Non-Commutative Key Exchange Protocol

We introduce a novel key exchange protocol based on non-commutative matrix multiplication defined in $\mathbb{Z}_p^{n \times n}$. The security of our method does not rely on computational problems as integer factorization or discrete logarithm whose difficulty is conjectured. We claim that the unique eavesdropper's opportunity to get the secret/private key is by means of an exhaustive search which is equivalent to the unsorted database search problem. Furthermore, we show that the secret/private keys become indistinguishable to the eavesdropper. Remarkably, to achieve a 512-bit security level, the keys (public/private) are of the same size when matrix multiplication is done over a reduced 8-bit size modulo. Also, we discuss how to achieve key certification and Perfect Forward Secrecy (PFS). Therefore, Lizama's algorithm becomes a promising candidate to establish shared keys and secret communication between (IoT) devices in the quantum era.

Non-Commutative Key Exchange Protocol

We introduce a novel key exchange protocol based on non-commutative matrix multiplication. The security of our method does not rely on computational problems as integer factorization or discrete logarithm whose difficulty is conjectured. We claim that the unique opportunity for the eavesdropper to get the private key is by means of an exhaustive search which is equivalent to searching an unsorted database problem. Therefore, the algorithm becomes a promising candidate to be used in the quantum era to establish shared keys and achieve secret communication. Furthermore, to establish a 256-bit secret key the size of the public key only requires 256 bits while the private key occupies just 384 bits. Matrix multiplications can be done over a reduced 4-bit size modulo. Also, we show that in a generalized method, private numbers become indistinguishable and we discuss how to achieve Perfect Forward Secrecy (PFS). As a consequence, Lizama's protocol becomes a promising alternative for Internet-of-Things (IoT) computational devices in the quantum era.

Karakteristik Matriks sebagai Daerah Asal Suatu Logaritma

Abstrak Rumus umum fungsi logaritma asli dengan daerah asal suatu matriks adalah ln⁡A=T S_((J_A ) ) {ln⁡〖(λ_1 I^((p_1 ) )+H^((p_1 ) ) ),ln⁡(λ_2 I^((p_2 ) )+H^((p_2 ) ) ),…,ln⁡(λ_u I^((p_u ) )+H^((p_u ) ) ) 〗 } 〖S_((J_A ) )〗^(-1) T^(-1) dengan T adalah matriks non-singular dimana A=TJ_A T^(-1), S_((J_A ) )adalah sebarang matriks yang komutatif dengan J_A, J_A adalah matriks Jordan dari matriks A, λ_i adalah nilai karakteristik dari pembagi elementer A, I adalah matriks identitas dan H^((p)) adalah matriks berukuran p×p yang mempunyai 1 sebagai anggota pada superdiagonal pertama dan 0 untuk lainnya. Karakteristik matriks A sebagai daerah asal suatu fungsi logaritma adalah matriks persegi yang non-singular dengan nilai-nilai karakteristik real positif Kata Kunci: matriks, daerah asal, logaritma asli Abstract The general formula of the natural logarithm function with domain of a matrix is ln⁡A=T S_((J_A ) ) {ln⁡〖(λ_1 I^((p_1 ) )+H^((p_1 ) ) ),ln⁡(λ_2 I^((p_2 ) )+H^((p_2 ) ) ),…,ln⁡(λ_u I^((p_u ) )+H^((p_u ) ) ) 〗 } 〖S_((J_A ) )〗^(-1) T^(-1) with T is the non-singular matrix which A=TJ_A T^(-1), S_((J_A ) ) is any commutative matrix with J_A, J_Ais the Jordan matrix of the matrix A, λ_i is the characteristic value of the elementary divider A, I is the identity matrix and H^((p)) is a square matrix which has 1 as a member of the first superdiagonal and 0 for other. The characteristic of matrix A as domain of a natural logarithm function is a non-singular square matrix with real positive characteristic values Keywords: matrix, domain, natural logarithm