Principles of Public Key Cryptography. Applications of Public Key Cryptography. Security in Public Key Algorithms

Size: px
Start display at page:

Download "Principles of Public Key Cryptography. Applications of Public Key Cryptography. Security in Public Key Algorithms"

Transcription

1 Principles of Public Key Cryptography Chapter : Security Techniques Background Secret Key Cryptography Public Key Cryptography Hash Functions Authentication Chapter : Security on Network and Transport Layer Chapter : Security on the Application Layer Chapter : Security Concepts for Networks.: Public Key Cryptography Principles of public key cryptography Number theory and algebraic foundations Classical public key cryptography Newer public key cryptography Also called asymmetric cryptography Different from secret key cryptography, algorithms for encoding and decoding differ considerably Working with two keys A private key d (known only to the owner) A public key e (known by possibly everyone) Public key cryptography principle (e.g. RSA): plaintext cipher text encryption public key e private key d decryption cipher text plaintext More easily configurable than secret key cryptography, but slower Often combined with secret key: authentication and distribution of a secret key (e.g. Diffie-Hellman for establishment of a shared secret) Page Page Applications of Public Key Cryptography Digital signatures (e.g. RSA, ElGamal, DSS) Associate a value with a message, like a checksum This value can only be generated by using the private key d ( decryption) It is readable for everyone knowing the public key e ( encryption) Similar to hand-written signature (authenticity without the chance to forge it) plaintext signed message signing private key d public key e verification signed message plaintext Authentication (zero knowledge proof systems) A generates a random number and encrypts it with the public key of B B decrypts the message with its private key and sends back the random number to A If A gets back the original random number, B is authenticated Security in Public Key Algorithms Security in many public key algorithms is based on the difficulty to factorise and compute discrete logarithms Factorising Find the prime factors for a given number One of the oldest problems in number theory, very time consuming Most popular method: Quadratic Sieve Discrete logarithm Problem to find the inverse to modular exponentiation: Find an x with a x b mod n for given a and b Not all discrete logarithms have solutions Very time consuming process to find solutions for big numbers Frequently used method: Index-Calculus method Page Page

2 Basics for Public Key Cryptography: Number Theory / Modular Arithmetic Number theory provides basic knowledge to understand how and why public key algorithms work Necessary concepts for understanding public key algorithms Most public key algorithms are based on modular arithmetic Modular arithmetic Operates on a ring (Z n, +, ), where Z n is a set of non-negative integers smaller than some positive integer n +: Z n Z n Z n is a function that is associative and commutative has a neutral element Z n has a inverse element x- to each x Z n, i.e. x + x- : Z n Z n Z n is an associative function (it is not necessarily commutative) + and have left and right exchangeability Needed for public key cryptography: addition, multiplication, exponentiation Computations of these functions are performed modulo n Arithmetic Operations modulo n Arithmetic computing modulo n Arithmetic operations are performed as usual, but the result is replaced by its remainder when divided by n (e.g. + 9 mod ) Modular addition Given: c x + k mod n, with c, x, k Zn if x + k < n : c a + b if x + k n : c j, where x + k i n + j and j < n Can be used to encrypt digits: each number x out of a range of numbers is unambiguously mapped onto another number c from this range Caesar Cipher: add a constant k to each number Decryption needs subtraction. This can be replaced by an addition of the inverse value Page Page Arithmetic Operations modulo n Modular multiplication * 7 9 Given: c x k mod n, with c, x, k Z n if x k < n : c x k 9 7 if x k n : c j, 7 9 where x k i n + j and j < n Encryption only works with special keys k Example for n : only k {,, 7, 9} is usable as (simple) cipher key only for these values the mapping is unambiguous for other values of k, an information loss occurs Only use keys k relatively prime to n k and n share no other common factor than Decryption works by multiplication of cipher text c with the multiplicative inverse k -, i.e. k k - mod n (e.g. 7 - mod, because 7 mod ) Multiplicative inverse for n only exists for,,7, and 9 Arithmetic Operations modulo n Modular exponential Given: c x k mod n, with c, x, k Z n if x k < n : c x k if x k n : c j, where x k i n + j and j < n Note: difference to modular multiplication: x k mod n x k+n mod n Encryption only works with special keys k Decryption needs an inverse k - with x k k- But: inverse k - does not exist in each case x y Page 7 Page

3 Finding Modular Inverses The Euclidean Algorithm Finding multiplicative inverses to x is a very time consuming process If x has digits, no Brute Force attack is possible Useful: x relatively prime to n a multiplicative inverse x- mod n exists Computing multiplicative inverse by the Euclidean Algorithm Euclidean algorithm Determines the greatest common divisor (gcd) of x and n Given x and n, it finds an y with x y mod n (if one exists) If x is relatively prime to n: gcd(x, n) Idea: Replace x and n with smaller numbers with the same gcd If one number becomes zero, the other one is the gcd Faster algorithm: the smaller the numbers are, the faster the computation of gcd is. Replace the bigger number with its remainder divided by the smaller number Example: gcd(, )? gcd(, -) gcd(,) gcd(,) gcd(,) gcd(,) gcd(,) The algorithm Note: gcd(, y) y In general: if d denotes a divisor of x and y x i d, y j d x - y i d - j d (i - j) d If x >, replace gcd(x, y) with gcd(x-y, y) Efficiency: x and y should be as small as possible Assume, d is the maximum of all divisors (achieved by division x mod y) gcd(x, y) gcd(x mod y, y) If y > x, exchange x and y function int gcd(int x, int y) begin int r x; int r y; int q; int help; while (r > ) begin q r / r; help r; r r % r // (r mod r) r help; end return r; end Page 9 Page Multiplicative Inverse by Euclidean Algorithm Computing the Multiplicative Inverse How to find a multiplicative inverse x - to x mod n, such that x x - mod n, with the euclidean algorithm? Multiplicative inverse for x mod n: a u exists with u x mod n u x differs from by a multiple of n There is a v with u x + v n Computing gcd(x, n) can compute such a v and a u, if gcd(x, n) If gcd(x, n), u is the multiplicative inverse to x Could there be more than one u mod n with u x mod n? Suppose: m x mod n m x u u mod n But u x mod n m u mod n m u Initialisation: u -, v -, u -, v -, r - x, r - y, i Repeat: if r n- gcd(x, y) r n- else divide r n- by r n- to get quotient q n and remainder r n Keep track of: u i u i- - q i u i-, v i v i- - q i v i- Example: r gcd(7, 9) r, multiplicative inverse u ( 7- mod 9) i q i r i u i v i Page Page

4 Finding Prime Numbers Problem with Euclidean algorithm: how to find x mod n with gcd(x, n)? Naive method: divide x by all numbers n Takes too long of your lifetime Practical solutions: there is no hundred percent that large number is prime But: there are tests for determining that a number is probably prime Use properties.) gcd(x, n), if x and n are relatively prime (x and n are relatively prime, if there are integers u and v with u x + v n ).) Φ(n), the totient function, denotes the number of integers relatively prime to n Page The Euler Function Φ(n) Computing Φ(n) If n is prime all numbers,..., n - are relatively prime to n Φ(n) n - If n is a product of primes p and q There are p q candidates {(j p + i q) i..q, j..p} for numbers relatively prime to n But from them, there are p multiples of q and q multiples of p (p + q - ) numbers are not relatively prime to n Φ(n) p q -(p + q -) (p -) (q -) i i + If n is a prime or a product of different primes x y mod n x y mod Φ(n) mod n Example for n ( ) Relatively prime to n: {,, 7, 9} Φ(n) ( - ) ( - ) Column i + is the same as column i Important special case: y mod Φ(n) for any x: x y x mod Φ(n) x mod n x y Page Euler's Theorem and Fermat's Theorem Euler's Theorem For any a relatively prime to n holds: a Φ(n) mod n If n is prime: Φ(n) n -. In this case: Fermat's Theorem If n is a prime and < a < n a n - mod n Good rule for determining primes But: what about n with a n - mod n, where n is no prime? Find primes by a simple prime test Choose an a with a < n and compute a n - mod n. If the result is not, n is no prime If the result is, n may be a prime (e.g., if n has digits, the probability for n to be no prime is - ) Prime Tests If the simple prime test fails: A cryptosystem like RSA might fail, a message cannot be decrypted An attacker might be able to compute keys easier "Solution": test n with other values for a Problem: Carmichael numbers (very rare) No primes, but for all a holds: a n - mod n Enhanced prime test is needed: Miller-Rabin prime test Improved method to find prime numbers Probabilistic prime test Basic foundation: for a prime n holds: Some Carmichael numbers: ) n - can always be expressed by b c, where c is an odd number.) Each square root (modulo n) of can only be ± (e.g. is a square root of mod, because mod, thus can not be a prime) Page Page

5 Miller-Rabin Algorithm Miller-Rabin Algorithm - Example Use Fermat's theorem: a n - mod n Pick a random number n and test if it is prime Test n with the division by smaller primes to speed up the process If you think a prime has been found: pick an a by random Miller-Rabin algorithm: compute r a c mod n if r mod n // is the first mod n-square root? n is prime // else: a n- only can become by squaring - in else for i to b - do // one of the b square operations if r - mod n // now: test on allowed square root. Because the n is prime // result before was not, it only can become else // by squaring -. Search for a - r r mod n // prepare testing the next square root n is not prime // only non-allowed square roots found Choose n as a possible prime n - 7 b, c 7 Pick randomly a Compute a c 7 7 mod (this is not nor -, and: mod ) no prime found Other variant: pick randomly a Compute 7 mod (this is not nor -, and: mod ) This means, is a square root of mod no prime found Choose n as a possible prime n - b, c Pick randomly a Compute mod Compute - mod - is an allowed square root of, thus is (possibly) prime Other try: pick randomly a compute 7 mod is (possibly) prime Page 7 Page Classical Public Key Cryptography RSA Developed by Rivest, Shamir, and Adleman RSA Public-key cryptography standard (PKCS) Rabin cryptosystem Diffie-Hellman cryptosystem ElGamal cryptosystem Merkle-Hellman cryptosystem Purpose: encryption and decryption of data Variable key length Long key used for high security needs Short key used for efficient encryption processes Common key length: bit Variable plaintext length Must be shorter than the key Cipher text blocks Length of the key Much slower than secret key algorithms like DES or IDEA Only used for short messages Important purpose: transmission of secret keys Page 9 Page

6 RSA Key Generation Usage Scenarios for RSA Generate a public key and a corresponding private key.) Choose two large primes p and q of bit each (p and q must be a secret!).) Compute n p q.) Compute Φ(n) (p -) (q -).) Choose e relatively prime to Φ(n).) Find d with d e mod Φ(n) (d is the multiplicative inverse to e) <e, n> is public key <d, n> is private key Why do these keys work? We use modular arithmetic (mod n) with p q n d and e were chosen to be d e mod Φ(n) Because n is product of distinct primes, for all x: x d e x mod Φ(n) x mod n n is public, but factorisation into p and q is computationally infeasible Encryption and decryption Encrypt message m using the public key of the receiver: c m e mod n Decrypt cipher text c with the private key of the receiver: m c d mod n Digital signatures Similar to encryption/decryption process Sender encrypts message m with his private key: s m d mod n Each receiver can read the signed message using the public key of the sender: m s e mod n Page Page Why is RSA (relatively) secure? How to determine p, q, e and d Breaking RSA means finding d from knowing e and n Attacker only knows: d is the exponential inverse to e mod Φ(n) Simple approach: knowing p and q you can compute Φ(n) (this is a kind of trapdoor) However: an attacker does not know p and q Attacker needs to factorise n to obtain p and q Factorising large numbers is difficult The best algorithms are too slow And: Brute Force attack is less efficient than factorising But it is possible to misuse RSA! Assume that an attacker knows the context of a message from A The attacker could encrypt messages with the public key e A If a match is found, the attacker has found the message.) Finding big primes p and q For a -digit number, the chance of finding a prime is in For a -digit number, the chance is only in Pick random numbers until you find a prime Use Fermat's theorem and the Rabin-Miller algorithm to test if a random number is prime.) Finding d and e for p and q Choose e as relatively prime to (p - ) (q -) a.) by choosing e at random and test if it is relatively prime to (p - ) (q -) b.) by choosing e first and then determine matching p and q RSA is not less secure if always the same e is chosen If e is small or its binary representation has few ''s, the operations for encryption and signature verification will become much more efficient Use Euclidean algorithm to determine d with e d mod Φ(n) Note: do not choose a small d; d is a secret, thus it should be hard to determine Page Page

7 Using small public keys Let e be a small constant Public key operations become faster, while leaving private key operations unchanged Popular values for e are and 7 Case of e Maximizes performance Apparently it does not weaken security of RSA (when some practical constraints on its use are considered) Problems with e Small messages m with m mod n m. Problem: it only takes the cubic root to decrypt Solution: padding message with a random number before encryption If a message is sent to or more receivers, m can be derived from the three encrypted values and the public keys of the receivers Find p and q so that is relatively prime to (p -) (q ) (practical problem: there are many numbers which is not relatively prime to) Using small public keys Case of e 7 Is equivalent to +, and it is prime The binary representation contains only two s Only 7 multiplications are necessary to to compute any m e Much faster than the 7 (on the average) multiplications necessary for a randomly chosen bit value The problems mentioned for e are avoided Page Page Public Key Cryptography Standard (PKCS) Example: PKCS# How could different implementations interwork? Standards for encoding of information that will be encrypted or signed Public Key Cryptography Standard Set of standards PKCS# - PKCS#9 Definition of encoding RSA public keys, RSA private keys, RSA signatures, short RSA-encrypted messages (typically secret keys), and short RSA-signed messages (typically a message digest) Designed to deal with Encrypting guessable messages Signing smooth numbers Multiple recipients of a message for e For e, encrypting messages that are less than a third of the length of n For e, signing messages where the information is in the high-order part PKCS# (encryption) Standard format for messages to be encrypted with RSA Consists of Preceding : the message remains smaller than the modulus : denotes a message which is to be encrypted Random bytes (padding): Each byte is chosen independently to make it harder to guess the message Independent padding for each recipient Make message long enough to avoid problems with m < n for e Next : marks the beginning data random non-zero bytes data Page 7 Page

8 Example: PKCS# PKCS# (signature) Standard format for messages to be signed with RSA Data are typically a Message Digest of Bit Padding is required Consists of: Preceding : the message remains smaller than the modulus : denotes a message which is to be signed Random bytes (padding): make the data bigger than byte Next : marks the begin of data Digest type standardises, how to tell another party which digest function was used Rabin Cryptosystem Rabin cryptosystem Secure because of the difficulty to find square roots modulo a composite number Nearly as difficult as factorising large numbers Rabin algorithm Choose primes p and q, both congruent to mod p and q form the private key n p q is the public key Encryption of message m in the range {,..., n -} c m mod n bytes of ff digest type and message digest Page 9 Page Decryption in the Rabin Cryptosystem Diffie-Hellman Cryptosystem Decryption is more complex Receiver knows p and q Solve the two congruencies using the so-called Chinese remainder problem Compute:t c (p + ) / mod p t p - c (p + ) / mod p t c (q + ) / mod q t q - c (q + ) / mod q Choose integers a q (q - mod p) and b p (p - mod q) Possible solutions are m (a t + b t ) mod n m (a t + b t ) mod n m (a t + b t ) mod n m (a t + b t ) mod n One of these results equals m If m is normal text, it is no problem to find the right m i Otherwise, add a known header to m before encryption Page Oldest public key cryptosystem Offers better performance than RSA Less general than RSA (does neither encryption nor signatures) Purpose: two persons can agree upon a secret number (e.g. a shared key), which cannot be computed by intercepting the publicly exchanged messages After the exchange of two public messages both communication partners know a secret number Having agreed on a secret number, they can use e.g. DES for communication Diffie-Hellman actually used for key establishment Remaining problem: no authentication between the partners Page

9 Diffie-Hellman Algorithm Algorithm for key establishment Choose a prime p with bit Choose a number g < p with some restrictions p and g are public! A randomly chooses a bit number S a and computes T a g Sa mod p B randomly chooses a bit number S b and computes T b g Sb mod p S a and S b are secret Aand B exchange T a and T b A computes k AB T Sa b mod p g Sa Sb mod p B computes k AB T Sb a mod p g Sa Sb mod p A and B both compute the same secret key gsa Sb Note: It is impossible to compute g Sa Sb fast enough knowing only T a and T b due to the difficulty to compute discrete logarithms, i.e. to compute S a from knowing g Sa Bucket-Brigade Attack on Diffie-Hellman Problem in Diffie-Hellman: no authentication between Aand B If A obtains T b, he cannot know for sure if B has sent it Bucket-Brigade attack An attacker O obtains T a and establishes a common secret with A Attack method: p and q are known publicly A sends g So to O (but believes it is sent to B) Ocomputes g Sx and sends it to B B computes g Sb and sends it to O O sends g So back to A O establishes k AO and k BO Aand B communicate via O A g Sa 9 9 Diffie-Hellman is only secure against passive attacks (i.e. just watching messages) Protection against active attacks: use trustful and public location to publish g Si for all persons I in advance O g So B g Sb shared key k AO 7 Sa 9 So shared key k BO 97 So 7 Sb Page Page Diffie-Hellman for Encryption ElGamal Cryptosystem Encryption algorithm using Diffie-Hellman Each participant chooses a private key S i Each participant computes a public key <p, g, T i > with T i g Si mod p Publish all public keys at a trusted public place Assume, B publishes <p, g, T b > A computes k AB T Sa b mod p A uses k AB as secret key with B to compute a cipher text A transmits the cipher text and g Sa mod p to B B computes k AB to decrypt the message The secret key is transmitted only together with the message For a better security, p and g should have these properties: p should be a strong prime number, i.e. (p-)/ is prime, too It is desirable to have g x mod p, x mod (p - ) [if p is a strong prime number, this is true for all g - mod p with g (p - ) / - mod p) But: this is a costly way for choosing p and g! Mainly used for digital signatures Secure because of the difficulty to calculate discrete logarithms in a finite field Uses same kind of key as Diffie-Hellman Additionally provides a scheme for signatures Each person has a long-time key public key: <g, p, T> private key: S with g S mod p T For each message m to be signed, a new key pair S m, <g, p, T m > has to be generated For the message m to be signed, compute a message digest d m MD(m T m ) Compute the signature X S m + d m S mod (p - ) Transmit m together with X and T m To verify signature, compute g X, d m, and T m T dm mod p Check: g X g Sm + dm S g Sm g dm S T m T dm mod p Page Page

10 Digital Signature Standard (DSS) Digital Signatures with DSS DSS algorithm is called Digital Signature Algorithm (DSA) Algorithm to create digital signatures based on ElGamal Difference to ElGamal is the speed of operations ( times faster): Instead using a p of bit, for some operations only use a prime q of bit, for which holds: p k p + Note: using ElGamal means to generate a key pair <S m, T m > for each message m which has to be signed If a pair of keys is used only for two different messages, it would expose the signer's private key: With only two uses, S m can be deducted By knowing S m, the secret key S easily can be computed Page 7 Digital Signature Algorithm Digital Signature Algorithm Generate and publish a -bit prime p and a -bit prime q with p k q + Generate and publish a g with g q mod p (use Fermat's theorem) Note: g must not be Generate a long-term public/private key pair <T, S> as in ElGamal For each message m generate a separate key pair <T m, S m > by choosing S m and compute T m ((g Sm mod p) mod q) For m, compute the message digest d m Compute the signature X S - m (d m + S T m ) mod q Transmit m, T m, and X Signature verification Calculate the mod q inverse of the signature, X - Calculate the message digest d m Calculate x d m X - mod q and y T m X - mod q Calculate z (g x T y mod p ) mod q If z T m, the signature is verified Page Merkle-Hellman Cryptosystem Merkle-Hellman in Cryptography Knapsack Problem Pack a knapsack optimally with n objects of different weights a,..., a n and overall size g n Search for an order (k i ), k i {, } for i,..., n with ai ki g i This is an NP hard problem Merkle-Hellman cryptosystem Based on the knapsack problem Special type of knapsack problem: The sizes of the objects form a fast growing sequence with a There is a solution in O(n): Start with the biggest object and find a new smaller knapsack with one object less j+ i > ai j Page 9 Principle: Use a simple Knapsack problem as private key and transform it into a hard one which is used as public key. A message m (m, m,..., m n,...) is seen as a solution for the problem, i.e. if m i, m i is in the knapsack Example: A chooses a Knapsack problem a with a (a i ) (,, 9,,,,,, 9) as key A chooses a prime p and a number k 9 A generates a hard Knapsack problem e (e i ) with e i k a i mod p e (7,,,, 9, 9, 7,, 7) B encrypts a message m (,,,,,,,, ) to A by using e c (this value is transmitted) A computes g k - c mod p 7 mod A solves for (a i ) by choosing the biggest fitting number in (a i ) till is reached: (,, 9,,,,,, 9) Chapter The.: original Public message Key Cryptography is given by the elements of a Page

11 Modern Public Key Cryptosystems Elliptic Curves Definition Classic public key cryptosystems are well analysed The performance of classic public-key cryptosystems is acceptable Security: classic public key cryptosystems are not perfectly secure, but computationally secure Modern public key cryptosystems improve the classic ones: Performance: modern public key cryptosystems have a better performance than the classic ones Security: modern public key cryptosystems also offer better security (with the same key length) Example: Elliptic Curve Cryptosystem Provide security equivalent to classical public key schemes Shorter key lengths, resulting in faster computing, less complex chips Definition: Let p > be prime. The elliptic curve y x + ax + b over Z p is the set of solutions (x,y) Z p Z p for the congruence y x + ax + b (mod p), where a, b Z p are constants, so that a + 7 b O (mod p), together with a special point O called the point of infinity. Page Page Addition Operation Elliptic Curve - Example Points on the elliptic curve E: y x +x+ in Z Let E be an elliptic curve over Z p, P (x, y ), Q (x, y ). If x x and y -y, then Q -P, P + Q : O; otherwise P + Q : (x, y ), with x λ x x y λ(x x ) y and y - y, if P Q x - x λ x + a, if P Q y Finally, P + O O + P P. x 7 9 x +x+ mod 9 7 in QR()? no no yes yes no yes no yes yes no y [no solution], 7,, 9, 9, E {O, (,), (,7), (,), (,), (,), (,9), (7,), (7,9), (,), (,), (,), (,9)} Let α (,7). Then α is a primitive element: α α α 7α 9α α (,7) (,) (,) (7,) (,9) (,9) α α α α α α (,) α + α (,) (7,9) (,) (,) (,) yes, 9 i.e. (x,y) (,) and (x,y) (,) are points on the elliptic curve Page Page

12 Example: ElGamal Encryption with Elliptic Curves Let α (,7) and a, so β α (,) (a is secret and for large numbers can t be obtained from α and β in reasonable time) The encryption operation is e k (x,r) (r α, x + r β) (y, y ), e k (x,r) (r (,7), x + r (,)), where x E and r and the decryption operation is d k (y,y ) y ay y y Alice wants to send x (7,9) to Bob; she chooses the random value r 7. She then computes y 7(,7) (7,) y (7,9) + 7(,) (7,9) + (,9) (,) Bob receives y ((7,),(,)) and obtains x (,) (7,) (,) (,9) (,) + (,) (7,9) Page

Public Key Cryptography: RSA and Lots of Number Theory

Public Key Cryptography: RSA and Lots of Number Theory Public Key Cryptography: RSA and Lots of Number Theory Public vs. Private-Key Cryptography We have just discussed traditional symmetric cryptography: Uses a single key shared between sender and receiver

More information

Elements of Applied Cryptography Public key encryption

Elements of Applied Cryptography Public key encryption Network Security Elements of Applied Cryptography Public key encryption Public key cryptosystem RSA and the factorization problem RSA in practice Other asymmetric ciphers Asymmetric Encryption Scheme Let

More information

Discrete Mathematics, Chapter 4: Number Theory and Cryptography

Discrete Mathematics, Chapter 4: Number Theory and Cryptography Discrete Mathematics, Chapter 4: Number Theory and Cryptography Richard Mayr University of Edinburgh, UK Richard Mayr (University of Edinburgh, UK) Discrete Mathematics. Chapter 4 1 / 35 Outline 1 Divisibility

More information

CSCE 465 Computer & Network Security

CSCE 465 Computer & Network Security CSCE 465 Computer & Network Security Instructor: Dr. Guofei Gu http://courses.cse.tamu.edu/guofei/csce465/ Public Key Cryptogrophy 1 Roadmap Introduction RSA Diffie-Hellman Key Exchange Public key and

More information

Public Key Cryptography and RSA. Review: Number Theory Basics

Public Key Cryptography and RSA. Review: Number Theory Basics Public Key Cryptography and RSA Murat Kantarcioglu Based on Prof. Ninghui Li s Slides Review: Number Theory Basics Definition An integer n > 1 is called a prime number if its positive divisors are 1 and

More information

Outline. Computer Science 418. Digital Signatures: Observations. Digital Signatures: Definition. Definition 1 (Digital signature) Digital Signatures

Outline. Computer Science 418. Digital Signatures: Observations. Digital Signatures: Definition. Definition 1 (Digital signature) Digital Signatures Outline Computer Science 418 Digital Signatures Mike Jacobson Department of Computer Science University of Calgary Week 12 1 Digital Signatures 2 Signatures via Public Key Cryptosystems 3 Provable 4 Mike

More information

Overview of Public-Key Cryptography

Overview of Public-Key Cryptography CS 361S Overview of Public-Key Cryptography Vitaly Shmatikov slide 1 Reading Assignment Kaufman 6.1-6 slide 2 Public-Key Cryptography public key public key? private key Alice Bob Given: Everybody knows

More information

RSA and Primality Testing

RSA and Primality Testing and Primality Testing Joan Boyar, IMADA, University of Southern Denmark Studieretningsprojekter 2010 1 / 81 Correctness of cryptography cryptography Introduction to number theory Correctness of with 2

More information

Study of algorithms for factoring integers and computing discrete logarithms

Study of algorithms for factoring integers and computing discrete logarithms Study of algorithms for factoring integers and computing discrete logarithms First Indo-French Workshop on Cryptography and Related Topics (IFW 2007) June 11 13, 2007 Paris, France Dr. Abhijit Das Department

More information

The application of prime numbers to RSA encryption

The application of prime numbers to RSA encryption The application of prime numbers to RSA encryption Prime number definition: Let us begin with the definition of a prime number p The number p, which is a member of the set of natural numbers N, is considered

More information

Cryptography and Network Security

Cryptography and Network Security Cryptography and Network Security Fifth Edition by William Stallings Chapter 9 Public Key Cryptography and RSA Private-Key Cryptography traditional private/secret/single key cryptography uses one key shared

More information

Lecture Note 5 PUBLIC-KEY CRYPTOGRAPHY. Sourav Mukhopadhyay

Lecture Note 5 PUBLIC-KEY CRYPTOGRAPHY. Sourav Mukhopadhyay Lecture Note 5 PUBLIC-KEY CRYPTOGRAPHY Sourav Mukhopadhyay Cryptography and Network Security - MA61027 Modern/Public-key cryptography started in 1976 with the publication of the following paper. W. Diffie

More information

Digital Signature. Raj Jain. Washington University in St. Louis

Digital Signature. Raj Jain. Washington University in St. Louis Digital Signature Raj Jain Washington University in Saint Louis Saint Louis, MO 63130 Jain@cse.wustl.edu Audio/Video recordings of this lecture are available at: http://www.cse.wustl.edu/~jain/cse571-11/

More information

CIS 6930 Emerging Topics in Network Security. Topic 2. Network Security Primitives

CIS 6930 Emerging Topics in Network Security. Topic 2. Network Security Primitives CIS 6930 Emerging Topics in Network Security Topic 2. Network Security Primitives 1 Outline Absolute basics Encryption/Decryption; Digital signatures; D-H key exchange; Hash functions; Application of hash

More information

Number Theory. Proof. Suppose otherwise. Then there would be a finite number n of primes, which we may

Number Theory. Proof. Suppose otherwise. Then there would be a finite number n of primes, which we may Number Theory Divisibility and Primes Definition. If a and b are integers and there is some integer c such that a = b c, then we say that b divides a or is a factor or divisor of a and write b a. Definition

More information

Public Key Cryptography. c Eli Biham - March 30, 2011 258 Public Key Cryptography

Public Key Cryptography. c Eli Biham - March 30, 2011 258 Public Key Cryptography Public Key Cryptography c Eli Biham - March 30, 2011 258 Public Key Cryptography Key Exchange All the ciphers mentioned previously require keys known a-priori to all the users, before they can encrypt

More information

The Mathematics of the RSA Public-Key Cryptosystem

The Mathematics of the RSA Public-Key Cryptosystem The Mathematics of the RSA Public-Key Cryptosystem Burt Kaliski RSA Laboratories ABOUT THE AUTHOR: Dr Burt Kaliski is a computer scientist whose involvement with the security industry has been through

More information

Software Implementation of Gong-Harn Public-key Cryptosystem and Analysis

Software Implementation of Gong-Harn Public-key Cryptosystem and Analysis Software Implementation of Gong-Harn Public-key Cryptosystem and Analysis by Susana Sin A thesis presented to the University of Waterloo in fulfilment of the thesis requirement for the degree of Master

More information

RSA Encryption. Tom Davis tomrdavis@earthlink.net http://www.geometer.org/mathcircles October 10, 2003

RSA Encryption. Tom Davis tomrdavis@earthlink.net http://www.geometer.org/mathcircles October 10, 2003 RSA Encryption Tom Davis tomrdavis@earthlink.net http://www.geometer.org/mathcircles October 10, 2003 1 Public Key Cryptography One of the biggest problems in cryptography is the distribution of keys.

More information

Table of Contents. Bibliografische Informationen http://d-nb.info/996514864. digitalisiert durch

Table of Contents. Bibliografische Informationen http://d-nb.info/996514864. digitalisiert durch 1 Introduction to Cryptography and Data Security 1 1.1 Overview of Cryptology (and This Book) 2 1.2 Symmetric Cryptography 4 1.2.1 Basics 4 1.2.2 Simple Symmetric Encryption: The Substitution Cipher...

More information

Applied Cryptography Public Key Algorithms

Applied Cryptography Public Key Algorithms Applied Cryptography Public Key Algorithms Sape J. Mullender Huygens Systems Research Laboratory Universiteit Twente Enschede 1 Public Key Cryptography Independently invented by Whitfield Diffie & Martin

More information

A Factoring and Discrete Logarithm based Cryptosystem

A Factoring and Discrete Logarithm based Cryptosystem Int. J. Contemp. Math. Sciences, Vol. 8, 2013, no. 11, 511-517 HIKARI Ltd, www.m-hikari.com A Factoring and Discrete Logarithm based Cryptosystem Abdoul Aziz Ciss and Ahmed Youssef Ecole doctorale de Mathematiques

More information

Notes on Network Security Prof. Hemant K. Soni

Notes on Network Security Prof. Hemant K. Soni Chapter 9 Public Key Cryptography and RSA Private-Key Cryptography traditional private/secret/single key cryptography uses one key shared by both sender and receiver if this key is disclosed communications

More information

NEW DIGITAL SIGNATURE PROTOCOL BASED ON ELLIPTIC CURVES

NEW DIGITAL SIGNATURE PROTOCOL BASED ON ELLIPTIC CURVES NEW DIGITAL SIGNATURE PROTOCOL BASED ON ELLIPTIC CURVES Ounasser Abid 1, Jaouad Ettanfouhi 2 and Omar Khadir 3 1,2,3 Laboratory of Mathematics, Cryptography and Mechanics, Department of Mathematics, Fstm,

More information

RSA Attacks. By Abdulaziz Alrasheed and Fatima

RSA Attacks. By Abdulaziz Alrasheed and Fatima RSA Attacks By Abdulaziz Alrasheed and Fatima 1 Introduction Invented by Ron Rivest, Adi Shamir, and Len Adleman [1], the RSA cryptosystem was first revealed in the August 1977 issue of Scientific American.

More information

Computer Security: Principles and Practice

Computer Security: Principles and Practice Computer Security: Principles and Practice Chapter 20 Public-Key Cryptography and Message Authentication First Edition by William Stallings and Lawrie Brown Lecture slides by Lawrie Brown Public-Key Cryptography

More information

Digital Signatures. (Note that authentication of sender is also achieved by MACs.) Scan your handwritten signature and append it to the document?

Digital Signatures. (Note that authentication of sender is also achieved by MACs.) Scan your handwritten signature and append it to the document? Cryptography Digital Signatures Professor: Marius Zimand Digital signatures are meant to realize authentication of the sender nonrepudiation (Note that authentication of sender is also achieved by MACs.)

More information

Lecture 13 - Basic Number Theory.

Lecture 13 - Basic Number Theory. Lecture 13 - Basic Number Theory. Boaz Barak March 22, 2010 Divisibility and primes Unless mentioned otherwise throughout this lecture all numbers are non-negative integers. We say that A divides B, denoted

More information

Factoring Algorithms

Factoring Algorithms Factoring Algorithms The p 1 Method and Quadratic Sieve November 17, 2008 () Factoring Algorithms November 17, 2008 1 / 12 Fermat s factoring method Fermat made the observation that if n has two factors

More information

A SOFTWARE COMPARISON OF RSA AND ECC

A SOFTWARE COMPARISON OF RSA AND ECC International Journal Of Computer Science And Applications Vol. 2, No. 1, April / May 29 ISSN: 974-13 A SOFTWARE COMPARISON OF RSA AND ECC Vivek B. Kute Lecturer. CSE Department, SVPCET, Nagpur 9975549138

More information

Cryptography and Network Security Chapter 9

Cryptography and Network Security Chapter 9 Cryptography and Network Security Chapter 9 Fifth Edition by William Stallings Lecture slides by Lawrie Brown (with edits by RHB) Chapter 9 Public Key Cryptography and RSA Every Egyptian received two names,

More information

Digital signatures. Informal properties

Digital signatures. Informal properties Digital signatures Informal properties Definition. A digital signature is a number dependent on some secret known only to the signer and, additionally, on the content of the message being signed Property.

More information

RSA Question 2. Bob thinks that p and q are primes but p isn t. Then, Bob thinks Φ Bob :=(p-1)(q-1) = φ(n). Is this true?

RSA Question 2. Bob thinks that p and q are primes but p isn t. Then, Bob thinks Φ Bob :=(p-1)(q-1) = φ(n). Is this true? RSA Question 2 Bob thinks that p and q are primes but p isn t. Then, Bob thinks Φ Bob :=(p-1)(q-1) = φ(n). Is this true? Bob chooses a random e (1 < e < Φ Bob ) such that gcd(e,φ Bob )=1. Then, d = e -1

More information

1720 - Forward Secrecy: How to Secure SSL from Attacks by Government Agencies

1720 - Forward Secrecy: How to Secure SSL from Attacks by Government Agencies 1720 - Forward Secrecy: How to Secure SSL from Attacks by Government Agencies Dave Corbett Technical Product Manager Implementing Forward Secrecy 1 Agenda Part 1: Introduction Why is Forward Secrecy important?

More information

Public Key (asymmetric) Cryptography

Public Key (asymmetric) Cryptography Public-Key Cryptography UNIVERSITA DEGLI STUDI DI PARMA Dipartimento di Ingegneria dell Informazione Public Key (asymmetric) Cryptography Luca Veltri (mail.to: luca.veltri@unipr.it) Course of Network Security,

More information

U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009. Notes on Algebra

U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009. Notes on Algebra U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009 Notes on Algebra These notes contain as little theory as possible, and most results are stated without proof. Any introductory

More information

159.334 Computer Networks. Network Security 1. Professor Richard Harris School of Engineering and Advanced Technology

159.334 Computer Networks. Network Security 1. Professor Richard Harris School of Engineering and Advanced Technology Network Security 1 Professor Richard Harris School of Engineering and Advanced Technology Presentation Outline Overview of Identification and Authentication The importance of identification and Authentication

More information

How To Know If A Message Is From A Person Or A Machine

How To Know If A Message Is From A Person Or A Machine The RSA Algorithm Evgeny Milanov 3 June 2009 In 1978, Ron Rivest, Adi Shamir, and Leonard Adleman introduced a cryptographic algorithm, which was essentially to replace the less secure National Bureau

More information

Network Security. Computer Networking Lecture 08. March 19, 2012. HKU SPACE Community College. HKU SPACE CC CN Lecture 08 1/23

Network Security. Computer Networking Lecture 08. March 19, 2012. HKU SPACE Community College. HKU SPACE CC CN Lecture 08 1/23 Network Security Computer Networking Lecture 08 HKU SPACE Community College March 19, 2012 HKU SPACE CC CN Lecture 08 1/23 Outline Introduction Cryptography Algorithms Secret Key Algorithm Message Digest

More information

Lukasz Pater CMMS Administrator and Developer

Lukasz Pater CMMS Administrator and Developer Lukasz Pater CMMS Administrator and Developer EDMS 1373428 Agenda Introduction Why do we need asymmetric ciphers? One-way functions RSA Cipher Message Integrity Examples Secure Socket Layer Single Sign

More information

CIS 5371 Cryptography. 8. Encryption --

CIS 5371 Cryptography. 8. Encryption -- CIS 5371 Cryptography p y 8. Encryption -- Asymmetric Techniques Textbook encryption algorithms In this chapter, security (confidentiality) is considered in the following sense: All-or-nothing secrecy.

More information

Cryptography and Network Security Department of Computer Science and Engineering Indian Institute of Technology Kharagpur

Cryptography and Network Security Department of Computer Science and Engineering Indian Institute of Technology Kharagpur Cryptography and Network Security Department of Computer Science and Engineering Indian Institute of Technology Kharagpur Module No. # 01 Lecture No. # 05 Classic Cryptosystems (Refer Slide Time: 00:42)

More information

Advanced Cryptography

Advanced Cryptography Family Name:... First Name:... Section:... Advanced Cryptography Final Exam July 18 th, 2006 Start at 9:15, End at 12:00 This document consists of 12 pages. Instructions Electronic devices are not allowed.

More information

Network Security. Chapter 2 Basics 2.2 Public Key Cryptography. Public Key Cryptography. Public Key Cryptography

Network Security. Chapter 2 Basics 2.2 Public Key Cryptography. Public Key Cryptography. Public Key Cryptography Chair for Network Architectures and Services Department of Informatics TU München Prof. Carle Encryption/Decryption using Public Key Cryptography Network Security Chapter 2 Basics 2.2 Public Key Cryptography

More information

CS 758: Cryptography / Network Security

CS 758: Cryptography / Network Security CS 758: Cryptography / Network Security offered in the Fall Semester, 2003, by Doug Stinson my office: DC 3122 my email address: dstinson@uwaterloo.ca my web page: http://cacr.math.uwaterloo.ca/~dstinson/index.html

More information

SECURITY IMPROVMENTS TO THE DIFFIE-HELLMAN SCHEMES

SECURITY IMPROVMENTS TO THE DIFFIE-HELLMAN SCHEMES www.arpapress.com/volumes/vol8issue1/ijrras_8_1_10.pdf SECURITY IMPROVMENTS TO THE DIFFIE-HELLMAN SCHEMES Malek Jakob Kakish Amman Arab University, Department of Computer Information Systems, P.O.Box 2234,

More information

Lecture 6 - Cryptography

Lecture 6 - Cryptography Lecture 6 - Cryptography CSE497b - Spring 2007 Introduction Computer and Network Security Professor Jaeger www.cse.psu.edu/~tjaeger/cse497b-s07 Question 2 Setup: Assume you and I don t know anything about

More information

Digital Signatures. Meka N.L.Sneha. Indiana State University. nmeka@sycamores.indstate.edu. October 2015

Digital Signatures. Meka N.L.Sneha. Indiana State University. nmeka@sycamores.indstate.edu. October 2015 Digital Signatures Meka N.L.Sneha Indiana State University nmeka@sycamores.indstate.edu October 2015 1 Introduction Digital Signatures are the most trusted way to get documents signed online. A digital

More information

EXAM questions for the course TTM4135 - Information Security May 2013. Part 1

EXAM questions for the course TTM4135 - Information Security May 2013. Part 1 EXAM questions for the course TTM4135 - Information Security May 2013 Part 1 This part consists of 5 questions all from one common topic. The number of maximal points for every correctly answered question

More information

Public Key Cryptography Overview

Public Key Cryptography Overview Ch.20 Public-Key Cryptography and Message Authentication I will talk about it later in this class Final: Wen (5/13) 1630-1830 HOLM 248» give you a sample exam» Mostly similar to homeworks» no electronic

More information

Cryptography and Network Security Chapter 10

Cryptography and Network Security Chapter 10 Cryptography and Network Security Chapter 10 Fifth Edition by William Stallings Lecture slides by Lawrie Brown (with edits by RHB) Chapter 10 Other Public Key Cryptosystems Amongst the tribes of Central

More information

ECE 842 Report Implementation of Elliptic Curve Cryptography

ECE 842 Report Implementation of Elliptic Curve Cryptography ECE 842 Report Implementation of Elliptic Curve Cryptography Wei-Yang Lin December 15, 2004 Abstract The aim of this report is to illustrate the issues in implementing a practical elliptic curve cryptographic

More information

LUC: A New Public Key System

LUC: A New Public Key System LUC: A New Public Key System Peter J. Smith a and Michael J. J. Lennon b a LUC Partners, Auckland UniServices Ltd, The University of Auckland, Private Bag 92019, Auckland, New Zealand. b Department of

More information

Cryptography and Network Security Number Theory

Cryptography and Network Security Number Theory Cryptography and Network Security Number Theory Xiang-Yang Li Introduction to Number Theory Divisors b a if a=mb for an integer m b a and c b then c a b g and b h then b (mg+nh) for any int. m,n Prime

More information

2. Cryptography 2.4 Digital Signatures

2. Cryptography 2.4 Digital Signatures DI-FCT-UNL Computer and Network Systems Security Segurança de Sistemas e Redes de Computadores 2010-2011 2. Cryptography 2.4 Digital Signatures 2010, Henrique J. Domingos, DI/FCT/UNL 2.4 Digital Signatures

More information

Integer Factorization using the Quadratic Sieve

Integer Factorization using the Quadratic Sieve Integer Factorization using the Quadratic Sieve Chad Seibert* Division of Science and Mathematics University of Minnesota, Morris Morris, MN 56567 seib0060@morris.umn.edu March 16, 2011 Abstract We give

More information

Math 319 Problem Set #3 Solution 21 February 2002

Math 319 Problem Set #3 Solution 21 February 2002 Math 319 Problem Set #3 Solution 21 February 2002 1. ( 2.1, problem 15) Find integers a 1, a 2, a 3, a 4, a 5 such that every integer x satisfies at least one of the congruences x a 1 (mod 2), x a 2 (mod

More information

CS549: Cryptography and Network Security

CS549: Cryptography and Network Security CS549: Cryptography and Network Security by Xiang-Yang Li Department of Computer Science, IIT Cryptography and Network Security 1 Notice This lecture note (Cryptography and Network Security) is prepared

More information

CHAPTER 5. Number Theory. 1. Integers and Division. Discussion

CHAPTER 5. Number Theory. 1. Integers and Division. Discussion CHAPTER 5 Number Theory 1. Integers and Division 1.1. Divisibility. Definition 1.1.1. Given two integers a and b we say a divides b if there is an integer c such that b = ac. If a divides b, we write a

More information

Elliptic Curve Cryptography Methods Debbie Roser Math\CS 4890

Elliptic Curve Cryptography Methods Debbie Roser Math\CS 4890 Elliptic Curve Cryptography Methods Debbie Roser Math\CS 4890 Why are Elliptic Curves used in Cryptography? The answer to this question is the following: 1) Elliptic Curves provide security equivalent

More information

Secure Network Communication Part II II Public Key Cryptography. Public Key Cryptography

Secure Network Communication Part II II Public Key Cryptography. Public Key Cryptography Kommunikationssysteme (KSy) - Block 8 Secure Network Communication Part II II Public Key Cryptography Dr. Andreas Steffen 2000-2001 A. Steffen, 28.03.2001, KSy_RSA.ppt 1 Secure Key Distribution Problem

More information

Index Calculation Attacks on RSA Signature and Encryption

Index Calculation Attacks on RSA Signature and Encryption Index Calculation Attacks on RSA Signature and Encryption Jean-Sébastien Coron 1, Yvo Desmedt 2, David Naccache 1, Andrew Odlyzko 3, and Julien P. Stern 4 1 Gemplus Card International {jean-sebastien.coron,david.naccache}@gemplus.com

More information

An Introduction to the RSA Encryption Method

An Introduction to the RSA Encryption Method April 17, 2012 Outline 1 History 2 3 4 5 History RSA stands for Rivest, Shamir, and Adelman, the last names of the designers It was first published in 1978 as one of the first public-key crytographic systems

More information

1. The RSA algorithm In this chapter, we ll learn how the RSA algorithm works.

1. The RSA algorithm In this chapter, we ll learn how the RSA algorithm works. MATH 13150: Freshman Seminar Unit 18 1. The RSA algorithm In this chapter, we ll learn how the RSA algorithm works. 1.1. Bob and Alice. Suppose that Alice wants to send a message to Bob over the internet

More information

MATH 168: FINAL PROJECT Troels Eriksen. 1 Introduction

MATH 168: FINAL PROJECT Troels Eriksen. 1 Introduction MATH 168: FINAL PROJECT Troels Eriksen 1 Introduction In the later years cryptosystems using elliptic curves have shown up and are claimed to be just as secure as a system like RSA with much smaller key

More information

Cryptography and Network Security

Cryptography and Network Security Cryptography and Network Security Spring 2012 http://users.abo.fi/ipetre/crypto/ Lecture 7: Public-key cryptography and RSA Ion Petre Department of IT, Åbo Akademi University 1 Some unanswered questions

More information

CRYPTOGRAPHY IN NETWORK SECURITY

CRYPTOGRAPHY IN NETWORK SECURITY ELE548 Research Essays CRYPTOGRAPHY IN NETWORK SECURITY AUTHOR: SHENGLI LI INSTRUCTOR: DR. JIEN-CHUNG LO Date: March 5, 1999 Computer network brings lots of great benefits and convenience to us. We can

More information

Implementation of Elliptic Curve Digital Signature Algorithm

Implementation of Elliptic Curve Digital Signature Algorithm Implementation of Elliptic Curve Digital Signature Algorithm Aqeel Khalique Kuldip Singh Sandeep Sood Department of Electronics & Computer Engineering, Indian Institute of Technology Roorkee Roorkee, India

More information

Primality Testing and Factorization Methods

Primality Testing and Factorization Methods Primality Testing and Factorization Methods Eli Howey May 27, 2014 Abstract Since the days of Euclid and Eratosthenes, mathematicians have taken a keen interest in finding the nontrivial factors of integers,

More information

Digital Signature CHAPTER 13. Review Questions. (Solution to Odd-Numbered Problems)

Digital Signature CHAPTER 13. Review Questions. (Solution to Odd-Numbered Problems) CHAPTER 13 Digital Signature (Solution to Odd-Numbered Problems) Review Questions 1. We mentioned four areas in which there is a differences between a conventional and a digital signature: inclusion, verification,

More information

Cryptography and Network Security Chapter 8

Cryptography and Network Security Chapter 8 Cryptography and Network Security Chapter 8 Fifth Edition by William Stallings Lecture slides by Lawrie Brown (with edits by RHB) Chapter 8 Introduction to Number Theory The Devil said to Daniel Webster:

More information

7! Cryptographic Techniques! A Brief Introduction

7! Cryptographic Techniques! A Brief Introduction 7! Cryptographic Techniques! A Brief Introduction 7.1! Introduction to Cryptography! 7.2! Symmetric Encryption! 7.3! Asymmetric (Public-Key) Encryption! 7.4! Digital Signatures! 7.5! Public Key Infrastructures

More information

Breaking The Code. Ryan Lowe. Ryan Lowe is currently a Ball State senior with a double major in Computer Science and Mathematics and

Breaking The Code. Ryan Lowe. Ryan Lowe is currently a Ball State senior with a double major in Computer Science and Mathematics and Breaking The Code Ryan Lowe Ryan Lowe is currently a Ball State senior with a double major in Computer Science and Mathematics and a minor in Applied Physics. As a sophomore, he took an independent study

More information

CSC474/574 - Information Systems Security: Homework1 Solutions Sketch

CSC474/574 - Information Systems Security: Homework1 Solutions Sketch CSC474/574 - Information Systems Security: Homework1 Solutions Sketch February 20, 2005 1. Consider slide 12 in the handout for topic 2.2. Prove that the decryption process of a one-round Feistel cipher

More information

Mathematics of Internet Security. Keeping Eve The Eavesdropper Away From Your Credit Card Information

Mathematics of Internet Security. Keeping Eve The Eavesdropper Away From Your Credit Card Information The : Keeping Eve The Eavesdropper Away From Your Credit Card Information Department of Mathematics North Dakota State University 16 September 2010 Science Cafe Introduction Disclaimer: is not an internet

More information

Public Key Cryptography of Digital Signatures

Public Key Cryptography of Digital Signatures ACTA UNIVERSITATIS APULENSIS No 13/2007 MATHEMATICAL FOUNDATION OF DIGITAL SIGNATURES Daniela Bojan and Sidonia Vultur Abstract.The new services available on the Internet have born the necessity of a permanent

More information

Computer and Network Security

Computer and Network Security MIT 6.857 Computer and Networ Security Class Notes 1 File: http://theory.lcs.mit.edu/ rivest/notes/notes.pdf Revision: December 2, 2002 Computer and Networ Security MIT 6.857 Class Notes by Ronald L. Rivest

More information

Primes in Sequences. Lee 1. By: Jae Young Lee. Project for MA 341 (Number Theory) Boston University Summer Term I 2009 Instructor: Kalin Kostadinov

Primes in Sequences. Lee 1. By: Jae Young Lee. Project for MA 341 (Number Theory) Boston University Summer Term I 2009 Instructor: Kalin Kostadinov Lee 1 Primes in Sequences By: Jae Young Lee Project for MA 341 (Number Theory) Boston University Summer Term I 2009 Instructor: Kalin Kostadinov Lee 2 Jae Young Lee MA341 Number Theory PRIMES IN SEQUENCES

More information

8 Primes and Modular Arithmetic

8 Primes and Modular Arithmetic 8 Primes and Modular Arithmetic 8.1 Primes and Factors Over two millennia ago already, people all over the world were considering the properties of numbers. One of the simplest concepts is prime numbers.

More information

Network Security. Security Attacks. Normal flow: Interruption: 孫 宏 民 hmsun@cs.nthu.edu.tw Phone: 03-5742968 國 立 清 華 大 學 資 訊 工 程 系 資 訊 安 全 實 驗 室

Network Security. Security Attacks. Normal flow: Interruption: 孫 宏 民 hmsun@cs.nthu.edu.tw Phone: 03-5742968 國 立 清 華 大 學 資 訊 工 程 系 資 訊 安 全 實 驗 室 Network Security 孫 宏 民 hmsun@cs.nthu.edu.tw Phone: 03-5742968 國 立 清 華 大 學 資 訊 工 程 系 資 訊 安 全 實 驗 室 Security Attacks Normal flow: sender receiver Interruption: Information source Information destination

More information

An Overview of Integer Factoring Algorithms. The Problem

An Overview of Integer Factoring Algorithms. The Problem An Overview of Integer Factoring Algorithms Manindra Agrawal IITK / NUS The Problem Given an integer n, find all its prime divisors as efficiently as possible. 1 A Difficult Problem No efficient algorithm

More information

Basic Algorithms In Computer Algebra

Basic Algorithms In Computer Algebra Basic Algorithms In Computer Algebra Kaiserslautern SS 2011 Prof. Dr. Wolfram Decker 2. Mai 2011 References Cohen, H.: A Course in Computational Algebraic Number Theory. Springer, 1993. Cox, D.; Little,

More information

CS 348: Computer Networks. - Security; 30 th - 31 st Oct 2012. Instructor: Sridhar Iyer IIT Bombay

CS 348: Computer Networks. - Security; 30 th - 31 st Oct 2012. Instructor: Sridhar Iyer IIT Bombay CS 348: Computer Networks - Security; 30 th - 31 st Oct 2012 Instructor: Sridhar Iyer IIT Bombay Network security Security Plan (RFC 2196) Identify assets Determine threats Perform risk analysis Implement

More information

Crittografia e sicurezza delle reti. Digital signatures- DSA

Crittografia e sicurezza delle reti. Digital signatures- DSA Crittografia e sicurezza delle reti Digital signatures- DSA Signatures vs. MACs Suppose parties A and B share the secret key K. Then M, MAC K (M) convinces A that indeed M originated with B. But in case

More information

MATH 537 (Number Theory) FALL 2016 TENTATIVE SYLLABUS

MATH 537 (Number Theory) FALL 2016 TENTATIVE SYLLABUS MATH 537 (Number Theory) FALL 2016 TENTATIVE SYLLABUS Class Meetings: MW 2:00-3:15 pm in Physics 144, September 7 to December 14 [Thanksgiving break November 23 27; final exam December 21] Instructor:

More information

Part VII. Digital signatures

Part VII. Digital signatures Part VII Digital signatures CHAPTER 7: Digital signatures Digital signatures are one of the most important inventions/applications of modern cryptography. The problem is how can a user sign a message such

More information

Cryptography: Authentication, Blind Signatures, and Digital Cash

Cryptography: Authentication, Blind Signatures, and Digital Cash Cryptography: Authentication, Blind Signatures, and Digital Cash Rebecca Bellovin 1 Introduction One of the most exciting ideas in cryptography in the past few decades, with the widest array of applications,

More information

Embedding more security in digital signature system by using combination of public key cryptography and secret sharing scheme

Embedding more security in digital signature system by using combination of public key cryptography and secret sharing scheme International Journal of Computer Sciences and Engineering Open Access Research Paper Volume-4, Issue-3 E-ISSN: 2347-2693 Embedding more security in digital signature system by using combination of public

More information

Signature Schemes. CSG 252 Fall 2006. Riccardo Pucella

Signature Schemes. CSG 252 Fall 2006. Riccardo Pucella Signature Schemes CSG 252 Fall 2006 Riccardo Pucella Signatures Signatures in real life have a number of properties They specify the person responsible for a document E.g. that it has been produced by

More information

Cryptographic hash functions and MACs Solved Exercises for Cryptographic Hash Functions and MACs

Cryptographic hash functions and MACs Solved Exercises for Cryptographic Hash Functions and MACs Cryptographic hash functions and MACs Solved Exercises for Cryptographic Hash Functions and MACs Enes Pasalic University of Primorska Koper, 2014 Contents 1 Preface 3 2 Problems 4 2 1 Preface This is a

More information

Number Theory and the RSA Public Key Cryptosystem

Number Theory and the RSA Public Key Cryptosystem Number Theory and the RSA Public Key Cryptosystem Minh Van Nguyen nguyenminh2@gmail.com 05 November 2008 This tutorial uses to study elementary number theory and the RSA public key cryptosystem. A number

More information

EXAM questions for the course TTM4135 - Information Security June 2010. Part 1

EXAM questions for the course TTM4135 - Information Security June 2010. Part 1 EXAM questions for the course TTM4135 - Information Security June 2010 Part 1 This part consists of 6 questions all from one common topic. The number of maximal points for every correctly answered question

More information

Implementation and Comparison of Various Digital Signature Algorithms. -Nazia Sarang Boise State University

Implementation and Comparison of Various Digital Signature Algorithms. -Nazia Sarang Boise State University Implementation and Comparison of Various Digital Signature Algorithms -Nazia Sarang Boise State University What is a Digital Signature? A digital signature is used as a tool to authenticate the information

More information

Some practice problems for midterm 2

Some practice problems for midterm 2 Some practice problems for midterm 2 Kiumars Kaveh November 15, 2011 Problem: What is the remainder of 6 2000 when divided by 11? Solution: This is a long-winded way of asking for the value of 6 2000 mod

More information

Shor s algorithm and secret sharing

Shor s algorithm and secret sharing Shor s algorithm and secret sharing Libor Nentvich: QC 23 April 2007: Shor s algorithm and secret sharing 1/41 Goals: 1 To explain why the factoring is important. 2 To describe the oldest and most successful

More information

ALGEBRAIC APPROACH TO COMPOSITE INTEGER FACTORIZATION

ALGEBRAIC APPROACH TO COMPOSITE INTEGER FACTORIZATION ALGEBRAIC APPROACH TO COMPOSITE INTEGER FACTORIZATION Aldrin W. Wanambisi 1* School of Pure and Applied Science, Mount Kenya University, P.O box 553-50100, Kakamega, Kenya. Shem Aywa 2 Department of Mathematics,

More information

A New Generic Digital Signature Algorithm

A New Generic Digital Signature Algorithm Groups Complex. Cryptol.? (????), 1 16 DOI 10.1515/GCC.????.??? de Gruyter???? A New Generic Digital Signature Algorithm Jennifer Seberry, Vinhbuu To and Dongvu Tonien Abstract. In this paper, we study

More information

Network Security. Abusayeed Saifullah. CS 5600 Computer Networks. These slides are adapted from Kurose and Ross 8-1

Network Security. Abusayeed Saifullah. CS 5600 Computer Networks. These slides are adapted from Kurose and Ross 8-1 Network Security Abusayeed Saifullah CS 5600 Computer Networks These slides are adapted from Kurose and Ross 8-1 Public Key Cryptography symmetric key crypto v requires sender, receiver know shared secret

More information

Elliptic Curve Cryptography

Elliptic Curve Cryptography Elliptic Curve Cryptography Elaine Brow, December 2010 Math 189A: Algebraic Geometry 1. Introduction to Public Key Cryptography To understand the motivation for elliptic curve cryptography, we must first

More information

Cryptography Lecture 8. Digital signatures, hash functions

Cryptography Lecture 8. Digital signatures, hash functions Cryptography Lecture 8 Digital signatures, hash functions A Message Authentication Code is what you get from symmetric cryptography A MAC is used to prevent Eve from creating a new message and inserting

More information