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Public-Key Cryptography and RSA

Explore the concept of public-key cryptography and the RSA algorithm, including attacks on RSA, key distribution, and Chinese remainder theorem in mathematics. Learn how to set up an RSA cryptosystem and achieve fast decryption using CRT.

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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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