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ABSTRACT (original article): Post-quantum cryptography is essential for securing digital communications against threats posed by quantum computers. Researchers have focused on developing algorithms that can withstand attacks from both classical and quantum computers, thereby ensuring the security of data transmissions over public networks. A critical component of this security is the key agreement protocol, which allows two parties to establish a shared secret key over an insecure channel. This paper introduces two novel post-quantum key agreement protocols that can be easily implemented on standard computers using rectangular or rank-deficient matrices, exploiting the generalizations of the matrix power function, which is a generator of NP-hard problems. We provide basic concepts and proofs, pseudocodes, and examples, along with a discussion of complexity. CITATION (original article): Hecht, J. P., & Scolnik, H. D. (2025). Post-Quantum Key Agreement Protocols Based on Modified Matrix-Power Functions over Singular Random Integer Matrix Semirings. Computer Networks and Communications, 3(1), 1–18. https://doi.org/10.37256/cnc.3120256112

This is a simple, secure and fast program to generate session keys. Obviously in a real production environment, it must be translated into a fast language such as RUST, Java or C++. The algebraic methods on which it is based are sophisticated but easy to use. All details can be found in the published work.
The notebook has two parts, in the first one it generates the keys in numerical form (in matrix format). In the second part, the core of the algorithm used is developed in symbolic form, which allows to appreciate the computational complexity that blocks both classical and quantum cryptanalytic attacks.

Definitions


Multi σ-MPF square rank-deficient BENCH

Generates the keys in numerical form (in matrix format).

In[]:=

Randomize;Print["....................................."];Print["dim = ",rows=cols=5];Print["prime = ",prime=65537];Print["Zlimit = ",Zlimit=2^10];sigma=RandomInteger[{-Zlimit,+Zlimit}];Print["sigma = ",sigma];Print["expTop = ",expTop=10];Print["total rounds = ",rounds=1];Print["verbose = ",verbose=True];Print["mini verbose = ",miniverbose=True];Print["....................................."];(*Wfullrank*)W=Rmat;If[verbose,Print["W = ",MatrixForm[W]]];BaseXU=RmatDef;If[verbose,Print["BaseXU = ",MatrixForm[BaseXU]]];BaseYV=RmatDef;If[verbose,Print["BaseYV = ",MatrixForm[BaseYV]]];Print["....................................."];Alist=Blist=KeyAlist=KeyBlist={};Timing[Module[{round=0,zerokeys=0},Label[begin];round+=1;Label[again];X=Mod[MatrixPower[BaseXU,randX=RandomInteger[{2,expTop}]],prime-1];If[verbose,Print["randX = ",randX]];If[verbose,Print["X = ",MatrixForm[X]]];Y=Mod[MatrixPower[BaseYV,randY=RandomInteger[{2,expTop}]],prime-1];If[verbose,Print["randY = ",randY]];If[verbose,Print["Y = ",MatrixForm[Y]]];U=Mod[MatrixPower[BaseXU,randU=RandomInteger[{2,expTop}]],prime-1];If[verbose,Print["randU = ",randU]];If[verbose,Print["U = ",MatrixForm[U]]];V=Mod[MatrixPower[BaseYV,randV=RandomInteger[{2,expTop}]],prime-1];If[verbose,Print["randV = ",randV]];If[verbose,Print["V = ",MatrixForm[V]]];A=MPFint[X,W,Y];If[verbose,Print["Token A = ",MatrixForm[A]]];Alist=Flatten[Append[Alist,A]];If[ANullMatrix,Goto[again]];B=MPFint[U,W,V];If[verbose,Print["Token B = ",MatrixForm[B]]];If[BNullMatrix,Goto[again]];Blist=Flatten[Append[Blist,B]];KeyA=MPFint[X,B,Y];If[verbose,Print["KeyA = ",MatrixForm[KeyA]]];If[KeyANullMatrix,Goto[begin]];KeyB=MPFint[U,A,V];If[verbose,Print["KeyB = ",MatrixForm[KeyB]]];If[KeyBNullMatrix,Goto[begin]];KeyAlist=Flatten[Append[KeyAlist,KeyA]];KeyBlist=Flatten[Append[KeyBlist,KeyB]];If[miniverbose,Print["keys agree ? ",KeyAKeyB,"....end of round = ",round]];If[miniverbose,Print["zerokeys = ",zerokeys]];If[round<rounds,Goto[begin]];]]If[verbose,Print["A-Token combined list = ",Alist]];If[verbose,Print["Length (numbers) = ",Length[Alist]]];If[verbose,Print["B-Token combined list = ",Blist]];If[verbose,Print["Length (numbers) = ",Length[Blist]]];Print["A-Keys combined list = ",KeyAlist];Print["Length (numbers) = ",Length[KeyAlist]];Print["B-Keys combined list = ",KeyBlist];Print["Length (numbers) = ",Length[KeyBlist]];Print["K session key from Alice = ",Hash[KeyAlist,"SHA3-512","HexString"]];Print["K session key from Bob = ",Hash[KeyBlist,"SHA3-512","HexString"]];

.....................................

dim = 5

prime = 65537

Zlimit = 1024

sigma = -806

expTop = 10

total rounds = 1

verbose = True

mini verbose = True

.....................................

W =

41697	6151	23262	52928	20614
5676	21811	33623	45340	53343
61042	58833	56804	41789	668
60637	13311	26580	48400	50086
61394	1049	30438	1371	11740

BaseXU =

46720	51449	38914	20462	10656
46720	51449	38914	20462	10656
60437	45705	59501	38633	57401
56999	62679	41221	12429	38072
50382	38044	33734	9703	10423

BaseYV =

89	21802	29582	59064	50633
49736	54349	23099	51946	41545
49736	54349	23099	51946	41545
10112	59634	49093	2639	34525
43025	32273	38261	47271	41217

.....................................

randX = 6

X =

52010	62187	10332	55150	7012
52010	62187	10332	55150	7012
64801	3839	37955	30696	59833
53283	28707	31823	34437	9263
39145	31951	33959	11649	21974

randY = 4

Y =

28873	7404	7823	60954	52229
57549	22914	59540	27329	8802
57549	22914	59540	27329	8802
353	48263	46698	32902	12506
25717	39534	13699	949	20110

randU = 5

U =

31584	10045	55138	46692	16126
31584	10045	55138	46692	16126
43070	7410	38990	54915	59493
56305	20975	53313	1134	24714
17970	28680	43688	57507	32554

randV = 7

V =

41289	54213	4773	20544	3362
51370	27328	34755	31082	63948
51370	27328	34755	31082	63948
16562	64681	27989	48735	8956
58466	37601	22737	46082	54275

Token A =

46148	5279	54674	41606	20857
46148	5279	54674	41606	20857
37287	19632	13311	60086	24268
44682	647	55760	7591	28641
6801	11757	65048	46504	8267

Token B =

16067	49747	5953	28444	5963
16067	49747	5953	28444	5963
14216	10093	24482	26391	424
20479	48558	10694	46525	43719
11722	19309	20064	30593	50794

KeyA =

12795	35494	23840	6271	17390
12795	35494	23840	6271	17390
34099	22115	43780	25140	33482
56338	7456	15812	32860	43719
37196	33376	21272	9198	50284

KeyB =

12795	35494	23840	6271	17390
12795	35494	23840	6271	17390
34099	22115	43780	25140	33482
56338	7456	15812	32860	43719
37196	33376	21272	9198	50284

keys agree ? True....end of round = 1

zerokeys = 0

Out[]=

{6.07813,Null}

A-Token combined list = {46148,5279,54674,41606,20857,46148,5279,54674,41606,20857,37287,19632,13311,60086,24268,44682,647,55760,7591,28641,6801,11757,65048,46504,8267}

Length (numbers) = 25

Symbolic Computation

The core of the algorithm used is developed in symbolic form.

CITE THIS NOTEBOOK

How to arrange a secret between two remote entities resistant to future quantum computer attacks
by Juan Pedro Hecht and Hugo Daniel Scolnik
Wolfram Community, STAFF PICKS, January 23, 2025
https://community.wolfram.com/groups/-/m/t/3363308

Definitions

Multi σ-MPF square rank-deficient BENCH

Symbolic Computation

CITE THIS NOTEBOOK

Definitions
