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Cryptography and Cybersecurity

18 Jul 202523 min read

See also Reverse Engineering

Steganography

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Articles

Concepts

Mathematical prerequisites

See also the general section on Math for more.

Symmetric Cryptography

Advanced Encryption Standard (AES)

  • Block cipher
    • block length 128 bits
    • arranged in grid of 4 x 4 bytes
    • column-major order
  • 10, 12, 14 rounds according to key bit length - resp. 128, 192 and 256-bit keys
  • key schedule allows derivation of round keys from the user-specified encryption key
    • 128-bit key arranged as 4x4 byte array - column “words” are expanded into “schedule” of 44 words (i.e. 11x expansion)
    • each round consumes 4 “words” from key schedule - i.e. 1 of the 11x expanded keys
  • notion of “word” as 4 bytes - i.e. columns and rows are words
  • subsitution-permutation network (c.f. DES which is based on the Feistel Network - Feistel cipher - Wikipedia)
  • byte-oriented cipher (c.f. DES - bit-oriented) enables fast + convenient implementation
  • Based on “wide trail strategy” - as described in Security of a Wide Trail Design by Joan Daemen and Vincent Rijmen
    • combines (1) local non-linear step - confusion with (2) linear (global) mixing transformation - diffusion
    • phrase “wide trail” refers to dispersal of the probabilities that one can associate with the bits at certain specific positions in a bit block as it propagates through the rounds
  • AES Algorithm (high-level)
    • first expanded key words XOR’d with the input state array before moving onto the 10 rounds of processing for the 128-bit key
      • decryption: input ciphertext state array XOR’d with last four words of key schedule
    • 10x rounds (for 128-bit key) of (encryption):
      1. substitute bytes
      2. shift rows
      3. mix columns
        • based on multiplicative inverses in - Galois field where each bit is interpreted as (binary) coefficient in degree- polynomial
      4. “Add” round key (XOR)
    • Decryption:
      • Inverse shift rows
      • Inverse substitute bytes
      • XOR with (“Add”) round key
      • Inverse mix columns
    • Note: No (invert) mix columns step in final round (resp. decryption)

Data Encryption Standard (DES)

  • bit-oriented
  • DES is based on the Feistel structure in which, for each round, one-half of the block passes through unchanged and the other half goes through a transformation that depends on the S-boxes and the round key. Key alternating ciphers lend themselves well to theoretical analysis of the security of the ciphers. — Lecture 8: AES: The Advanced Encryption Standard by Avi Kak p. 7

Modes of Operation for Block Ciphers

  1. Electronic Code Book (ECB)
  2. Cipher Block Chaining Mode (CBC)
  3. Cipher Feedback Mode (CFB)
  4. Output Feedback Mode (OFB)
  5. Counter Mode (CTR)
  6. Galois/Counter Mode (GCM)

Hash Algorithms & Cryptographic Hash Functions

  • MurmurHash - Wikipedia used by
  • SHA-1 - Wikipedia
    • message digest (output or “signature”) length: 160-bit
    • block size: 512-bit blocks
    • initialized based on constants (see standard)
    • works on 5 32-bit words
    • 80 rounds
    • Merkle-Damgard (iterate on input until expended)
    • Collision resistance:
    • Marc Stevens showed collisions for SHA-1 when attacker can choose IV in 2015 and with large GPU compute (from Google) with SHA-1 IV in 2017 in: The first collision for full SHA-1
  • RIPEMD - Wikipedia - Developed by Dobbertin, Bossaelaers, Preneel in 1996 (developed in the open academic community, in contrast to the NSA-designed SHA-1 and SHA-2 hash algorithms)
    • Block size: 512 bits
    • Message digest length (n): 128, 160, 256, 320 bits
    • Number of rounds: 80
    • ĐąŃƒŃ€Đ”: Merkle-Damgard
    • Collision resistance: (due to birthday paradox)
    • (RIPEMD-160 is the only remaining 160 bit digest size hash function that has not been broken yet)
    • Bitcoin script allows RIPEMD-160 with the opcode OP_RIPEMD160
    • Bitcoin uses RIPEMD-160 (after using SHA-256) for address generation with the opcode OP-HASH160
  • SHA-2 - Wikipedia - uses 256 or 512 bit internal state; uses Merkle–DamgĂ„rd construction with Davies–Meyer compression function
  • SHA-3 - Wikipedia - Keccak, winner of SHA-3 competition, is designed by Guido Bertoni, Joan Daemen, MichaĂ«l Peeters and Gilles Van Assche
  • Desiderata of secure hash functions
    1. Preimage resistance: Given it is computationally infeasible to find s.t.
      • is this equivalent to one-wayness?
      • Brute force attack requires operations
      • Practical Use Case: Password storage: Practically store i.e. concat password and random salt and perform hash several times (to slow attacker brute force attack)
    2. Second Preimage resistance: Given both and it is computationally infeasible to find s.t.
      • Brute force attack requires operations
    3. Collision resistance: Given the hash function, it should be computationally infeasible to find s.t.
      • Brute force attack requires operations due to birthday paradox
      • Practical Use Case: Hacker produces software executables and and sends to Microsoft who approve and store on Windows OS to allow to run, but hacker distributes with exploit which is cleared for execution by kernel since
      • Why SHA-256 is called 128-bit security because 256/2 = 128 due to the birthday paradox
  • Birthday Paradox: How many people should be in a room before it is more likely than not that two people share a birthday? (i.e.
  • Merkle–DamgĂ„rd construction - Wikipedia
    • SHA-1, SHA-2
  • Sponge Construction: Sponge function - Wikipedia
    • SHA-3 (Keccak)
  • Merkle-Damgard and Sponge Constructions for Hashing - Cihangir Tezcan
  • MD4, MD5, RIPEMD, SHA1, SHA2 in Cryptocurrencies - Cihangir Tezcan
  • Password Hashing Competition - there wasn’t an established standard to fulfill the needs of modern applications and to best protect against attackers. We started the Password Hashing Competition (PHC) to solve this problem.
    • PHC ran from 2013 to 2015 as an open competition—the same kind of process as NIST’s AES and SHA-3 competitions, and the most effective way to develop a crypto standard. We received 24 candidates, including many excellent designs, and selected one winner, Argon2, an algorithm designed by Alex Biryukov, Daniel Dinu, and Dmitry Khovratovich from University of Luxembourg.

Message Authentication Codes (MAC) and Hash-based MAC (HMAC)

Asymmetric (Public-Private Key) Cryptography

Digital Signatures

Elliptic Curve Cryptography

NOTE

I suppose this could be placed under Key Exchange or Assymetric Cryptography since elliptic curves are drop-in - or plug in - replacements for the discrete logarithm (under modulus) problem / one-way function of the original Diffie-Hellman key exchange formulation.

Notes on Motivations for use of Elliptic Curve Cryptography from National Security AgencyCentral Security Service Web Site

The two noteworthy first generation public key algorithms used to secure the Internet today are known as RSA and Diffie-Hellman (DH). The security of the first is based on the difficulty of factoring the product of two large primes. The second is related to a problem known as the discrete logarithm problem for finite groups. Both are based on the use of elementary number theory. Interestingly, the security of the two schemes, though formulated differently, is closely related.


elliptic curve cryptography has remained at its full strength since it was first presented in 1985



The length of a key, in bits, for a conventional encryption algorithm is a common measure of security. To attack an algorithm with a k-bit key it will generally require roughly 2k-1 operations. Hence, to secure a public key system one would generally want to use parameters that require at least 2k-1 operations to attack. The following table gives the key sizes recommended by the National Institute of Standards and Technology to protect keys used in conventional encryption algorithms like the (DES) and (AES) together with the key sizes for RSA, Diffie-Hellman and elliptic curves that are needed to provide equivalent security.

Symmetric Key Size (bits)RSA and Diffie-Hellman Key Size (bits)Elliptic Curve Key Size (bits)
801024160
1122048224
1283072256
1927680384
25615360521
Table 1: NIST Recommended Key Sizes

To use RSA or Diffie-Hellman to protect 128-bit AES keys one should use 3072-bit parameters: three times the size in use throughout the Internet today. The equivalent key size for elliptic curves is only 256 bits.

One can see that as symmetric key sizes increase the required key sizes for RSA and Diffie-Hellman increase at a much faster rate than the required key sizes for elliptic curve cryptosystems.

Key Exchange

Certification and Certificate Authorities

Generate a new certificate (locally) for testing purposes with OpenSSL (Lecture 13 of Avi Kak’s course pp. 27):

openssl req -new -newkey rsa:1024 -days 365 -nodes -x509 -keyout test.pem -out test.cert

Differential Privacy

Post-Quantum Cryptography

Blockchains and Cryptocurrencies

Bitcoin

Ethereum

  • Ethereum - Wikipedia was proposed in a 2013 white paper by Vitalik Buterin - Wikipedia
  • In September 2022, Ethereum moved from Proof of Work (PoW) to Proof of Stake (PoS)
    • Prior to September 2022 In its proof of work algorithm ethash, Ethereum uses Keccak but it should be noted that it is slightly different than the standardized version of Keccak, namely SHA-3. A late change in the standardization process of SHA-3 resulted in a different padding than the one used in ethash.
      • Thus, sometimes the used hash algorithms in ethash are referred to as Keccak-256 and Keccak-512, instead of SHA3-256 and SHA3-512.

Mining Puzzles with SHA-3 Competition Algorithms

Some cryptocurrencies use multiple hash functions for their hash puzzles. This complexity is expected to prevent efficient ASIC development:

  1. ï»żï»żX11: Blake, BMW, Groestl, JH, Keccak, Skein, Luffa, Cubehash, Shavite, Simd, Echo
  2. ï»żï»żX13: BLAKE, BMW, Groestl, JH, Keccak, Skein, Luffa, Cubehash, Shavite, Simd, Echo, Hamsi and Fugue
  3. X15: Blake, BMW, Groestl, JH, Keccak, Skein, Luffa, Cubehash, Shavite, Simd, Echo, Hamsi, Fugue, Shabal, and Whirlpool
  4. ï»żï»żX16: the same 15 algorithms used in X15 model + SHA512
  5. ï»żï»żX16r: rotates through 16 different algorithms in random
  6. ï»żï»żX17: Blake, BMW, Groestl, JH, Keccak, Skein, Luffa, Cubehash, Shavite, Simd, Echo, Hamsi, Fugue, Shabal, Whirlpool, Loselosei, and Dib2

Blockchain & Cryptocurrency Technologies - Cihangir Tezcan

  1. Intro to Blockchain and Cryptocurrency Technologies
  2. Why You Do NOT Need a Blockchain
  3. Hashing in Blockchains and Cryptocurrencies
  4. Merkle-Damgard and Sponge Constructions for Hashing
  5. MD4, MD5, RIPEMD, SHA1, SHA2 in Cryptocurrencies
  6. SHA3 in Cryptocurrency Mining
  7. Public Key Cryptography for Blockchains
  8. Algebra for Blockchains & Cryptocurrencies
  9. Elliptic Curve Cryptography for Blockchains & Cryptocurrencies
  10. Discrete Logarithm Problem for Blockchains & Cryptocurrencies
  11. Digital Signatures for Blockchains & Cryptocurrencies
  12. Elliptic Curve Digital Signature Algorithm for Blockchains & Cryptocurrencies
  13. Bitcoin Signature Verification
  14. Blockchain Technology Introduction
  15. Blocks in Blockchains
  16. Consensus Models
  17. BITCOIN
  18. BITCOIN Blocks
  19. BITCOIN Wallets
  20. BITCOIN Transactions
  21. BITCOIN Pay to Public-key Hash
  22. BITCOIN Pay to Script Hash
  23. Soft and Hard BITCOIN Forks
  24. BITCOIN Mining
  25. Multi-Signatures in BITCOIN
  26. MuSig: Schnorr Multi-Signatures for BITCOIN
  27. How to Find Hidden Messages in Bitcoin Blockchain
  28. Alternative COINS
  29. ETH Ethereum
  30. NFT Non-Fungible Token
  31. NFT Non-Fungible Token Security Issues
  32. Stellar: Connecting the World’s Financial Infrastructure
  33. Ben Kriptocu GördĂŒm! - Aydaki Adam: Ä°lker Canikligil - Konuk: Cihangir Tezcan - B48

Randomness

Attack Types

Clickjacking

Clickjacking or formally UI Redressing: attacker “redresses” the interface seen by the user with an invisible frame that tricks them into doing things they wouldn’t ordinarily do

Secure Shell (SSH)

WiFi (Wireless) Security

Password managers

Pretty Good Privacy (PGP) and GNU Privacy Guard (GnuPG or GPG)

Implementations

👉 See Also

Graph View

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