Public-key cryptography and invariant theory

Public-key cryptosystems are suggested based on invariants of groups. We give also an overview of the known cryptosystems which involve groups.Comment: 10 pages, LaTe

Folding Alternant and Goppa Codes with Non-Trivial Automorphism Groups

The main practical limitation of the McEliece public-key encryption scheme is probably the size of its key. A famous trend to overcome this issue is to focus on subclasses of alternant/Goppa codes with a non trivial automorphism group. Such codes display then symmetries allowing compact parity-check or generator matrices. For instance, a key-reduction is obtained by taking quasi-cyclic (QC) or quasi-dyadic (QD) alternant/Goppa codes. We show that the use of such symmetric alternant/Goppa codes in cryptography introduces a fundamental weakness. It is indeed possible to reduce the key-recovery on the original symmetric public-code to the key-recovery on a (much) smaller code that has not anymore symmetries. This result is obtained thanks to a new operation on codes called folding that exploits the knowledge of the automorphism group. This operation consists in adding the coordinates of codewords which belong to the same orbit under the action of the automorphism group. The advantage is twofold: the reduction factor can be as large as the size of the orbits, and it preserves a fundamental property: folding the dual of an alternant (resp. Goppa) code provides the dual of an alternant (resp. Goppa) code. A key point is to show that all the existing constructions of alternant/Goppa codes with symmetries follow a common principal of taking codes whose support is globally invariant under the action of affine transformations (by building upon prior works of T. Berger and A. D{\"{u}}r). This enables not only to present a unified view but also to generalize the construction of QC, QD and even quasi-monoidic (QM) Goppa codes. All in all, our results can be harnessed to boost up any key-recovery attack on McEliece systems based on symmetric alternant or Goppa codes, and in particular algebraic attacks.Comment: 19 page

Isogeny-based post-quantum key exchange protocols

The goal of this project is to understand and analyze the supersingular isogeny Diffie Hellman (SIDH), a post-quantum key exchange protocol which security lies on the isogeny-finding problem between supersingular elliptic curves. In order to do so, we first introduce the reader to cryptography focusing on key agreement protocols and motivate the rise of post-quantum cryptography as a necessity with the existence of the model of quantum computation. We review some of the known attacks on the SIDH and finally study some algorithmic aspects to understand how the protocol can be implemented

Post Quantum Cryptography from Mutant Prime Knots

By resorting to basic features of topological knot theory we propose a (classical) cryptographic protocol based on the `difficulty' of decomposing complex knots generated as connected sums of prime knots and their mutants. The scheme combines an asymmetric public key protocol with symmetric private ones and is intrinsecally secure against quantum eavesdropper attacks.Comment: 14 pages, 5 figure

Algorithms for Determining the Order of the Group of Points on an EllipticCurve with Application in Cryptography

Eliptické křivky jsou rovinné křivky, jejíž body vyhovují Weierstrassově rovnici. Jejich hlavní využití je v kryptografii, kde představují důležitý nástroj k tvorbě těžko rozluštitelných kódů bez znalosti klíče, který je v porovnání s ostatními šifrovacími systémy krátký. Díky těmto přednostem jsou hojně využívány. Abychom mohli kódovat a dekódovat zprávy v systému eliptických křivek, musíme znát řád dané eliptické křivky. K jeho získání se mimo jiné používá Shanksův algoritmus a jeho vylepšená varianta, Mestreho algoritmus.The elliptic curves are plane curves whose points satisfy the Weierstrass equation. Their main application is in the cryptography, where they represent an important device for creating code which is hard to break without knowing the key and which is short in comparison with other encoding methods. The elliptic curves are widely used thanks to these advantages. To be able to code and decode in the elliptic curve cryptography we must know the order of the given elliptic curve. The Shank's algorithm and its improved version, the Mestre's algorithm, are used for its determining.