P1. Public-Key Cryptography and RSA 5351: Introduction to Cryptography Spring 2013.

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Presentation transcript:

p1. Public-Key Cryptography and RSA 5351: Introduction to Cryptography Spring 2013

p2. 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.

p3. Why Public-Key Cryptography? Developed to address two main issues: –key distribution –digital signatures Invented by Diffie & Hellman in 1976.

p4.

p5.

p6.

p7.

p8.

p9. Modular Arithmetic

p10.

p11.

p12.

p13.

p14.

p15.

p16.

p17.

p18.

p19.

p20. The Chinese Remainder Problem A problem described in an ancient Chinese arithmetic book, Sun Tze Suan Ching, by Sun Tze (around 300AD, author of The Art of War). Problem: We have a number of objects, but we do not know exactly how many. If we count them by threes we have two left over. If we count them by fives we have three left over. If we count them by sevens we have two left over. How many objects are there?

p21.

p22. Example: Chinese remainder theorem

p23.

p24.

p25.

p26.

p27. Algorithms

p28.

p29.

p30.

p31.

p32.

p33.

p34. The RSA Cryptosystem RSA Encryption RSA Digital Signature

p35.

p36.

p37.

p38.

p39.

p40.

p41.

p42.

p43.

p44.

p45. Attacks on RSA

p46.

p47.

p48.

p49.

p50.

p51. 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.

p52.

p53.

p54.

p55.

p56.

p57.

p58.

p59.

p60.

p61.

p62.

p63.

p64.

p65.

p66. Padded RSA

p67.

p68.

p69.

p70.

p71.

p72.

p73. 0 n 2n 3n 4n ns 2B 3B

p74. 2B 3B

p75.

p76. CCA-Secure RSA in the Random Oracle Model

p77.

p78.

p79.

p80.

p81.

p82.

p83.

p84. Digital Signatures

p85. Message m MAC k (m) Message m Sig sk (m)

p86.

p87. MCED PK Bob SK Bob Alice Bob M M SED PK Bob SK Bob Alice Bob Verify the signature Sign Encryption (using RSA): Signing (using RSA -1 ): E(S) =M?

p88.

p89.

p90.

p91.

p92.

p93.

p94.

p95.

p96.

p97.

p98.

p99.

p100.

p101.

p102.

p103.

p104. Generating large primes To set up an RSA cryptosystem, we need two large primes p and q.

p105.

p106.

p107.

p108.

p109.

p110.

p111.

p112.

p113.

p114.

p115.

p116.

p117.

p118.

p119.

p120.

p121.

p122.