CIS 5371 Cryptography
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Transcript of CIS 5371 Cryptography
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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 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 be a polynomial and a deterministic polynomial-time algorithm that on input any will output string of length . is a pseudorandom generator if:
• ∀ PPT distinguishers D, where is uniform random string of length is uniform random of length and the probabilities are taken over the coins used by and the choices of .
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A secure fixed length encryption scheme𝑘
𝑝𝑙𝑎𝑖𝑛𝑡𝑒𝑥𝑡 h𝑐𝑖𝑝 𝑒𝑟𝑡𝑒𝑥𝑡𝑋𝑂𝑅
𝑝𝑎𝑑
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A secure fixed length encryption Protocol
Let be a pseudorandom generator with expansion factor . Define a private-key encryption scheme for messages of length as follows• Gen: on input choose uniformly at random and output as key.• Enc: on input a key and a message m output the ciphertext
• Dec: on input a key and a ciphertext c output the plaintext
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A secure fixed length encryption Theorem
If 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 (Protocol ) 𝑤
𝑏 ′
𝑐𝑏
𝑚0 ,𝑚1
1𝑛
1 if 0 if
choose a random bit compute
Suppose that A succeeds with probability
Adversary A’ (Distinguisher D)
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A secure fixed length encryption Proof
Let Pr . Then, • when is uniform random we have .• when we have
Pr .
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A secure fixed length encryption Proof
Therefore when is chosen uniformly in
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Variable output length pseudorandom generators
A deterministic polynomial-time algorithm is a variable output-length pseudorandom generator if:1. Let be a string and an integer. Then outputs a
string of length . 2. For all with the string is a prefix of .
Define . Then for every polynomial it holds that is a pseudorandom generator with expansion factor .
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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,
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 802.11
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
3..
DefinitionA private-key encryption scheme =(Gen,Enc,Dec) that has indistinguishable multiple encryptions in the presence of an eavesdropper satisfies:
: where the probability is taken over the random coins of , 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 for the pair of vector messages and
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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.
where the initial vector 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
1. . .
Indistinguishable encryptions under CPA
Definition
A private-key encryption scheme has indistinguishable encryptions under CPA if
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 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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