This document provides an overview of cryptography and this course on the subject. It discusses the history of cryptography, examples throughout history of codes being invented and then broken, and modern cryptography since the 1970s which focuses on provable security. The course will cover foundations and principles, definitions and proofs of security, applications, and advanced topics like public key cryptography and zero-knowledge proofs. It provides information about lectures, prerequisites, readings, assignments, and notes that cryptography is a challenging subject involving mathematical proofs.
This document discusses cryptography and the Caesar cipher. It begins by defining cryptography as the encoding of messages to achieve secure communication and outlines its goals of confidentiality, integrity, and availability. The document then describes the Caesar cipher technique, in which each letter is shifted a fixed number of positions in the alphabet. It provides an example of encrypting a message with a shift of 11. The document explains that the Caesar cipher is vulnerable to brute force and statistical cryptanalysis due to its small key space and predictable letter frequencies. It concludes that more advanced algorithms are needed for secure encryption in the digital age.
The document discusses classical encryption techniques such as substitution ciphers like the Caesar cipher and monoalphabetic cipher, transposition ciphers like the rail fence cipher and row transposition cipher, and polyalphabetic ciphers like the Vigenere cipher. It introduces basic concepts and terminology in cryptography such as plaintext, ciphertext, encryption, decryption, and secret keys. The goals are to introduce basic concepts and terminology of encryption and to prepare for studying modern cryptography.
This document provides an introduction to cryptography and cryptanalysis. It contains a table of contents outlining the topics to be covered, which include the history and concepts of cryptography, symmetric and public key cryptosystems, cryptanalysis techniques, and applications of cryptography such as digital signatures and internet security protocols. The author thanks several people who provided input and acknowledges that any mistakes are their own. It also includes a crash course on basic number theory concepts relevant to cryptography.
Symmetric encryption uses a single, shared key between the sender and receiver to encrypt and decrypt messages. Common symmetric algorithms are DES, 3DES, and AES. The main drawback is securely exchanging the encryption key between parties. Cryptanalysis is the study of decrypting ciphertext without knowing the key, and involves cryptanalytic attacks or brute-force attacks to discover the plaintext or key.
Quantum cryptography uses principles of quantum mechanics to securely distribute encryption keys between two parties. It allows Alice and Bob to detect if an eavesdropper (Eve) is trying to intercept the key during transmission. Eve's attempt to measure the quantum states used to transmit the key would introduce detectable errors. The document discusses the history and principles of quantum cryptography, including types like discrete and continuous variable QKD. It also covers desirable attributes like confidentiality and rapid key delivery, providing an example of how quantum key distribution works between Alice and Bob.
- The document provides an overview of the schedule and topics for a cryptography class, including an introduction to cryptography today, Elliptic Curve Cryptography and signatures on Wednesday, and a checkup on the first three classes next Monday.
- It also lists the assigned readings for chapters 1-4 of the textbook and provides information about the backgrounds of students in the class.
- The remainder of the document discusses setting up a Bitcoin wallet, downloading the blockchain, hierarchical deterministic wallets, and provides a recap of the concepts from the previous class around what makes something a currency and how ownership of digital goods can be established.
This document discusses the use of number theory in cryptography. It begins by describing several historical ciphers such as the Caesar cipher, Morse code, the Enigma machine, public key cryptography, and the Scytale cipher. It then explains how number theory is applied in some of these ciphers, such as how the Caesar cipher uses modulo arithmetic, how RSA public key cryptography is based on the difficulty of factoring large numbers, and how the Enigma machine's encryption can be expressed mathematically as a product of permutations. The document concludes by noting the vast number of possible settings for the Enigma machine.
This document provides an overview of cryptography and this course on the subject. It discusses the history of cryptography, examples throughout history of codes being invented and then broken, and modern cryptography since the 1970s which focuses on provable security. The course will cover foundations and principles, definitions and proofs of security, applications, and advanced topics like public key cryptography and zero-knowledge proofs. It provides information about lectures, prerequisites, readings, assignments, and notes that cryptography is a challenging subject involving mathematical proofs.
This document discusses cryptography and the Caesar cipher. It begins by defining cryptography as the encoding of messages to achieve secure communication and outlines its goals of confidentiality, integrity, and availability. The document then describes the Caesar cipher technique, in which each letter is shifted a fixed number of positions in the alphabet. It provides an example of encrypting a message with a shift of 11. The document explains that the Caesar cipher is vulnerable to brute force and statistical cryptanalysis due to its small key space and predictable letter frequencies. It concludes that more advanced algorithms are needed for secure encryption in the digital age.
The document discusses classical encryption techniques such as substitution ciphers like the Caesar cipher and monoalphabetic cipher, transposition ciphers like the rail fence cipher and row transposition cipher, and polyalphabetic ciphers like the Vigenere cipher. It introduces basic concepts and terminology in cryptography such as plaintext, ciphertext, encryption, decryption, and secret keys. The goals are to introduce basic concepts and terminology of encryption and to prepare for studying modern cryptography.
This document provides an introduction to cryptography and cryptanalysis. It contains a table of contents outlining the topics to be covered, which include the history and concepts of cryptography, symmetric and public key cryptosystems, cryptanalysis techniques, and applications of cryptography such as digital signatures and internet security protocols. The author thanks several people who provided input and acknowledges that any mistakes are their own. It also includes a crash course on basic number theory concepts relevant to cryptography.
Symmetric encryption uses a single, shared key between the sender and receiver to encrypt and decrypt messages. Common symmetric algorithms are DES, 3DES, and AES. The main drawback is securely exchanging the encryption key between parties. Cryptanalysis is the study of decrypting ciphertext without knowing the key, and involves cryptanalytic attacks or brute-force attacks to discover the plaintext or key.
Quantum cryptography uses principles of quantum mechanics to securely distribute encryption keys between two parties. It allows Alice and Bob to detect if an eavesdropper (Eve) is trying to intercept the key during transmission. Eve's attempt to measure the quantum states used to transmit the key would introduce detectable errors. The document discusses the history and principles of quantum cryptography, including types like discrete and continuous variable QKD. It also covers desirable attributes like confidentiality and rapid key delivery, providing an example of how quantum key distribution works between Alice and Bob.
- The document provides an overview of the schedule and topics for a cryptography class, including an introduction to cryptography today, Elliptic Curve Cryptography and signatures on Wednesday, and a checkup on the first three classes next Monday.
- It also lists the assigned readings for chapters 1-4 of the textbook and provides information about the backgrounds of students in the class.
- The remainder of the document discusses setting up a Bitcoin wallet, downloading the blockchain, hierarchical deterministic wallets, and provides a recap of the concepts from the previous class around what makes something a currency and how ownership of digital goods can be established.
This document discusses the use of number theory in cryptography. It begins by describing several historical ciphers such as the Caesar cipher, Morse code, the Enigma machine, public key cryptography, and the Scytale cipher. It then explains how number theory is applied in some of these ciphers, such as how the Caesar cipher uses modulo arithmetic, how RSA public key cryptography is based on the difficulty of factoring large numbers, and how the Enigma machine's encryption can be expressed mathematically as a product of permutations. The document concludes by noting the vast number of possible settings for the Enigma machine.
This document discusses the application of number theory in cryptography. It begins by describing several historical ciphers such as the Caesar cipher, Morse code, the Enigma machine, and public key cryptography. It then examines how number theory underpins various ciphers, such as how the Caesar cipher uses modular arithmetic and how the RSA algorithm relies on the difficulty of factoring large numbers. The document concludes by discussing future work exploring other ciphers and their implementation in programming languages like MATLAB.
The document summarizes classical encryption techniques, including:
- Symmetric encryption uses a shared key between sender and receiver for encryption/decryption.
- Early techniques included the Caesar cipher (shifting letters), monoalphabetic cipher (mapping each letter to another), and Playfair cipher (encrypting letter pairs).
- The Vigenère cipher improved security by using a keyword to select different Caesar ciphers for successive letters, making it a polyalphabetic cipher.
This document discusses cryptography and how it can be used to own digital goods like cryptocurrency. It begins by introducing key concepts in cryptography like cryptosystems, attacks, and asymmetry. It then discusses how early systems like Jefferson's wheel cipher provided security through obscurity of algorithms and keys. The document explores how brute force attacks become impractical as key sizes increase due to the vast amounts of energy required. It introduces public key cryptography and how RSA provides asymmetry through a trapdoor function. The document explains how asymmetric cryptography can be used for signatures and confidentiality. It concludes by noting how cryptography achieves the scarcity needed for digital ownership of coins.
Public-key cryptography uses two keys: a public key for encryption and digital signatures, and a private key for decryption and signature verification. RSA is the most widely used public-key cryptosystem, using large prime factorization and modular exponentiation. It allows secure communication without prior key exchange. While brute force attacks on RSA are infeasible due to large key sizes, its security relies on the difficulty of factoring large numbers.
This document provides an overview of cryptography and its applications. It discusses the history of cryptography beginning in ancient Egypt. It defines basic cryptography terminology like plaintext, ciphertext, cipher, key, encryption, decryption, cryptography, and cryptanalysis. It describes classical ciphers like the Caesar cipher and substitution ciphers. It also discusses cryptanalysis techniques, transposition ciphers, modern symmetric ciphers, public key cryptography including RSA, key distribution methods, and hybrid encryption.
Mathematics Towards Elliptic Curve Cryptography-by Dr. R.Srinivasanmunicsaa
The document provides an overview of elliptic curve cryptography including:
1. It discusses the evolution of cryptography from ancient times to modern algorithms like RSA, AES, and Diffie-Hellman key exchange.
2. It introduces elliptic curve cryptography as an alternative that provides the same level of security with smaller key sizes due to the difficulty of solving the elliptic curve discrete logarithm problem.
3. It provides examples of elliptic curve groups over prime fields and binary fields, showing how points on the curve satisfy the elliptic curve equation over a finite field.
This document proposes enhancing an existing "Odd-Even Transposition Technique" for encrypting plaintext. The enhancement applies the "Rail-Fence Technique" to the cipher text for added complexity. The procedure involves numbering words in the plaintext as odd or even, arranging characters in a table based on their number, then reversing columns and words to generate the cipher text. Applying Rail-Fence Technique to the resulting cipher text further scrambles the text. The example provided encrypts the plaintext "change never informs its arrival" using this two-step process.
This document provides an overview of cryptography from classical to modern times. It discusses the history and evolution of cryptographic techniques including substitution ciphers, transposition ciphers, codes, public key cryptography, digital signatures, and key distribution problems. The document also summarizes the four main topics that will be covered in the course: the history and foundations of modern cryptography, using cryptography in practice, the theory of cryptography including proofs and definitions, and a special topic in cryptography.
This document summarizes classical encryption techniques discussed in Chapter 2. It describes symmetric encryption methods that use a shared secret key, such as the Caesar cipher and monoalphabetic ciphers. It also covers the Playfair cipher, polyalphabetic ciphers like the Vigenère cipher, and transposition ciphers. More complex techniques are discussed like product ciphers implemented using rotor machines. The document also defines cryptography terminology and approaches to cryptanalysis like frequency analysis.
Cryptography, Classical Encryption
Breaking the Cryptosystem
Review the Simple attack to break the cryptosystem
Modular Arithmetic, Groups and Rings
One example each in classical substitutive and transposition ciphering.
Caesar/Affine Cipher –Worksheet and Lab Program
Basic Talk. 90 minute talk to an audience of Freshmen and Sophomores of IIT Bombay on 23/02/10 as a part of Science Week. Organised by Web and Coding Club. Place: GG 101 (Elec Department)
This document provides an overview of cryptography concepts and techniques. It defines cryptography and its principles such as symmetric and asymmetric ciphers. It then describes various classical encryption techniques like the Caesar cipher, monoalphabetic and polyalphabetic ciphers, the Playfair cipher, Hill cipher, and the Vernam cipher. For each technique, it explains the encryption and decryption algorithms and provides examples to illustrate how they work. The document also discusses cryptanalysis techniques like brute force attacks that can be used to break certain ciphers.
Cryptography is the process of securing communication and information. This document discusses several methods of cryptography including symmetric and public key cryptography. It provides examples of classical cryptography techniques like the Caesar cipher, transposition cipher, substitution cipher and the Vigenere cipher. It also discusses modern symmetric key algorithms like the Data Encryption Standard (DES) and the Advanced Encryption Standard (AES) which are widely used today. The one-time pad is described as theoretically unbreakable but impractical to implement. Block ciphers and padding methods are also summarized.
This document discusses classical encryption techniques such as symmetric encryption, where a shared key is used for encryption and decryption. It defines terminology like plaintext, ciphertext, encryption, and decryption. Symmetric ciphers require a strong algorithm and secret key. Classical ciphers discussed include the Caesar cipher, monoalphabetic ciphers, Playfair cipher, Vigenère cipher, and the one-time pad. It also covers transposition ciphers like the rail fence cipher and steganography.
The document provides an overview of cryptography concepts including encryption, decryption, symmetric cryptosystems, block ciphers, substitution ciphers, the one-time pad, and algorithms such as DES, Triple DES, AES, and others. Key points covered include Kerckhoffs's principle of keeping algorithms public and keys private, how symmetric encryption works between two parties with a shared key, methods of encrypting plaintext in blocks or as a bit stream, techniques like substitution and transposition ciphers, weaknesses of approaches like the Hill cipher, and the history and operation of standard block ciphers.
Bob and Alice want to securely communicate messages between each other over an insecure channel. Cryptography allows them to encrypt messages using public key encryption so that only the intended recipient can decrypt it. The document discusses the basics of public key cryptography including how it works, the RSA algorithm, key generation process, and approaches to attacking public key cryptography like brute force attacks or mathematical attacks like integer factorization to derive the private key.
Public-key cryptography uses two keys: a public key to encrypt messages and verify signatures, and a private key for decryption and signing. RSA is the most widely used public-key cryptosystem, using large prime factorization and exponentiation modulo n for encryption and decryption. While faster than brute-force, breaking RSA remains computationally infeasible with sufficiently large key sizes over 1024 bits.
The document discusses classical cryptography techniques such as substitution ciphers, transposition ciphers, and product ciphers. It then covers topics like cryptanalysis techniques including brute force attacks and statistical attacks. Modern symmetric key algorithms like AES and cryptographic hash functions are also mentioned.
it describes the bony anatomy including the femoral head , acetabulum, labrum . also discusses the capsule , ligaments . muscle that act on the hip joint and the range of motion are outlined. factors affecting hip joint stability and weight transmission through the joint are summarized.
This document discusses the application of number theory in cryptography. It begins by describing several historical ciphers such as the Caesar cipher, Morse code, the Enigma machine, and public key cryptography. It then examines how number theory underpins various ciphers, such as how the Caesar cipher uses modular arithmetic and how the RSA algorithm relies on the difficulty of factoring large numbers. The document concludes by discussing future work exploring other ciphers and their implementation in programming languages like MATLAB.
The document summarizes classical encryption techniques, including:
- Symmetric encryption uses a shared key between sender and receiver for encryption/decryption.
- Early techniques included the Caesar cipher (shifting letters), monoalphabetic cipher (mapping each letter to another), and Playfair cipher (encrypting letter pairs).
- The Vigenère cipher improved security by using a keyword to select different Caesar ciphers for successive letters, making it a polyalphabetic cipher.
This document discusses cryptography and how it can be used to own digital goods like cryptocurrency. It begins by introducing key concepts in cryptography like cryptosystems, attacks, and asymmetry. It then discusses how early systems like Jefferson's wheel cipher provided security through obscurity of algorithms and keys. The document explores how brute force attacks become impractical as key sizes increase due to the vast amounts of energy required. It introduces public key cryptography and how RSA provides asymmetry through a trapdoor function. The document explains how asymmetric cryptography can be used for signatures and confidentiality. It concludes by noting how cryptography achieves the scarcity needed for digital ownership of coins.
Public-key cryptography uses two keys: a public key for encryption and digital signatures, and a private key for decryption and signature verification. RSA is the most widely used public-key cryptosystem, using large prime factorization and modular exponentiation. It allows secure communication without prior key exchange. While brute force attacks on RSA are infeasible due to large key sizes, its security relies on the difficulty of factoring large numbers.
This document provides an overview of cryptography and its applications. It discusses the history of cryptography beginning in ancient Egypt. It defines basic cryptography terminology like plaintext, ciphertext, cipher, key, encryption, decryption, cryptography, and cryptanalysis. It describes classical ciphers like the Caesar cipher and substitution ciphers. It also discusses cryptanalysis techniques, transposition ciphers, modern symmetric ciphers, public key cryptography including RSA, key distribution methods, and hybrid encryption.
Mathematics Towards Elliptic Curve Cryptography-by Dr. R.Srinivasanmunicsaa
The document provides an overview of elliptic curve cryptography including:
1. It discusses the evolution of cryptography from ancient times to modern algorithms like RSA, AES, and Diffie-Hellman key exchange.
2. It introduces elliptic curve cryptography as an alternative that provides the same level of security with smaller key sizes due to the difficulty of solving the elliptic curve discrete logarithm problem.
3. It provides examples of elliptic curve groups over prime fields and binary fields, showing how points on the curve satisfy the elliptic curve equation over a finite field.
This document proposes enhancing an existing "Odd-Even Transposition Technique" for encrypting plaintext. The enhancement applies the "Rail-Fence Technique" to the cipher text for added complexity. The procedure involves numbering words in the plaintext as odd or even, arranging characters in a table based on their number, then reversing columns and words to generate the cipher text. Applying Rail-Fence Technique to the resulting cipher text further scrambles the text. The example provided encrypts the plaintext "change never informs its arrival" using this two-step process.
This document provides an overview of cryptography from classical to modern times. It discusses the history and evolution of cryptographic techniques including substitution ciphers, transposition ciphers, codes, public key cryptography, digital signatures, and key distribution problems. The document also summarizes the four main topics that will be covered in the course: the history and foundations of modern cryptography, using cryptography in practice, the theory of cryptography including proofs and definitions, and a special topic in cryptography.
This document summarizes classical encryption techniques discussed in Chapter 2. It describes symmetric encryption methods that use a shared secret key, such as the Caesar cipher and monoalphabetic ciphers. It also covers the Playfair cipher, polyalphabetic ciphers like the Vigenère cipher, and transposition ciphers. More complex techniques are discussed like product ciphers implemented using rotor machines. The document also defines cryptography terminology and approaches to cryptanalysis like frequency analysis.
Cryptography, Classical Encryption
Breaking the Cryptosystem
Review the Simple attack to break the cryptosystem
Modular Arithmetic, Groups and Rings
One example each in classical substitutive and transposition ciphering.
Caesar/Affine Cipher –Worksheet and Lab Program
Basic Talk. 90 minute talk to an audience of Freshmen and Sophomores of IIT Bombay on 23/02/10 as a part of Science Week. Organised by Web and Coding Club. Place: GG 101 (Elec Department)
This document provides an overview of cryptography concepts and techniques. It defines cryptography and its principles such as symmetric and asymmetric ciphers. It then describes various classical encryption techniques like the Caesar cipher, monoalphabetic and polyalphabetic ciphers, the Playfair cipher, Hill cipher, and the Vernam cipher. For each technique, it explains the encryption and decryption algorithms and provides examples to illustrate how they work. The document also discusses cryptanalysis techniques like brute force attacks that can be used to break certain ciphers.
Cryptography is the process of securing communication and information. This document discusses several methods of cryptography including symmetric and public key cryptography. It provides examples of classical cryptography techniques like the Caesar cipher, transposition cipher, substitution cipher and the Vigenere cipher. It also discusses modern symmetric key algorithms like the Data Encryption Standard (DES) and the Advanced Encryption Standard (AES) which are widely used today. The one-time pad is described as theoretically unbreakable but impractical to implement. Block ciphers and padding methods are also summarized.
This document discusses classical encryption techniques such as symmetric encryption, where a shared key is used for encryption and decryption. It defines terminology like plaintext, ciphertext, encryption, and decryption. Symmetric ciphers require a strong algorithm and secret key. Classical ciphers discussed include the Caesar cipher, monoalphabetic ciphers, Playfair cipher, Vigenère cipher, and the one-time pad. It also covers transposition ciphers like the rail fence cipher and steganography.
The document provides an overview of cryptography concepts including encryption, decryption, symmetric cryptosystems, block ciphers, substitution ciphers, the one-time pad, and algorithms such as DES, Triple DES, AES, and others. Key points covered include Kerckhoffs's principle of keeping algorithms public and keys private, how symmetric encryption works between two parties with a shared key, methods of encrypting plaintext in blocks or as a bit stream, techniques like substitution and transposition ciphers, weaknesses of approaches like the Hill cipher, and the history and operation of standard block ciphers.
Bob and Alice want to securely communicate messages between each other over an insecure channel. Cryptography allows them to encrypt messages using public key encryption so that only the intended recipient can decrypt it. The document discusses the basics of public key cryptography including how it works, the RSA algorithm, key generation process, and approaches to attacking public key cryptography like brute force attacks or mathematical attacks like integer factorization to derive the private key.
Public-key cryptography uses two keys: a public key to encrypt messages and verify signatures, and a private key for decryption and signing. RSA is the most widely used public-key cryptosystem, using large prime factorization and exponentiation modulo n for encryption and decryption. While faster than brute-force, breaking RSA remains computationally infeasible with sufficiently large key sizes over 1024 bits.
The document discusses classical cryptography techniques such as substitution ciphers, transposition ciphers, and product ciphers. It then covers topics like cryptanalysis techniques including brute force attacks and statistical attacks. Modern symmetric key algorithms like AES and cryptographic hash functions are also mentioned.
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it describes the bony anatomy including the femoral head , acetabulum, labrum . also discusses the capsule , ligaments . muscle that act on the hip joint and the range of motion are outlined. factors affecting hip joint stability and weight transmission through the joint are summarized.
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cryptography_priceton_university_fall_2007.ppt
1. Princeton University • COS 433 • Cryptography • Fall 2007 • Boaz Barak
COS 433: Cryptography
Princeton University
Fall 2007
Boaz Barak
Two important quick notes:
Slides will be on course web site
Please stop me if you have questions!
2. 2
Cryptography
History of 2500- 4000 years.
Recurring theme: (until 1970’s)
Secret code invented
Typically claimed “unbreakable” by inventor
Used by spies, ambassadors, kings, generals for crucial tasks.
Broken by enemy using cryptanalysis.
Throughout most of this history:
cryptography = “secret writing”:
“Scramble” (encrypt) text such that it is hopefully unreadable by
anyone except the intended receiver that can decrypt it.
3. 3
Examples
1587: Ciphers from Mary of Scots plotting assassination of queen
Elizabeth broken; used as evidence to convict her of treason.
1860’s (civil war): Confederacy used good cipher (Vigenere) in a bad
way. Messages routinely broken by team of young union
cryptanalysts; in particular leading to a Manhattan manufacturer of
plates for printing rebel currency.
1878: New York Tribune decodes telegram proving Democrats’ attempt
to buy an electoral vote in presidential election for $10K.
1914: With aid of partial info from sunken German ships, British
intelligence broke all German codes.
Cracked telegram of German plan to form alliance with Mexico and
conquer back territory from U.S. As a result, U.S. joined WWI.
WWII: Cryptanalysis used by both sides. Polish & British cryptanalysts
break supposedly unbreakable Enigma cipher using mix of
ingenuity, German negligence, and mechanical computation.
Churchill credits cryptanalysts with winning the war.
4. 4
This Course
What you’ll learn:
Foundations and principles of the science
Definitions and proofs of security
High-level applications
Critical view of security suggestions and products
What you will not learn:
The most efficient and practical versions of components.
Designing secure systems*
“Hacking” – breaking into systems.
Everything important about crypto
Basic primitives and components.
Viruses, worms, Windows/Unix bugs, buffer overflow etc..
Buzzwords
Will help you avoid designing
insecure systems.
5. 5
This Course
Modern (post 1970’s) cryptography:
Provable security – breaking the “invent-break-tweak” cycle
Perfect security (Shannon) and its limitations
Computational security
Pseudorandom generators, one way functions
Beyond encryption – public-key crypto and other wonderful creatures
Public-key encryption based on factoring and RSA problem
Digital signatures, hash functions
Zero-knowledge proofs
Active security – Chosen-Ciphertext Attack
Advanced topics (won’t have time for all )
The SSL Protocol and attacks on it
Secret Sharing
Multi-party secure computation
Quantum cryptography
Password-based key-exchange, broadcast encryption, obfuscation
6. 6
Administrative Info
Lectures: Tue,Thu 1:30-2:50pm (start on time!)
Instructor: Boaz Barak: boaz@cs
Web page: http://www.cs.princeton.edu/courses/archive/fall07/cos433/
Or: Google “Boaz Barak” and click “courses”
TA: Rajsekar Manokaran ( rajsekar@cs )
Important: join mailing list, email me to set appointment
before next class
Office hrs: Thu after class (3pm) or by appointment.
Precepts: ---
Office hrs: ---
7. 7
Prerequisites
1. Ability to read and write mathematical proofs and definitions.
2. Familiarity with algorithms – proving correctness and
analyzing running time (O notation).
Required:
Helpful but not necessary:
Complexity. NP-Completeness, reductions, P, BPP, P/poly
Probabilistic Algorithms. Primality testing, hashing,
Number theory. Modular arithmetic, prime numbers
See web-site for links and resources.
3. Familiarity with basic probability theory (random variables,
expectations – see handout).
8. 8
Reading
Foundations of Cryptography / Goldreich.
Graduate-level text, will be sometimes used.
Lecture notes on web: (links on web site)
Computational Intro to Algebra and Number Theory / Shoup.
(Available also on the web)
Introduction to the Theory of Computation / Sipser.
For complexity background
Introduction to Modern Cryptography / Katz & Lindell
Main text used, though not 100% followed
9. 9
Requirements
Exercises: Weekly from Thursday till Thursday before class.
Submit by email / mailbox / in class to Rajsekar.
Flexibility: 4 late days, bonus questions
Take home final.
Final grade: 50% homework, 50% final
Honor code. Collaboration on homework with other students encouraged.
However, write alone and give credit.
Work on final alone and as directed.
10. 10
This course is hard
Challenging weekly exercises
Emphasis on mathematical proofs
Counterintuitive concepts.
Extensive use of quantifiers/probability
But it’s not my fault :)
Good coverage of crypto (meat, vegetables and desert) takes a year.
Simulation / experimentation can’t be used to show security.
Need to acquire “crypto-intuition”
Quantifiers, proofs by contradiction, reductions, probability are inherent.
Mitigating hardness
Avoid excessive exercises – only questions that teach you something.
Try best to explain intuition behind proofs
Me and Rajsekar available for any questions and clarifications.
11. 11
Encryption Schemes
Alice wants to send Bob a secret message.
They agree in advance on 3 components:
Encryption algorithm: E
Decryption algorithm: D
Secret key: k
To encrypt plaintext m, Alice sends c = E(m,k) to Bob.
To decrypt a cyphertext c, Bob computes m’ = D(c,k).
c = E(m,k)
c
m’ = D(c,k)
A scheme is valid if m’=m
Intuitively, a scheme is secure if eavesdropper can not learn m from c.
12. 12
Example 1: Caesar’s Cipher
Key: k = no. between 0 and 25.
Encryption: encode the ith letter as the (i+k) th letter.
(working mod 26: z+1=a )
Decryption: decode the jth letter to the (j-k) th letter.
S E N D R E I N F O R C E M E N T
Plain-text:
Key: 2
Cipher-text: U G P F T F K P H Q T E G O G P V
Problem: only 26 possibilities for key – can be broken in short time.
In other words: “security through obscurity” does not work.
Kerchoff’s Principle (1883): System should be secure even if
algorithms are known, as long as key is secret.
13. 13
Example 2: Substitution Cipher
Key: k = table mapping each letter to another letter
A B C Z
U R B E
Encryption and decryption: letter by letter according to table.
# of possible keys: 26! ( = 403,291,461,126,605,635,584,000,000 )
However – substitution cipher is still insecure!
Key observation: can recover plaintext using statistics on letter
frequencies.
LIVITCSWPIYVEWHEVSRIQMXLEYVEOIEWHRXEXIPFEMVEWHKVSTYLX
ZIXLIKIIXPIJVSZEYPERRGERIMWQLMGLMXQERIWGPSRIHMXQEREKI
He e e e h e t t ht
ethe eet e e h h t e e t e
I – most common letter
LI – most common pair
XLI – most common triple
Here e r e h e t t r r ht
ethe eet e r e h h t e e t e
I=e L=h X=t
Here e ra a e ha a ea tat a ra r ht
ethe eet e r a a e h h t a e e t a a e
V=r E=a Y=g
HereUpOnLeGrandAroseWithAGraveAndStatelyAirAndBrought
MeTheBeetleFromAGlassCaseInWhichItWasEnclosedItWasABe
14. 14
Example 3- Vigenere
“Multi-Caesar Cipher” – A statefull cipher
Key: k = (k1,k2,…,km) list of m numbers between 0 and 25
Encryption: 1st letter encoded as Caesar w/ key=k1 : i I + k1 (mod 26)
2nd letter encoded as Caesar w/ key=k2 : i I + k2 (mod 26)
mth letter encoded as Caesar w/ key=km : i I + km (mod 26)
m+1th letter encoded as Caesar w/ key=k1 : i I + k1 (mod 26)
Decryption: In the natural way
…
Important Property: Can no longer break using letter frequencies alone.
‘e’ will be mapped to ‘e’+k1,‘e’+k2,…,‘e’+km according to location.
nth letter encoded w/ key=k(n mod m) : i I + k(n mod m) (mod 26)
Considered “unbreakable” for 300 years (broken by Babbage, Kasiski 1850’s)
(Belaso, 1553)
15. 15
Example 3- Vigenere
“Multi-Caesar Cipher” – A statefull cipher
Key: k = (k1,k2,…,km) list of m numbers between 0 and 25
Encryption:
Breaking Vigenere:
nth letter encoded w/ key=k(n mod m) : i I + k(n mod m) (mod 26)
(Belaso, 1553)
LIVITCSWPIYVEWHEVSRIQMXLEYVEOIEWHRXEXIPFEM VEWHKV
Step 1: Guess the length of the key m
Step 2: Group together positions {1, m+1, 2m+1, 3m+1,…}
{m-1, 2m+m-1, 3m+m-1,…}
Decryption: In the natural way
…
{2, m+2, 2m+2, 3m+2,…}
16. 16
Example 3- Vigenere
“Multi-Caesar Cipher” – A statefull cipher
Key: k = (k1,k2,…,km) list of m numbers between 0 and 25
Encryption:
Breaking Vigenere:
nth letter encoded w/ key=k(n mod m) : i i + k(n mod m) (mod 26)
(Belaso, 1553)
LIVITC
SWPIYV
EWHEVS
RIQMXL
EYVEOI
EWHRXE
XIPFEM
VEWHKV
Step 1: Guess the length of the key m
Step 2: Group together positions 1, m+1, 2m+1, 3m+1,…
Step 3: Frequency-analyze each group independently.
Decryption: In the natural way
{m-1, 2m+m-1, 3m+m-1,…}
…
{2, m+2, 2m+2, 3m+2,…}
17. 17
Example 4 - The Enigma
A mechanical statefull cipher.
Roughly: composition of 3-5 substitution ciphers
implemented by wiring.
Wiring on rotors moving in different schedules,
making cipher statefull
Key: 1) Wiring of machine (changed infrequently)
2) Daily key from code books
3) New operator-chosen key for each message
Tools used by Poles & British to break Enigma:
1) Mathematical analysis combined w/ mechanical computers
2) Captured machines and code-books
3) German operators negligence
4) Known plaintext attacks (greetings, weather reports)
5) Chosen plaintext attacks
Used by Germany in WWII for top-secret communication.
18. 18
Post 1970’s Crypto
Two major developments:
1) Provably secure cryptography
Encryptions w/ mathematical proof that are unbreakable*
* Currently use conjectures/axioms,
however defeated all cryptanalysis effort so far.
2) Cryptography beyond “secret writing”
Public-key encryptions
Digital signatures
Zero-knowledge proofs
Anonymous electronic elections
Privacy-preserving data mining
e-cash
…
19. 19
Review of Encryption Schemes
Alice wants to send Bob a secret message.
Encryption algorithm: E
Decryption algorithm: D
Secret key: k
To encrypt m, Alice sends c = E(m,k) to Bob.
To decrypt c, Bob computes m’ = D(c,k).
c = E(m,k)
c
m’ = D(c,k)
Q: Can Bob send Alice the secret key over the net?
A: Of course not!! Eve could decrypt c!
Q: What if Bob could send Alice a “crippled key”
useful only for encryption but no help for decryption
20. 20
Public Key Cryptography [DH76,RSA77]
Alice wants to send Bob a secret message.
Encryption algorithm: E
Decryption algorithm: D
To encrypt m, Alice sends c = E(m,e) to Bob.
To decrypt c, Bob computes m’ = D(c,d).
c = E(m,e)
c
m’ = D(c,d)
Key: Bob chooses two keys: Secret key d for decrypting messages.
Public key e for encrypting messages.
choose d,e
e
Should be safe to send e “in the clear”!
A scheme is valid if m’=m
Intuitively, a scheme is secure if eavesdropper can not learn m from c.
Even if Eve knows the key e!
21. 21
Other Crypto Wonders
Digital Signatures. Electronically sign documents in unforgeable way.
Zero-knowledge proofs. Alice proves to Bob that she earns <$50K
without Bob learning her income.
Privacy-preserving data mining. Bob holds DB. Alice gets answer to one
query, without Bob knowing what she asked.
Playing poker over the net. Alice, Bob, Carol and David can play poker
over the net without trusting each other or any central server.
Distributed systems. Distribute sensitive data to 7 servers
s.t. as long as <3 are broken, no harm to security occurs.
Electronic auctions. Can run auctions s.t. no one (even not seller)
learns anything other than winning party and bid.
22. 22
Cryptography & Security
Prev slides: Have provably secure algorithm for every crypto task imaginable.
Q: How come nothing is secure?
A1: Not all of these are used or used correctly:
Strange tendency to use “home-brewed” cryptosystems.
Combining secure primitives in insecure way
Strict efficiency requirements for crypto/security:
Many provably secure algs not efficient enough
The cost is visible but benefit invisible.
Easy to get implementation wrong – many subtleties
Compatibility issues, legacy systems,
Misunderstanding properties of crypto components.
23. 23
Cryptography & Security
Prev slides: Have provably secure algorithm for every crypto task imaginable.
Q: How come nothing is secure?
A2: Cryptography is only part of designing secure systems
Chain is only as strong as weakest link.
A “dormant bug” is often a security hole.
Security is hard to “modularize”
Human element
(hard to add to existing system, changes in system
features can have unexpected consequences)
Many subtle issues (e.g., caching & virtual memory, side
channel attacks)
Key storage and protection issues.
Hinweis der Redaktion
1-32
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