Quantum-Safe Cryptography
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Quantum computing is a model of computing based on the quantum physics, which works differently than classical computers and can do things that classical computers can’t, such as breaking RSA and ECC efficiently. Quantum computers are not "faster computers" and they are not all-powerful and cannot do any computing job faster. Quantum computers are very efficient for certain problems and quite weak for others.
It is well known in computer science that quantum computers will break some cryptographic algorithms, especially the public-key cryptosystems like RSA, ECC and ECDSA that rely on the IFP (integer factorization problem), the DLP (discrete logarithms problem) and the ECDLP (elliptic-curve discrete logarithm problem). Quantum algorithms will not be the end of cryptography, because:
    Only some cryptosystems are quantum-unsafe (like RSA, DHKE, ECC, ECDSA and ECDH).
    Some cryptosystems are quantum-safe and will be only slightly affected (like cryptographic hashes, MAC algorithms and symmetric key ciphers).
Let's discuss this in details.

Quantum-Safe and Quantum-Broken Crypto Algorithms

Most cryptographic hashes (like SHA2, SHA3, BLAKE2), MAC algorithms (like HMAC and CMAK), key-derivation functions (bcrypt, Scrypt, Argon2) are basically quantum-safe (only slightly affected by quantum computing).
    Use 384-bits or more to be quantum-safe (256-bits should be enough for long time)
Symmetric ciphers (like AES-256, Twofish-256) are quantum-safe.
    Use 256-bits or more as key length (don't use 128-bit AES)
Most popular public-key cryptosystems (like RSA, DSA, ECDSA, EdDSA, DHKE, ECDH, ElGamal) are quantum-broken!
    Most digital signature algorithms (like RSA, ECDSA, EdDSA) are quantum-broken!
    Quantum-safe signature algorithms and public-key cryptosystems are already developed (e.g. lattice-based or hash-based signatures), but are not massively used, because of longer keys and longer signatures than ECC.
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Quantum-Resistant Crypto Algorithms
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ECC Cryptography and Most Digital Signatures are Quantum-Broken!

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A k-bit number can be factored in time of order O(k^3) using a quantum computer of 5k+1 qubits (using Shor's algorithm).
256-bit number (e.g. Bitcoin public key) can be factorized using 1281 qubits in 72*256^3 quantum operations.
    ~ 1.2 billion operations == ~ less than 1 second using good machine
ECDSA, DSA, RSA, ElGamal, DHKE, ECDH cryptosystems are all quantum-broken
Conclusion: publishing the signed transactions (like Ethereum does) is not quantum safe -> avoid revealing the ECC public key

Hashes are Quantum Safe

Cryptographic hashes (like SHA2, SHA3, BLAKE2) are considered quantum-safe:
    On traditional computer, finding a collision for 256-bit hash takes √2^256 steps (using the birthday attack) -> SHA256 has 2^128 crypto-strength
    Quantum computers might find hash collisions in ∛2^256 operations (see the BHT algorithm), but this is disputed (see [Bernstein 2009] - http://cr.yp.to/hash/collisioncost-20090823.pdf
    On theory it might take 2^85 quantum operations to find SHA256 / SHA3-256 collision, but in practice it may cost significantly more.
Conclusion: SHA256 / SHA3-256 are most probably quantum-safe
    SHA384, SHA512 and SHA3-384, SHA3-512 are quantum-safe
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Symmetric Ciphers are Quantum Safe

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Most symmetric ciphers (like AES and ChaCha20) are quantum-safe:
    [Grover's algorithm]([[[[[[https://en.wikipedia.org/wiki/Grover's_algorithm](https://en.wikipedia.org/wiki/Grover's_algorithm)](https://en.wikipedia.org/wiki/Grover's_algorithm](https://en.wikipedia.org/wiki/Grover's_algorithm))](https://en.wikipedia.org/wiki/Grover's_algorithm](https://en.wikipedia.org/wiki/Grover's_algorithm)](https://en.wikipedia.org/wiki/Grover's_algorithm](https://en.wikipedia.org/wiki/Grover's_algorithm)))](https://en.wikipedia.org/wiki/Grover's_algorithm](https://en.wikipedia.org/wiki/Grover's_algorithm)](https://en.wikipedia.org/wiki/Grover's_algorithm](https://en.wikipedia.org/wiki/Grover's_algorithm))](https://en.wikipedia.org/wiki/Grover's_algorithm](https://en.wikipedia.org/wiki/Grover's_algorithm)](https://en.wikipedia.org/wiki/Grover's_algorithm](https://en.wikipedia.org/wiki/Grover's_algorithm))))](https://en.wikipedia.org/wiki/Grover's_algorithm](https://en.wikipedia.org/wiki/Grover's_algorithm)](https://en.wikipedia.org/wiki/Grover's_algorithm](https://en.wikipedia.org/wiki/Grover's_algorithm))](https://en.wikipedia.org/wiki/Grover's_algorithm](https://en.wikipedia.org/wiki/Grover's_algorithm)](https://en.wikipedia.org/wiki/Grover's_algorithm](https://en.wikipedia.org/wiki/Grover's_algorithm)))](https://en.wikipedia.org/wiki/Grover's_algorithm](https://en.wikipedia.org/wiki/Grover's_algorithm)](https://en.wikipedia.org/wiki/Grover's_algorithm](https://en.wikipedia.org/wiki/Grover's_algorithm))](https://en.wikipedia.org/wiki/Grover's_algorithm](https://en.wikipedia.org/wiki/Grover's_algorithm)](https://en.wikipedia.org/wiki/Grover's_algorithm](https://en.wikipedia.org/wiki/Grover's_algorithm)))))](https://en.wikipedia.org/wiki/Grover's_algorithm]%28https://en.wikipedia.org/wiki/Grover's_algorithm%29]%28https://en.wikipedia.org/wiki/Grover's_algorithm]%28https://en.wikipedia.org/wiki/Grover's_algorithm%29%29]%28https://en.wikipedia.org/wiki/Grover's_algorithm]%28https://en.wikipedia.org/wiki/Grover's_algorithm%29]%28https://en.wikipedia.org/wiki/Grover's_algorithm]%28https://en.wikipedia.org/wiki/Grover's_algorithm%29%29%29]%28https://en.wikipedia.org/wiki/Grover's_algorithm]%28https://en.wikipedia.org/wiki/Grover's_algorithm%29]%28https://en.wikipedia.org/wiki/Grover's_algorithm]%28https://en.wikipedia.org/wiki/Grover's_algorithm%29%29]%28https://en.wikipedia.org/wiki/Grover's_algorithm]%28https://en.wikipedia.org/wiki/Grover's_algorithm%29]%28https://en.wikipedia.org/wiki/Grover's_algorithm]%28https://en.wikipedia.org/wiki/Grover's_algorithm%29%29%29%29]%28https://en.wikipedia.org/wiki/Grover's_algorithm]%28https://en.wikipedia.org/wiki/Grover's_algorithm%29]%28https://en.wikipedia.org/wiki/Grover's_algorithm]%28https://en.wikipedia.org/wiki/Grover's_algorithm%29%29]%28https://en.wikipedia.org/wiki/Grover's_algorithm]%28https://en.wikipedia.org/wiki/Grover's_algorithm%29]%28https://en.wikipedia.org/wiki/Grover's_algorithm]%28https://en.wikipedia.org/wiki/Grover's_algorithm%29%29%29]%28https://en.wikipedia.org/wiki/Grover's_algorithm]%28https://en.wikipedia.org/wiki/Grover's_algorithm%29]%28https://en.wikipedia.org/wiki/Grover's_algorithm]%28https://en.wikipedia.org/wiki/Grover's_algorithm%29%29]%28https://en.wikipedia.org/wiki/Grover's_algorithm]%28https://en.wikipedia.org/wiki/Grover's_algorithm%29]%28https://en.wikipedia.org/wiki/Grover's_algorithm]%28https://en.wikipedia.org/wiki/Grover's_algorithm%29%29%29%29%29)) finds AES secret key using √𝑁 quantum operations
    Quantum era will double the key size of the symmetric ciphers, see [http://cr.yp.to/codes/grovercode-20100303.pdf](http://cr.yp.to/codes/grovercode-20100303.pdf%29%29)
AES-256 in the post-quantum era is like AES-128 before
    128-bits or less symmetric ciphers are quantum-attackable
Conclusion: 256-bit symmetric ciphers are generally quantum safe
    AES-256, ChaCha20-256, Twofish-256, Camellia-256 are considered quantum-safe

Post-Quantum Cryptography

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Quantum-Safe key agreement: https://en.wikipedia.org/wiki/CECPQ1
Post-quantum signature scheme XMSS:
Post-quantum signatures and key agreements (XMSS, McEliece, NewHope): https://github.com/randombit/botan
QC-MDPC and libPQC are quantum-broken: https://eprint.iacr.org/2016/858.pdf

Hash-Based Public-Key Cryptography

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Code-Based Public-Key Cryptography

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Lattice-Based Public-Key Cryptography

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GLYPH signatures (lattice-based Ring-LWE Lattice, Ring-LWE, Ring Learning with Errors)
NewHope
XMSS
NTRU: NTRUEncrypt and NTRUSign

Zero-Knowledge Proof-Based

Multivariate-Quadratic-Equations Public-Key Cryptography

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Quantum-Resistant Cryptography - Libraries

The quantum-safe cryptography is still emerging, not mature, and still not widely supported by the most crypto-libraries and tools like Web browsers, OpenSSL, OpenSSH, etc. This is a list of well developed quantum crypto algorithm libraries:

SPHINCS+ Signatures in Python

NewHope Key Exchange in Python

Last modified 8d ago