Pedro Hecht

Pedro Hecht

Post-Quantum Cryptography PQC attempts to find cryptographic protocols resistant to attacks using Shors polynomial time algorithm for numerical field problems or Grovers algorithm to find the unique input to a black-box function that produces a particular output value. The use of non-standard algebraic structures like non-commutative or non-associative structures, combined with one-way trapdoor functions derived from combinatorial group theory, are mainly unexplored choices for these new kinds of protocols and overlooked in current PQC solutions. In this paper, we develop an algebraic extension ring framework who could be applied to different asymmetric protocols, i.e. key exchange, key transport, enciphering, digital signature, zero-knowledge authentication, oblivious transfer, secret sharing etc.. A valuable feature is that there is no need for big number libraries as all arithmetic is performed in F256 extension field operations (precisely the AES field). We assume that the new framework is cryptographical secure against strong classical attacks like the sometimes-useful length-based attack, Romankovs linearization attacks and Tsabans algebraic span attack. This statement is based on the non-linear structure of the selected platform which proved to be useful protecting the AES protocol. Otherwise, it could resist post-quantum attacks Grover, Shor and be particularly useful for computational platforms with limited capabilities like USB cryptographic keys or smartcards. Semantic security IND-CCA2 could also be inferred for this new platform.

Pedro Hecht

Post-Quantum Cryptography (PQC) attempts to find cryptographic protocols resistant to attacks using Shor polynomial time algorithm for numerical field problems or Grover search algorithm. A mostly overlooked but valuable line of solutions is provided by non-commutative algebraic structures, specifically canonical protocols that rely on one-way trapdoor functions (OWTF). Here we develop an algebraic framework who could be applied to different asymmetric protocols like D-H KE (Diffie-Hellman key exchange), Public Key Encryption, Digital Signature, ZKP (zero-knowledge proof) authentication, Oblivious Transfer, Multi-Party Computing, and so on. The trapdoor one-way functions selected are (a) Triple decomposition Problem (TDP) developed by Kurt, where a known element is factored into a product of three unknown factors and (b) a new version of conjugacy search that we refer from now on as Blind Conjugacy Search Problem (BCSP). Our platform structure is the general linear group GL(d,F_p) d-square non-singular matrices of prime field values. We give support to the fact that this framework is cryptographically secure against classical attacks like linear algebra attacks, length-based attacks, side-channel attacks against square (or duplicate) and multiply (or sum) algorithm, high sensitivity to pseudo random deterministic generators, etc. At same time it is immune against quantum attacks (using Grover and Shor), if the size parameters are carefully selected. Semantic security and IND-CCA2 compliance for this framework is discussed.

Pedro Hecht

Currently there is an active Post-Quantum Cryptography (PQC) solutions search, which attempts to find cryptographic protocols resistant to attacks by means of for instance Shor polynomial time algorithm for numerical field problems like integer factorization (IFP) or the discrete logarithm (DLP). The use of non-commutative or non-associative structures are, among others, valid choices for these kinds of protocols. In our case, we focus on a permutation subgroup of high order and belonging to the symmetric group S381. Using adequate one-way functions (OWF), we derived a Diffie-Hellman key exchange and an ElGamal ciphering procedure that only relies on combinatorial operations. Both OWF pose hard search problems which are assumed as not belonging to BQP time-complexity class. Obvious advantages of present protocols are their conceptual simplicity, fast throughput implementations, high cryptanalytic security and no need for arithmetic operations and therefore extended precision libraries. Such features make them suitable for low performance and low power consumption platforms like smart cards, USB-keys and cellphones.

Pedro Hecht

In this paper, we present a simple bare-bones solution of a Zero-Knowledge authentication protocol which uses non-commutative algebra and a variation of the generalized symmetric decomposition problem (GSDP) as a one-way function. The cryptographic security is assured as long the GSDP problem is computationally hard to solve in non-commutative algebraic structures and belongs currently to the PQC category as no quantum computer attack is likely to exists.

Pedro Hecht

Post-quantum cryptography (PQC) attempts to find cryptographic protocols resistant to attacks using for instance Shor's polynomial time algorithm for numerical field problems like integer factorization (IFP) or the discrete logarithm (DLP). Other aspects are the backdoors discovered in deterministic random generators or recent advances in solving some instances of DLP. Using alternative algebraic structures like non-commutative or non-associative partial groupoids, magmas, monoids, semigroups, quasigroups or groups, are valid choices for these new protocols. This paper focuses on an asymmetric cipher based on a generalized ElGamal non-arbitrated protocol using a non-commutative general linear group. The developed protocol forces a hard subgroup membership search problem into a non-commutative structure. The protocol involves at first a generalized Diffie-Hellman key interchange and further on the private and public parameters are recursively updated each time a new cipher session is launched. Security is based on a hard variation of the Generalized Symmetric Decomposition Problem (GSDP). Working with GL(8, F251) 64-bit security is achieved, and if GL(16, F251) is chosen, the security rises to 127-bit. An appealing feature is that there is no need for big number libraries as all arithmetic is performed in Z_251. Therefore the new protocol is particularly useful for computational platforms with very limited capabilities like smartphones or smartcards.