An introduction to post-quantum cryptography
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1 An introduction to post-quantum cryptography Andrew Savchenko NRNU MEPhI, Moscow, Russia LVEE Winter 2016
2 Outline 1 Terminology 2 Quantum computing 3 Impact on cryptographic algorithms 4 Free software solutions 5 Summary
3 What is what: Terminology Classical cryptography usual cryptography, designed to withstand cryptanalysis using classical computers Postquantum cryptography cryptography resilient to quantum computing Quantum cryptography has nothing to do with post-quantum cryptography It uses quantum mechanical properties of the matter for crypto applications, eg: secure key distribution using entangled particles protection from data copying Requires a very dedicated hardware and connection lines
4 What is what: Terminology Classical cryptography usual cryptography, designed to withstand cryptanalysis using classical computers Postquantum cryptography cryptography resilient to quantum computing Quantum cryptography has nothing to do with post-quantum cryptography It uses quantum mechanical properties of the matter for crypto applications, eg: secure key distribution using entangled particles protection from data copying Requires a very dedicated hardware and connection lines
5 What is what: Terminology Classical cryptography usual cryptography, designed to withstand cryptanalysis using classical computers Postquantum cryptography cryptography resilient to quantum computing Quantum cryptography has nothing to do with post-quantum cryptography It uses quantum mechanical properties of the matter for crypto applications, eg: secure key distribution using entangled particles protection from data copying Requires a very dedicated hardware and connection lines
6 Quantum computing Base elements: qubits (quantum bits) quantum logic gates quantum algorithm: sequence of quantum gates applied to qubits
7 Qubits Each classical bit can be either 0 or 1, but qubits: can be 0 and 1 at the same time contain wave function of both states (eg superposition + ) but with different probabilities usually electron spin is used, but any discernible quantum number is OK N qubits 2 N states at the same time vs N classical bits 1 state at once
8 Quantum gates Quantum logic gate: is a rotation (unitary transformation) applied to the wave functions (usually of 1 or 2 qubits) They provide full set of logical operations All quantum gates are reversible in contrast to classical gates
9 Quantum computing 2 Caveats: only N bit can be extracted from 2 N states measurement (wave function collapse) is probabilistic: = 5 OK! but P (2 + 2 = 4) > P (2 + 2 = 5) Summary: internal data reduction upon measurement results must be either: checked or repeated many times
10 Quantum computing 2 Caveats: only N bit can be extracted from 2 N states measurement (wave function collapse) is probabilistic: = 5 OK! but P (2 + 2 = 4) > P (2 + 2 = 5) Summary: internal data reduction upon measurement results must be either: checked or repeated many times
11 Solves integer factorisation problem: Shor s algorithm for known P find N 1, N 2 : N 1 N 2 = P, where N 1, N 2 primes Steps: (C) factoring problem function f period finding problem (Q) period finding: Hadamard transformation (equal probability superposition) apply f as quantum transform apply quantum Fourier transform measure the period (C) obtain prime from the period For details see [1]
12 Solves integer factorisation problem: Shor s algorithm for known P find N 1, N 2 : N 1 N 2 = P, where N 1, N 2 primes Steps: (C) factoring problem function f period finding problem (Q) period finding: Hadamard transformation (equal probability superposition) apply f as quantum transform apply quantum Fourier transform measure the period (C) obtain prime from the period For details see [1]
13 Factorisation complexity Complexity estimation for N : Algo ( Complexity Operations GNFS O e 19(ln N)1/3 (ln ln N) 2/3) ( ) Shor s O (ln N) 2 (ln ln N) (ln ln ln N) 10 8 GNFS General number field sieve, the fastest classical factorisation algorithm
14 Grover s algorithm A quantum brute-force (BF) algorithm Black box setup: 1 known output N unknown inputs Complexity estimation for N : Algo Complexity Operations BF O (N) 10 ( ) 77 Grover s O N For details see [2]
15 Complexity classes PSPACE problems NP Problems Grover NP Complete Shor P Problems BQP P easy to solve and verify NP hard to solve, but easy to verify BQP easy to solve on quantum computer and verify
16 Impact on crypto algos Symmetric crypto: Key sizes are halved Common asymmetric crypto: Elliptic curves are very dead RSA and alike are dead Quantum resistant asymmetric crypto: Hash-based [3] Lattice-based [3] Code-based [3] Multivariative quadratic equations [3] Supersingular elliptic curve isogeny [4]
17 Impact on crypto algos Symmetric crypto: Key sizes are halved Common asymmetric crypto: Elliptic curves are very dead RSA and alike are dead Quantum resistant asymmetric crypto: Hash-based [3] Lattice-based [3] Code-based [3] Multivariative quadratic equations [3] Supersingular elliptic curve isogeny [4]
18 Impact on crypto algos Symmetric crypto: Key sizes are halved Common asymmetric crypto: Elliptic curves are very dead RSA and alike are dead Quantum resistant asymmetric crypto: Hash-based [3] Lattice-based [3] Code-based [3] Multivariative quadratic equations [3] Supersingular elliptic curve isogeny [4]
19 D-Wave Systems Computing in adiabatic quantum approximation [5] Declared to be suitable only for discrete optimisation [5, 6] Operates just at 0015K :)
20 D-Wave Systems Timeline qubits:time qubits time Present: qubits from 2048-qubit chip
21 Timeline Year 2012 CIA $30m investment in D-Wave [7] Year = factored on 4 qubits [8] at 300 K! mr Snowden revealed that NSA spent $ 80m on a quantum computer development [9, 10] Year 2015 NSA announce development of Suite B algorithms resistible to quantum computing [11]
22 Free software solutions Codecrypt [12] GnuPG-like encryption tool Signatures: hash-tree based (Merkle-tree signature) Asymmetric encryption: code based McEliece cryptosystem [13]) Symmetric encryption: up to 4096-bit keys Keys generation: ccr --gen-key sig --name John Doe # signature key ccr --gen-key enc --name John Doe # encryption key # the same with manual key choise ccr -g FMTSEQ256H20C-CUBE512-CUBE256 -N username ccr -g MCEQCMDPC256FO-SHA512-CHACHA20 -N username
23 Free software solutions Codecrypt [12] GnuPG-like encryption tool Signatures: hash-tree based (Merkle-tree signature) Asymmetric encryption: code based McEliece cryptosystem [13]) Symmetric encryption: up to 4096-bit keys Keys generation: ccr --gen-key sig --name John Doe # signature key ccr --gen-key enc --name John Doe # encryption key # the same with manual key choise ccr -g FMTSEQ256H20C-CUBE512-CUBE256 -N username ccr -g MCEQCMDPC256FO-SHA512-CHACHA20 -N username
24 Free software solutions Sign and encrypt: ccr -se -r Frank < lettertxt > letterccr Decrypt and verify: ccr -dv -o replytxt < replyccr Upstream is very dynamic and responsive Quantum Computing Language: QCL [14] quantum assembler and routines emulator of a quantum computer
25 Free software solutions Sign and encrypt: ccr -se -r Frank < lettertxt > letterccr Decrypt and verify: ccr -dv -o replytxt < replyccr Upstream is very dynamic and responsive Quantum Computing Language: QCL [14] quantum assembler and routines emulator of a quantum computer
26 Summary Symmetric cryptography is still secure, but double key size Drop RSA and elliptic curves in the long run Use codecrypt and other systems, but with caution Combine multiple crypto systems Outlook: Popular projects (*SSL, GnuPG) should include post-quantum algos one day Raise post-quantum awareness Thank you for your attention!
27 Summary Symmetric cryptography is still secure, but double key size Drop RSA and elliptic curves in the long run Use codecrypt and other systems, but with caution Combine multiple crypto systems Outlook: Popular projects (*SSL, GnuPG) should include post-quantum algos one day Raise post-quantum awareness Thank you for your attention!
28 Bibliography I Shor s algorithm URL: s_algorithm Grover s algorithm URL: s_algorithm Bernstein Daniel J, Buchmann Johannes, Dahmen Erik Post-quantum cryptography Berlin : Springer, 2009 ISBN: URL: document/cda_downloaddocument/ c1pdf Supersingular isogeny Diffie Hellman key exchange URL: https: //enwikipediaorg/wiki/supersingular_isogeny_key_exchange D-Wave Systems URL: D-Wave Systems URL:
29 Bibliography II CIA and Amazon Founder Greedily Eye D-Wave s Quantum Computer URL: Greedily+Eye+DWaves+Quantum+Computer/article27866htm Dattani Nikesh S, Bryans Nathaniel Quantum factorization of with only 4 qubits 2014 URL: Snowden docs: NSA building encryption-cracking quantum computer URL: show_nsa_building_encryptioncracking_quantum_system/ NSA seeks to build quantum computer that could crack most types of encryption URL: nsa-seeks-to-build-quantum-computer-that-could-crack-most-types-o 2014/01/02/8fff297e e3-8def-a df2_storyhtml NSA Suite B Cryptography URL: https: //wwwnsagov/ia/programs/suiteb_cryptography/indexshtml
30 Bibliography III Codecrypt (post-quantum crypto suite) URL: McEliece cryptosystem URL: QCL (quantum computing language and quantum computer emulator) URL:
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