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Dec 30, 2023 •334 likes •501 views

Special Topics in Cryptography Mohammad Mahmoody Last time Secrecy based on (unproven) computational assumptions Pseudorandom generators How to encrypt longer messages in an ind-secure way using a PRG Today How to make PRGs

  1. Special Topics in Cryptography Mohammad Mahmoody

  2. Last time • Secrecy based on (unproven) computational assumptions • Pseudorandom generators • How to encrypt longer messages in an ind-secure way using a PRG Today • How to make PRGs stretch more • How to use Cryptographic Hash Functions to get PRGs • Chosen plain-text security • Pseudorandom generators (functions) -> CPA secure encryption

  3. Recall: using PRGs to encrypt longer messages • Key 𝑙 of length 𝑜 • Message 𝑛 of length 2𝑜 • A PRG 𝑕: 0,1 𝑜 → 0,1 2𝑜 • Enc 𝑙, 𝑛 = 𝑕 𝑙 ⊕ 𝑛 • Dec 𝑙, 𝑑 = 𝑑 ⊕ 𝑕(𝑙)

  4. How to make PRGs stretch the output more?

  5. Continuing Proof of security

  6. Two main questions: 1. How to get PRGs? 2. Is “indistinguishability - based security” enough in practice? a) How to define stronger security notions? b) How to achieve them again using PRGs!

  7. Cryptographic Hash Functions • Two general ways to talk about Hash functions: ℎ ∶ 0,1 ∗ → 0,1 𝑒 for a constant 𝑒 1. ℎ ∶ 0,1 𝑑 → 0,1 𝑒 for constants 𝑒, 𝑑 2. • The output is called the “message digest” • SHA1: 160-bit digest • SHA2: 224, 256, 384 or 512 bits • SHA3: digest size: arbitrary Key insight: a “secure” hash shall be unpredictable as it could be (practically like a random function) In particular, it should be pseudorandom! • http://www.sha1-online.com/ • https://emn178.github.io/online-tools/sha3_512.html

  9. Less Practical, but More Robust Constructions • PRGs based on “one way functions”

  10. Two main questions: 1. How to get PRGs? 2. Is “indistinguishability - based security” enough in practice? a) How to define stronger security notions? b) How to achieve them again using PRGs (or something similar!)

  11. What is wrong with Ind-based definition tailored to *one message* security games?

  12. Necessity for *Randomized* Encryption

  14. Security against Chosen Plaintext Attacks (CPA Security)

  15. Chosen Plaintext Security

  16. Next time 1. How to get PRGs? 2. Is “indistinguishability - based security” enough in practice? a) How to define stronger security notions? b) How to achieve CPA security using PRGs ( or something similar !)

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