This document discusses post-quantum cryptography and code-based cryptography as a potential solution. It provides an overview of cryptography, both symmetric and asymmetric, and explains how quantum computers could break many current systems by solving mathematical problems efficiently. Code-based cryptography is introduced as an alternative that does not rely on these vulnerable problems. The McEliece cryptosystem and Staircase code-based schemes are described. The document then outlines a project to implement a random split of Staircase codes to thwart information set decoding attacks, including researching the topic, developing implementations, validating the approach works as intended, and verifying the results against benchmarks. It emphasizes that development should begin now to have solutions ready when needed.

1. Post Quantum
Cryptography
With random split of
St-Gen Codes

2. Cryptography
● A very old science that has existed since the roman times.
● Nowadays it is deeply integrated into everyday life.
● Just a few out of many example uses:
○ Securing online sessions (SSL)
○ Present in almost every texting application.
○ The OpenPGP standard for encrypting.
● No longer only concerned with confidentiality of information.
● Digital signatures can provide authentication and integrity.
● Cryptographic onions can be used for anonymity.

3. Cryptography
● Symmetric Cryptography:
○ Makes use of a single key for encryption and decryption
○ Can operate on blocks or streams of bytes
○ Most popular examples are AES, DES, 3DES
● Asymmetric Cryptography:
○ Each party must generate two different key
○ Public key is given out to encrypt incoming messages
○ Private key is kept hidden to be used for decryption
○ Can be used for other primitives such as digital signature
○ Most famous example is the RSA algorithm

4. Cryptographic Problems
Security is achieved through hardness of mathematical problems.
The Factorization Problem:
Given n = pq where p and q are unique prime numbers. Find p and q in polynomial
time.
The Discrete Logarithm Problem:
Given β=αa
, find a in polynomial time.
No efficient classical algorithms exist for either of the two problems.

5. Quantum Computers
● Classical Computers use bits to
store information.
● Always in one of two states at any
point in time (0 or 1).
● In contrast, quantum computers
have Qubits.
● Can be in two states
simultaneously (0 and 1).
● A quantum computer with two
bits can be in and act upon 4
states at the same time.

6. Classical Cryptography Under Quantum
Attacks
● It started with Shor’s algorithm
developed by Peter Shor in 1994.
● Can factorize a composite number
N in polynomial time.
● Demonstrated that public key
cryptography algorithms can be
broken.

7. What now ? (Post Quantum Cryptography)
● We need to update our current cryptographic primitives to be able to deal with
the new threat.
● Fortunately we do not have to start from scratch.
● Cryptographic classes that do not rely on vulnerable mathematical problems
already exist.
○ Code based cryptography
○ Multivariate public key cryptography
○ Lattice based cryptography
○ Hash based cryptosystems

8. Code based cryptography
● Builds on the concepts introduced by
Claude Shannon in 1948.
● Coding theory was developed to be able
to retrieve the original message after
transmission through a noisy channel.
● A concept that is easily adaptable to
cryptography.
● Artificial noise can be applied to a
message to hide its contents.
● The original recipient can recover the
message by knowing additional
information about the encoding scheme.

9. Linear Codes and List Decoding
Linear Codes
● A commonly used subfamily of block
codes.
● The original message is represented by
K, a vector of length k.
● The output is N, a new vector of length
n where k < n.
● The encoding operation is represented
by c’=Gm where G is known as the
generator matrix.
● C = c’ + e.
List Decoding Algorithm
● There are several ways to retrieve
the original message from K.
● The List decoding algorithm allows
us to recover from larger number
of errors.
● It returns several possibilities for
the original message with one
having an overwhelming
probability.

10. The McEliece Cryptosystem
● Code based cryptosystem proposed by McEliece in 1978.
● Many following variations and attempted improvements including Niederreiter
in 1986.
● First successful digital signature scheme is as recent as 2001.
● While requiring some modifications over the years, it remains unbroken after
near 30 years of cryptanalysis.
● Despite faster encryption and decryption procedures, never received much
popularity.
● Key size is 32 KBytes compared to 4096 bits for RSA.

11. Staircase Generated Codes
● In 2014, a new family of linear codes was introduced as staircase generated
codes
● Based on it, a new variation of the McEliece cryptosystem was proposed
including an encryption and signature scheme.
● The new scheme imposes restrictions on the structure of the generator matrix
allowing for more efficient list decoding algorithm.
● It also gives the sender control over the noise generated by the “noisy channel”
by defining two parameters: density and granularity.
● Encryption scheme can be adapted directly into a signature scheme using the
decryption algorithm.

12. Random Split of St-Gen Codes
● A successful attack using Information Set Decoding was later demonstrated.
● ISD is a technique to recover the error vector used to encrypt the message.
● Which can in turn be used for practical key recovery.
● Exposing the staircase generator matrix allows for structural attacks.
● To thwart the ISD attack a new idea is introduced to split the public generator
matrix into s randomly generated matrices.
● With the random split, the probability of a successful attack becomes negligible.

13. Methodology
● This project is mainly concerned with the implementation of the newly
proposed techniques for thwarting ISD attacks against St-Gen Codes.
● Several implementations will be done using multiple platforms (python, C++)
● A set of evaluation metrics must be decided on for comparison with other
solutions.
● The validity of the ISD attacks must be retested against the new
implementations.
● Concrete security parameters will be tested against the expected security level.

14. Project Plan
● Phase 1 : Extensive Research
○ Study existing implementations
○ Study the proposed attack and the proposed solution
○ Further research on the inner workings of McEliece, St-Gen Codes and ISD attacks
● Phase 2 : Implementation
○ Start implementation of the encryption scheme using St-Gen codes with random splits.
○ Implement the Signature scheme based on the decryption algorithm.
● Phase 3 : Validation
○ Insure that the proposed solution holds against the proposed attacks.
○ Check for any newly created security holes.
● Phase 4 : Verification
○ Compare the final implementation against existing benchmarks.
○ Demonstrate the workings for the implementation using concrete parameters.

15. Why start now?
● Efficiency
○ Even with a secure working system, many improvements to speed and space requirements need
to be done.
● Confidence
○ Users require time to put their faith in a new cryptographic scheme.
○ Many mistakes should be expected.
● Usability
○ A new cryptographic scheme requires the development of a new infrastructure.
○ Padding schemes, handling of longer messages, software and hardware implementations.

16. Thank you.
Questions ?