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crypto-slides-14-pk-tutor.1x1 - Tutorial on Public Key...

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Tutorial on Public Key Cryptography – RSA c circlecopyrt Eli Biham - August 18, 2010 384 Tutorial on Public Key Cryptography – RSA (14)

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RSA – the Key Generation – Example 1. Randomly choose two prime numbers p and q . We choose p = 11 and q = 13. 2. Compute n = pq . We compute n = pq = 11 · 13 = 143. 3. Randomly choose an odd number e in the range 1 < e < ϕ ( n ) which is coprime to ϕ ( n ) (i.e., e Z ϕ ( n ) ). ϕ ( n ) = ϕ ( p ) · ϕ ( q ) = 10 · 12 = 120. Thus, we choose e = 7 ( e Z 120 ). 4. Compute d e 1 (mod ϕ ( n )) by Euclid’s algorithm. Thus, de 1 (mod ϕ ( n )). We compute d e 1 7 1 103 (mod ϕ (143) = 120). Check that 120 | 7 103 1 = 721 1 = 720. c circlecopyrt Eli Biham - August 18, 2010 385 Tutorial on Public Key Cryptography – RSA (14)
RSA – the Key Generation – Example (cont.) 5. Publish ( n, e ) as the public key, and keep d secret as the secret key. We publish ( n, e ) = (143 , 7) as the public key, and keeps d = 103 secret as the secret key. c circlecopyrt Eli Biham - August 18, 2010 386 Tutorial on Public Key Cryptography – RSA (14)

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RSA – Encryption/Decryption – Example The encryption algorithm E : Everybody can encrypt messages m (0 m < n A ) to user A by c = E A ( m ) = m e A mod n A . The ciphertext c (0 c < n A ) can be sent to A , and only A can decrypt. Encrypt m = 3: E A ( m ) m e A 3 7 42 (mod 143) c circlecopyrt Eli Biham - August 18, 2010 387 Tutorial on Public Key Cryptography – RSA (14)
RSA – Encryption/Decryption – Example (cont.) The decryption algorithm D : Only A knows his secret key d A and can decrypt: m = D A ( c ) = c d A mod n A . Decrypt c = 42: D A ( c ) c d A 42 103 3 (mod 143) Decrypt c = 2: D A ( c ) c d A 2 103 63 (mod 143) c circlecopyrt Eli Biham - August 18, 2010 388 Tutorial on Public Key Cryptography – RSA (14)

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Existential Forgery of an RSA Signature Given a public key ( n A , e A ) of user A, can another user B create a message m and a signature D A ( m ) m d A (mod n A )? User B can forge a signature in the following way: B Chooses y Z n and calculates x E A ( y ) = y e A (mod n A ). Now B can claim that y x d A (mod n A ) is A’s signature on x . c circlecopyrt Eli Biham - August 18, 2010 389 Tutorial on Public Key Cryptography – RSA (14)
Multiplication Property of RSA Multiplication Property : Given a public key ( n A , e A ) of user A, and m 1 , m 2 Z n then E A ( m 1 · m 2 ) E A ( m 1 ) · E A ( m 2 ) (mod n ) Proof : E A ( m 1 ) m e A 1 (mod n A ) E A ( m 2 ) m e A 2 (mod n A ) and, E A ( m 1 · m 2 ) ( m 1 · m 2 ) e A m e A 1 · m e A 2 E A ( m 1 ) · E A ( m 2 ) (mod n A ) QED c circlecopyrt Eli Biham - August 18, 2010 390 Tutorial on Public Key Cryptography – RSA (14)

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Random Self Reducibility of RSA
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