Pseudo Random Bit Generators: Definition, Blum-Blum-Shub Algorithm, and Encryption, Slides of Cryptography and System Security

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Pseudo Random Bit

Generators

2

Pseudo Random

Generator

PRG ’s also known as stream ciphers because they correspond to pseudo-random one-time pads. Intuitively, these are deterministic functions whose outputs cannot be differentiated from random bitstreams.

4

Blum-Blum-Shub

Official PRG

l ( k ) is any polynomial > k INPUT: random seed x of length k OUTPUT: bitstring s of length L Use 1st ¼ of x to generate p deterministically Use 2nd ¼ of x to generate q deterministically Let n = p ⋅ q , and r = 2nd ½ of x. Return BBS-PRG( n, r, l ( k )) // slide #5 from // “probabilistic encryption”

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PRG Stateful Private

Encryption

THM: A pseudo random bit generator exists iff a stateful symmetric encryption scheme exists with |M| > |K| that is computationally secure. ½ proof : PRG g Encryption : Use the pseudo random one time pad defined by

security parameter k chosen so l ( k ) ≥ | m |

G: K = (key K a rand. k -bit string)

Uk

EK

EK ( m ) = g ( K ) ⊕ m