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Mathematical Cryptography Diffie Hellman, Discrete Log Problem, Collision Algorithms Mentor: Tao Song , Mentee: Lisette del Pino University of Pennsylvania Directed Reading Program May 16, 2020 Mentor: Tao Song , Mentee: Lisette del Pino

  1. Mathematical Cryptography Diffie Hellman, Discrete Log Problem, Collision Algorithms Mentor: Tao Song , Mentee: Lisette del Pino University of Pennsylvania Directed Reading Program May 16, 2020 Mentor: Tao Song , Mentee: Lisette del Pino Mathematical Cryptography

  2. Diffie Hellman Public Key Exchange A Privacy Dillema You and your friend want to exchange secret messages, in this case, these numbers translate to words. You and your friend live very far away from each other and can’t meet up to share secrets. What do you do? Mentor: Tao Song , Mentee: Lisette del Pino Mathematical Cryptography

  3. Diffie Hellman Public Key Exchange A Privacy Dillema You and your friend want to exchange secret messages, in this case, these numbers translate to words. You and your friend live very far away from each other and can’t meet up to share secrets. What do you do? 1 First, pick prime p , and a nonzero integer g mod p. Mentor: Tao Song , Mentee: Lisette del Pino Mathematical Cryptography

  4. Diffie Hellman Public Key Exchange A Privacy Dillema You and your friend want to exchange secret messages, in this case, these numbers translate to words. You and your friend live very far away from each other and can’t meet up to share secrets. What do you do? 1 First, pick prime p , and a nonzero integer g mod p. 2 You: pick a secret integer a . Compute A ≡ g a mod p Mentor: Tao Song , Mentee: Lisette del Pino Mathematical Cryptography

  5. Diffie Hellman Public Key Exchange A Privacy Dillema You and your friend want to exchange secret messages, in this case, these numbers translate to words. You and your friend live very far away from each other and can’t meet up to share secrets. What do you do? 1 First, pick prime p , and a nonzero integer g mod p. 2 You: pick a secret integer a . Compute A ≡ g a mod p 3 Your friend: pick a secret integer b , Compute B ≡ g b mod p Mentor: Tao Song , Mentee: Lisette del Pino Mathematical Cryptography

  6. Diffie Hellman Public Key Exchange A Privacy Dillema You and your friend want to exchange secret messages, in this case, these numbers translate to words. You and your friend live very far away from each other and can’t meet up to share secrets. What do you do? 1 First, pick prime p , and a nonzero integer g mod p. 2 You: pick a secret integer a . Compute A ≡ g a mod p 3 Your friend: pick a secret integer b , Compute B ≡ g b mod p 4 Send A to your friend. Your friend sends B to you. Mentor: Tao Song , Mentee: Lisette del Pino Mathematical Cryptography

  7. Diffie Hellman Public Key Exchange A Privacy Dillema You and your friend want to exchange secret messages, in this case, these numbers translate to words. You and your friend live very far away from each other and can’t meet up to share secrets. What do you do? 1 First, pick prime p , and a nonzero integer g mod p. 2 You: pick a secret integer a . Compute A ≡ g a mod p 3 Your friend: pick a secret integer b , Compute B ≡ g b mod p 4 Send A to your friend. Your friend sends B to you. 5 In private, you compute B ′ ≡ B a mod p Mentor: Tao Song , Mentee: Lisette del Pino Mathematical Cryptography

  8. Diffie Hellman Public Key Exchange A Privacy Dillema You and your friend want to exchange secret messages, in this case, these numbers translate to words. You and your friend live very far away from each other and can’t meet up to share secrets. What do you do? 1 First, pick prime p , and a nonzero integer g mod p. 2 You: pick a secret integer a . Compute A ≡ g a mod p 3 Your friend: pick a secret integer b , Compute B ≡ g b mod p 4 Send A to your friend. Your friend sends B to you. 5 In private, you compute B ′ ≡ B a mod p 6 In private, your friend computes A ′ ≡ A b mod p Mentor: Tao Song , Mentee: Lisette del Pino Mathematical Cryptography

  9. Diffie Hellman Public Key Exchange A Privacy Dillema You and your friend want to exchange secret messages, in this case, these numbers translate to words. You and your friend live very far away from each other and can’t meet up to share secrets. What do you do? 1 First, pick prime p , and a nonzero integer g mod p. 2 You: pick a secret integer a . Compute A ≡ g a mod p 3 Your friend: pick a secret integer b , Compute B ≡ g b mod p 4 Send A to your friend. Your friend sends B to you. 5 In private, you compute B ′ ≡ B a mod p 6 In private, your friend computes A ′ ≡ A b mod p 7 This is the shared value. B ′ and A ′ are the same. Mentor: Tao Song , Mentee: Lisette del Pino Mathematical Cryptography

  10. Diffie Hellman Public Key Exchange A Privacy Dillema You and your friend want to exchange secret messages, in this case, these numbers translate to words. You and your friend live very far away from each other and can’t meet up to share secrets. What do you do? 1 First, pick prime p , and a nonzero integer g mod p. 2 You: pick a secret integer a . Compute A ≡ g a mod p 3 Your friend: pick a secret integer b , Compute B ≡ g b mod p 4 Send A to your friend. Your friend sends B to you. 5 In private, you compute B ′ ≡ B a mod p 6 In private, your friend computes A ′ ≡ A b mod p 7 This is the shared value. B ′ and A ′ are the same. 8 Proof: A ′ ≡ B a ≡ g ba ≡ A b ≡ B ′ mod p Mentor: Tao Song , Mentee: Lisette del Pino Mathematical Cryptography

  11. A Numerical Example of the DHKE Let’s try an explicit example with relatively small numbers. In practice, you should pick primes about 4000 bits long. Mentor: Tao Song , Mentee: Lisette del Pino Mathematical Cryptography

  12. A Numerical Example of the DHKE Let’s try an explicit example with relatively small numbers. In practice, you should pick primes about 4000 bits long. 1 First, pick prime p = 941, and a nonzero integer g = 627 mod 941. Recall eavesdroppers know these values : A , B , g a , g b , g , p And they need to find: g ab This problem is no harder than the Discrete Logarithm Problem Mentor: Tao Song , Mentee: Lisette del Pino Mathematical Cryptography

  13. A Numerical Example of the DHKE Let’s try an explicit example with relatively small numbers. In practice, you should pick primes about 4000 bits long. 1 First, pick prime p = 941, and a nonzero integer g = 627 mod 941. 2 You: pick a secret integer a = 347. Compute A = 390 Recall eavesdroppers know these values : A , B , g a , g b , g , p And they need to find: g ab This problem is no harder than the Discrete Logarithm Problem Mentor: Tao Song , Mentee: Lisette del Pino Mathematical Cryptography

  14. A Numerical Example of the DHKE Let’s try an explicit example with relatively small numbers. In practice, you should pick primes about 4000 bits long. 1 First, pick prime p = 941, and a nonzero integer g = 627 mod 941. 2 You: pick a secret integer a = 347. Compute A = 390 3 Your friend: pick a secret integer b = 781, Compute B = 691 Recall eavesdroppers know these values : A , B , g a , g b , g , p And they need to find: g ab This problem is no harder than the Discrete Logarithm Problem Mentor: Tao Song , Mentee: Lisette del Pino Mathematical Cryptography

  15. A Numerical Example of the DHKE Let’s try an explicit example with relatively small numbers. In practice, you should pick primes about 4000 bits long. 1 First, pick prime p = 941, and a nonzero integer g = 627 mod 941. 2 You: pick a secret integer a = 347. Compute A = 390 3 Your friend: pick a secret integer b = 781, Compute B = 691 4 Send A to your friend. Your friend sends B to you. Recall eavesdroppers know these values : A , B , g a , g b , g , p And they need to find: g ab This problem is no harder than the Discrete Logarithm Problem Mentor: Tao Song , Mentee: Lisette del Pino Mathematical Cryptography

  16. A Numerical Example of the DHKE Let’s try an explicit example with relatively small numbers. In practice, you should pick primes about 4000 bits long. 1 First, pick prime p = 941, and a nonzero integer g = 627 mod 941. 2 You: pick a secret integer a = 347. Compute A = 390 3 Your friend: pick a secret integer b = 781, Compute B = 691 4 Send A to your friend. Your friend sends B to you. 5 The shared value is 470 ≡ 627 347 ∗ 781 mod 941 Recall eavesdroppers know these values : A , B , g a , g b , g , p And they need to find: g ab This problem is no harder than the Discrete Logarithm Problem Mentor: Tao Song , Mentee: Lisette del Pino Mathematical Cryptography

  17. Discrete Logarithm Problem Recall that computing the shared value g ab is no harder than solving the Discrete Log Problem. Mentor: Tao Song , Mentee: Lisette del Pino Mathematical Cryptography

  18. Discrete Logarithm Problem Recall that computing the shared value g ab is no harder than solving the Discrete Log Problem. Fermat’s Little Theorem: if p is prime, any integer g gives us g p − 1 ≡ 1 mod p Mentor: Tao Song , Mentee: Lisette del Pino Mathematical Cryptography

  19. Discrete Logarithm Problem Recall that computing the shared value g ab is no harder than solving the Discrete Log Problem. Fermat’s Little Theorem: if p is prime, any integer g gives us g p − 1 ≡ 1 mod p recall elements of a finite multiplicative group F ∗ p with a generator g are: 1 , g 1 , g 2 , ..., g p − 2 Mentor: Tao Song , Mentee: Lisette del Pino Mathematical Cryptography

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