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Public-Key Cryptography and RSA

Public-Key Cryptography and RSA. Dr. Ayad I. Abdulsada 2017-2018. Abstract. We will discuss The concept of public-key cryptography RSA algorithm Attacks on RSA. Public-Key Cryptography. Also known as asymmetric-key cryptography.

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Public-Key Cryptography and RSA

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  1. Public-Key Cryptography and RSA Dr. Ayad I. Abdulsada 2017-2018

  2. Abstract • We will discuss • The concept of public-key cryptography • RSA algorithm • Attacks on RSA

  3. Public-Key Cryptography • Also known as asymmetric-key cryptography. • Each user has a pair of keys: a public key and a private key. • The public key is used for encryption. • The key is known to the public. • The private key is used for decryption. • The key is only known to the owner.

  4. Bob Alice

  5. Why Public-Key Cryptography? • Developed to address two main issues: • key distribution • digital signatures • Invented by Whitfield Diffie & Martin Hellman 1976.

  6. Modular Arithmetic Mathematics used in RSA

  7. Chinese remainder theorem (CRT)

  8. Example of CRT • Take 15 = 5 · 3, and consider Z*15 = {1, 2, 4, 7, 8, 11, 13, 14}. • The Chinese remainder theorem says that this group is isomorphic to Z*5 × Z*3. • Example: Compute 14 · 13 mod 15. • Solution: 14 ⟷(4, 2) and 13 ⟷ (3, 1). • Now, [14 · 13 mod 15] ⟷(4, 2) · (3, 1) = ([4 · 3 mod 5], [2 · 1 mod 3]) = (2, 2) • But (2, 2) ⟷ 2, which is the correct answer since 14 · 13 = 2 mod 15.

  9. Setting up an RSA Cryptosystem • A user wishing to set up an RSA cryptosystem will: • Choose a pair of public/private keys: (PU, PR). • Publish the public (encryption) key. • Keep secret the private (decryption) key.

  10. Fast Decryption with the Chinese Remainder Theorem • [cd mod N] ⟷ (cd mod p, cd mod q) the receiver can compute the partial results • And then combine these to obtain m ⟷ (mp, mq) as:

  11. RSA-200 = 27,997,833,911,221,327,870,829,467,638, 722,601,621,070,446,786,955,428,537,560, 009,929,326,128,400,107,609,345,671,052, 955,360,856,061,822,351,910,951,365,788, 637,105,954,482,006,576,775,098,580,557, 613,579,098,734,950,144,178,863,178,946, 295,187,237,869,221,823,983.

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