reasonable attacks attempt to find keys according to their probability, hopefully reducing the probability of many of them near 0 without actually testing each and every one. To do this one needs to create statistics of the ciphertext. Shannon recognizes that aside of brute force attacks, reasonable attacks attempt to find keys according to their probability, hopefully reducing the probability of many of them near 0 without actually testing each and every one. To do this one needs to create statistics of the ciphertext.

Designing cryptosystems that are hard to attack II Statistics must be: Simple to measure Depend more on the key (if we’re trying to find the key) than the message Useful – divide the key-space into areas of similar probability and eliminate most Usable – the separation of the key-space must be natural Statistics must be: Simple to measure Depend more on the key (if we’re trying to find the key) than the message Useful – divide the key-space into areas of similar probability and eliminate most Usable – the separation of the key-space must be natural

Confusion and Diffusion To make finding such statistics harder (without an ideal system) Shannon suggests: Diffusion: Spreading the information in such a way that it is hard to get exact results. Confusion: Make the natural separation of the key- space hard to use. (Make all parameters of key dependant in natural decryption). He believes that a combination of an initial transposition with alternating substitutions and linear operations may do the trick. To make finding such statistics harder (without an ideal system) Shannon suggests: Diffusion: Spreading the information in such a way that it is hard to get exact results. Confusion: Make the natural separation of the key- space hard to use. (Make all parameters of key dependant in natural decryption). He believes that a combination of an initial transposition with alternating substitutions and linear operations may do the trick.

Closing Thoughts

Effect of this Paper Paper did not bring forth an explosion similar to the 1947 paper “The problem of good cipher design is essentially one of finding difficult problems” This type of problem was made very public with the creation of DES. Both DES and AES use Shannon’s ideas of combining confusion and diffusion (although other ideas that he hadn’t mentioned appear in both). Paper did not bring forth an explosion similar to the 1947 paper “The problem of good cipher design is essentially one of finding difficult problems” This type of problem was made very public with the creation of DES. Both DES and AES use Shannon’s ideas of combining confusion and diffusion (although other ideas that he hadn’t mentioned appear in both).

Some closing thoughts Cryptography is always in the context of communication between agents. Not only what messages are transmitted but from whom to whom is important. We can hardly hide the size of messages. One can encrypt messages in ways that allow breaking them, to misinform. In huge information environments one could easily(?) conceal the existence of messages and the identities of the sender and the receiver. Cryptography is always in the context of communication between agents. Not only what messages are transmitted but from whom to whom is important. We can hardly hide the size of messages. One can encrypt messages in ways that allow breaking them, to misinform. In huge information environments one could easily(?) conceal the existence of messages and the identities of the sender and the receiver.