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1 CIS 5371 Cryptography 3b. Pseudorandomness. B ased on: Jonathan Katz and Yehuda Lindell Introduction to Modern Cryptography

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2 Pseudorandomness An introduction A distribution D is pseudorandom if no PPT distinguisher can detect if it a string sampled according to D or chosen uniformly at random. This is formalized by requiring that every PPT algorithm outputs 1 with almost the same probability when given a truly random string as when given a pseudorandom string.

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3 Pseudorandomness An introduction

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4 Existence of pseudorandom generators We cannot prove that pseudorandom generators exist! We believe that such generators can be constructed from one-way functions. There are some long-standing problems that have no efficient solution and it is believed that they are unsolvable in polynomial time.

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5 Pseudorandom generators informal definition A distribution D is pseudorandom if no PPT distinguisher can detect if it is given a string sampled according to D or a string chosen uniformly at random. This can be formalized by requiring that a PPT distinguisher D outputs 1 with almost the same probability when given a truly random string and when given a pseudorandom string.

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6 Pseudorandomness Definition

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7 A secure fixed length encryption scheme

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8 A secure fixed length encryption Protocol

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9 A secure fixed length encryption Theorem

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10 A secure fixed length encryption Reduction Adversary A ( Protocol ) Adversary A’ ( Distinguisher D )

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11 A secure fixed length encryption Proof

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12 A secure fixed length encryption Proof

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13 Variable output length pseudorandom generators

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14 Stream ciphers

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Discussion We use the term stream cipher for the PR stream generator, not the encryption algorithm. There are a number of practical constructions of stream ciphers that are extraordinarily fast, such as the stream cipher RC4. 15

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Discussion The WEP encryption protocol for used RC4 and was broken. But since then it is fixed---and the standard updated. If RC4 has to be used the first 1024 bits or so should be discarded. 16

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Discussion From a security point of view it is advocated to use block cipher constructions for constructing secure encryption schemes. This disadvantage is that this approach is less efficient when compared to using a dedicated stream cipher. 17

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Definition 19

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Indistinguishable single encryptions vs indistinguishable multi encryptions 20

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Secure multiple encryptions using a stream cipher Synchronized mode Communicating parties use a different part of the stream cipher output to encrypt a message. Useful for parties communicating in the same session. Communicating parties must maintain state between encryptions. 21

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Secure multiple encryptions using a stream cipher 22

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Security against Chosen- Plaintext Attack (CPA) We now consider a more powerful adversary that is active. The adversary can ask for the encryptions of some specific plaintext messages, as well as eavesdrop. 23

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Indistinguishable encryptions under CPA Definition 25

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CPA security for multiple encryptions 26

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