Public Key Encryption: Secure Communication without Sharing Keys, Slides of Cryptography and System Security

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Public Key Encryption

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Public Key Encryption

PROBLEM: Several individuals: Bob, Carla, David, ... wish to sent messages to Alice over insecure channel eavesdropped by Eve GOAL: Each individual encrypts their message without having to pre- establish secret key B A E C D

X

e b aY

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Asymmetric Key

Each key K splits up into two parts:

P ( K ) - public key used for encryption with the function

T (K) - trapdoor or private key used for decryption with the function P ( K ) should be impossible to compute from T ( K ) at a minimum ( key security ). ePK dT K

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RSA

K = ( p,q,e ) with p,q primes of equal bitlength, e is relatively prime to both p - 1 and q - 1

PK = P ( K ) = ( n , e ) with n = pq

TK = T ( K ) = ( n,d ) same n ,

P, C depend on K :

Encrypt by exponentiating:

Decrypt by extracting root (raise to the d ):

P = C = Z n

ePK ( x ) = x e mod n dT K ( y ) = y d mod n d = e − 1 mod !( n )

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Trap-Door PKE Issues

Trap-door functions as defined are deterministic. Since is public information, Eve can compute as well and compare to eavesdropped messages: E.g. suppose message space is limited to {ATTACK, RETREAT}. Eve pre-computes on each message and checks to see which one was sent by Bob. CONCLUSION: Any secure PKE system must be randomized. ePK ePK

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Security of RSA

Intuitive Security: No known method of extracting e ’th roots mod n without knowing CLAIM: For n = pq, computing is equivalent to factoring n. Key Security THM: If a BPP algorithm exists for finding a valid d from ( n,e ), then a BPP algorithm for factoring n = pq exists. Open Question: Can factoring be reduced to decrypting RSA? !( n ) !( n )

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Rabin

K = ( p,q ) with p,q primes of equal bitlength

PK = pq = n = product of the primes

TK = K = primes factors of n

P, C depend on K :

Encrypt by squaring:

Decrypt by square-roots:

1. Compute
2. Patch back with CRT to get 4 roots ePK ( x ) = x 2 mod n dT K ( x ) = √ x mod n ± y p + 1 (^4) mod p , ± y q + 1 (^4) mod q

P = C = Z n

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Rabin - Analysis

multivalued. FIX: Include uniquely identifying information in plaintext Decryption Security THM: If there is a BPP algorithm for decrypting Rabin, then there is a BPP algorithm for factoring such n. Negative THM: Key insecure under chosen ciphertext attack dT K ( x )