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**CIS 5371 Cryptography 3b. Pseudorandomness.**

Based on: Jonathan Katz and Yehuda Lindell Introduction to Modern Cryptography

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**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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**Pseudorandomness An introduction**

A pseudorandom generator is a deterministic algorithm that given a short truly random seed of length n will stretch it to into a longer string of length 𝑙(𝑛) that is pseudorandom.

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**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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**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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**Pseudorandomness Definition**

Let 𝑙(∙) be a polynomial and 𝐺 a deterministic polynomial-time algorithm that on input any 𝑠 𝜖 {0,1 } 𝑛 will output string of length 𝑙(𝑛). 𝐺 is a pseudorandom generator if: 𝑙 𝑛 >𝑛 ∀ PPT distinguishers D, ∃ 𝑎 negl function with: | Pr 𝐷 𝑟 =1 − Pr 𝐷 𝐺 𝑠 =1 ≤negl(n) where 𝑟 is uniform random string of length 𝑙 𝑛 , 𝑠 𝑖𝑠 is uniform random of length 𝑛 and the probabilities are taken over the coins used by 𝐷 and the choices of 𝑟,𝑠.

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**A secure fixed length encryption scheme**

𝑘 𝑝𝑙𝑎𝑖𝑛𝑡𝑒𝑥𝑡 𝑐𝑖𝑝ℎ𝑒𝑟𝑡𝑒𝑥𝑡 𝑋𝑂𝑅 𝑝𝑠𝑒𝑢𝑑𝑜𝑟𝑎𝑛𝑑𝑜𝑚 𝑔𝑒𝑛𝑒𝑟𝑎𝑡𝑜𝑟 𝑝𝑎𝑑

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**A secure fixed length encryption Protocol **

Let 𝐺 be a pseudorandom generator with expansion factor 𝑙. Define a private-key encryption scheme for messages of length 𝑙 as follows Gen: on input 1 𝑛 choose 𝑘 {0,1 } 𝑛 uniformly at random and output 𝑘 as key. Enc: on input a key 𝑘 {0,1 } 𝑛 and a message m{0,1 } 𝑙(𝑛) output the ciphertext 𝑐≔G 𝑘 𝑚 . Dec: on input a key 𝑘 {0,1 } 𝑛 and a ciphertext c{0,1 } 𝑙(𝑛) output the plaintext 𝑚≔G 𝑘 𝑐 .

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**A secure fixed length encryption Theorem**

If 𝐺 be a pseudorandom generator then protocol is a fixed-length private-key encryption scheme that has indistinguishable encryptions in the presence of an eavesdropper.

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**A secure fixed length encryption Reduction**

Adversary A’ (Distinguisher D) Adversary A (Protocol ) 𝑤 1 𝑛 𝑚 0 , 𝑚 1 choose a random bit 𝑏 compute 𝑐 𝑏 := w 𝑚 𝑏 Suppose that A succeeds with probability 𝜀(𝑛) 𝑐 𝑏 1 if 𝑏 ′ =𝑏 𝑏′ 0 if 𝑏 ′ 𝑏

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**A secure fixed length encryption Proof**

Let 𝜀 𝑛 = Pr[Priv K eav (𝐴, ) 𝑛 =1]− Then, when 𝑤 is uniform random we have Pr 𝐷 𝑤 =1 =Pr[Priv K eav (𝐴, ) 𝑛 =1]= when 𝑤=𝐺(𝑘) we have Pr 𝐷 𝑤 =1 = Pr 𝐷 𝐺 𝑘 =1 = Pr[Priv K eav (𝐴, ) 𝑛 =1]= 𝜀(𝑛).

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**A secure fixed length encryption Proof**

Therefore when 𝑤 is chosen uniformly in {0,1 } 𝑙 𝑛 : |Pr 𝐷 𝑤 =1 −Pr[𝐷 𝐺 𝑘 =1]|= (𝑛) .

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**Variable output length pseudorandom generators**

A deterministic polynomial-time algorithm 𝐺 is a variable output-length pseudorandom generator if: Let 𝑠 be a string and 𝑙>0 an integer. Then 𝐺 𝑠, 1 𝑙 outputs a string of length 𝑙. For all 𝑠,𝑙,𝑙′ with 𝑙< 𝑙 ′ , the string 𝐺 𝑠, 1 𝑙 is a prefix of 𝐺 𝑠, 1 𝑙 ′ . Define 𝐺 𝑙 𝑠 ≝ 𝐺 𝑠, 1 𝑙(|𝑠|) . Then for every polynomial it holds that 𝐺 𝑙 𝑠, 1 𝑙 is a pseudorandom generator with expansion factor 𝑙.

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Stream ciphers We can easily modify the earlier construction for the encryption scheme for variable output length PRG. In this case, 𝑐≔G 𝑘, 1 𝑚 𝑚 . 𝑚≔G 𝑘, 1 |𝑐| 𝑐 .

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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.

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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.

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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.

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**Multi-message eavesdropping experiment Priv K mult (𝐴,)(𝑛)**

The adversary 𝐴 is given input 1 𝑛 and outputs a pair of vectors of messages 𝑚 0 1 ,…, 𝑚 0 𝑡 and 𝑚 1 1 ,…, 𝑚 1 𝑡 witℎ |𝑚 0 𝑖 = 𝑚 1 𝑖 | for all 𝑖. A key 𝑘 is generated runnng 𝐺𝑒𝑛 1 𝑛 and a random bit 𝑏∈ 0,1 is chosen. For all 𝑖 the ciphertext 𝑐 𝑖 En 𝑐 𝑘 𝑚 𝑏 𝑖 is computed and the vector of ciphertexts 𝑐 𝑏 1 , …, 𝑐 𝑏 𝑡 is given to 𝐴. .𝐴 outputs a bit 𝑏 ′ . The output of the experiment i𝑠 1 if 𝑏 =𝑏 ′ and 0 otherwise.

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**Definition PPT Adversary 𝐴, a negligible function negl:**

A private-key encryption scheme =(Gen,Enc,Dec) that has indistinguishable multiple encryptions in the presence of an eavesdropper satisfies: PPT Adversary 𝐴, a negligible function negl: Pr[Priv K mult (𝐴, ) 𝑛 =1] ≤ negl 𝑛 , where the probability is taken over the random coins of 𝐴, and the experiment.

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**Indistinguishable single encryptions vs indistinguishable multi encryptions**

The secure fixed length encryption Protocol presented earlier is deterministic and cannot be used as a construction for a indistinguishable multi encryptions. To see why, we use the experiment Priv K mult for the pair of vector messages ( 0 𝑛 , 0 𝑛 ) and 0 𝑛 , 1 𝑛 .

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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.

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**Secure multiple encryptions using a stream cipher**

Unsynchronized mode Encryptions are carried out independently of one another. Communicating parties are not required to maintain state between encryptions. 𝐸𝑛 𝑐 𝑘 𝑚 ≔ 𝐼𝑉, 𝐺 𝑘,𝐼𝑉 𝑚 where the initial vector 𝐼𝑉 {0,1} 𝑛 is chosen at random.

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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.

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**The CPA indistinguishability experiment Priv K cpa (𝐴,)(𝑛)**

A key 𝑘 is generated runnng Gen 1 𝑛 . The adversary 𝐴 is given input 1 𝑛 and oracle access to En 𝑐 𝑘 ∙ , .and outputs a pair of messages 𝑚 0 , 𝑚 1 of equal length. A random bit 𝑏 0,1 is chosen and a ciphertext c En 𝑐 𝑘 𝑚 𝑏 is computed and given to 𝐴. Adversary 𝐴 continues to have oracle access to En 𝑐 𝑘 ∙ , and outputs a bit 𝑏 ′ . The output of the experiment i𝑠 1 if 𝑏 =𝑏 ′ and 0 otherwise.

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**Indistinguishable encryptions under CPA Definition**

A private-key encryption scheme = Gen,Enc,Dec has indistinguishable encryptions under CPA if ∀ PPT adversaries 𝐴, ∃ a negl function such that, Pr[Priv Kcpa 𝐴, 𝑛 =1] ≤ negl 𝑛 , where the probability is taken over the coins of A and those of the experiment.

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**CPA security for multiple encryptions**

As for single encryption, extend the experiment to Priv K cpa in which the adversary outputs a pair of vectors of plaintext. Any private-key encryption scheme that has indistinguishable encryptions under CPA also has indistinguishable multiple encryptions under CPA

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