Simplest (3,2) Threshold Secret Sharing Schemes Based on EXCLUSIVE-OR Function

Indian Journal of Cryptography and Network Security (IJCNS) ISSN: 2582-9238 (Online), Volume-5 Issue-2, November 2025 12 Published By: Lattice Science Publication (LSP) © Copyright: All rights reserved. Retrieval Number:100.1/ijcns.B144205021125 DOI: 10.54105/ijcns.B1442.05021125 Journal Website: www.ijcns.latticescipub.com Simplest (3,2) Threshold Secret Sharing Schemes Based on EXCLUSIVE-OR Function Sergei Kulikov Abstract: This paper presents a new XOR-based secret sharing scheme designed for simplicity, efficiency, and strong security guarantees. Unlike traditional threshold schemes that rely on matrix operations and work on large data blocks, the proposed method operates on a per-byte basis, applying lightweight bytewise XOR calculations. The goal of this research is to create a minimalistic yet robust (3,2) threshold secret sharing algorithm that is practical for real-world systems with constrained computational resources. The proposed scheme divides a secret file into three independent shadow shares using a simple transformation: each share is generated as the XOR of the secret with a random byte string and a shifted version of that string. Specifically, the algorithm uses one random string and constructs the second and third shares by XOR-ing the secret with the random string and with the string shifted by one byte in opposite directions. This process is applied independently to each byte of the secret file. The resulting scheme is both perfect and ideal: each share is the same size as the original file, and no information about the secret can be obtained from a single share. Any two out of three shares are sufficient to reconstruct the original data, while fewer reveal nothing. The encoding and decoding procedures are transparent, computationally efficient, and well-suited for implementation in hardware or constrained environments. The algorithm was implemented in Fortran and can be easily integrated into secure cloud storage systems, distributed communication protocols, or embedded devices. This work contributes a lightweight alternative to existing XOR-based sharing methods, providing a novel balance between cryptographic strength and algorithmic simplicity. Keywords: Secret Sharing. Cryptographic Protocols, Real-Time Systems Abbreviations: SSS: Secret Sharing Schemes I. INTRODUCTION Secret sharing schemes (SSS) based on the Exclusive-OR (XOR) operation have been explored in several studies [1]. XOR-based (k,n)-threshold schemes allow secure distribution of a secret among n participants such that any k or more can reconstruct the secret, while fewer than k reveal no information. These schemes are ideal, meaning the size of each share equals the size of the secret, and are computationally efficient due to the lightweight XOR operation [2]. However, most existing XOR-based schemes operate on the whole secret or large blocks, often involving matrix operations [3]. We propose a more memory-efficient alternative that works byte-by-byte. II. PROCEDURE We present a (3,2) threshold XOR-based scheme with a streaming, byte-level approach. Each byte of the secret is processed independently, resulting in low memory overhead and fast execution. Let S = (s1,s2, …, st) is a secret file of t bytes. Wj = (w1, w2, …, wt) are 3 shadows, j=1,2,3 ⊕ denotes a bit-wise exclusive-OR operation. The procedures for Share and Recover are as follows. A. Sharing Procedure (S -> W1, W2, W3) 1. Generate an initial random byte: x0= random (0255), w10 = x0, 2. For each byte i=1 to t xi=random (0-255), w1i = xi , w2i = si ⊕ xi , w3i = si ⊕ xi-1 Thus - the first shadow W1 is just the random string of length t+1, - the second shadow W2 is the XOR of the secret and the first shadow of length t, - The third W3 shadow is the XOR of the secret and the first shadow shifted back by one byte, of length t. The sharing procedure is illustrated in Figure 1. Figure 1. Sharing the procedure scheme. Secret S1 S2 S3 S4 … St Shadow 1 w0 w1 w2 w3 w4 … wt <- random numbers Shadow 2 S1⊕w1 S2⊕w2 S3⊕w3 S4⊕w4 … St⊕wt Shadow 3 S1⊕w0 S2⊕w1 S3⊕w2 S4⊕w3 … St⊕wt-1 Manuscript received on 25 July 2025 | First Revised Manuscript received on 02 August 2025 | Second Revised Manuscript received on 17 October 2025 | Manuscript Accepted on 15 November 2025 | Manuscript published on 30 November 2025. *Correspondence Author(s) Sergei Kulikov*, Head, Department of Biostatistics, National Research Center for Hematology, Novozykovsky pr. 4A, MOSCOW, Russian Federation (RUS), Russia. Email ID: [email protected], ORCID ID: 0000-0002-6288-7570 © The Authors. Published by Lattice Science Publication (LSP). This is an open-access article under the CC-BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) B. Recovery Procedures 1. W1, W2 -> S. If we recode the secret by using the first and second shadows, we just combine them by XOR: For i=1 to t si = w1i ⊕ w2i 2. W1, W3 -> S. If we recode the secret by using the first and third shadows, we just combine them by XOR. First shadow is shifted by one bite back: Simplest (3,2) Threshold Secret Sharing Schemes Based on EXCLUSIVE-OR Function 13 Published By: Lattice Science Publication (LSP) © Copyright: All rights reserved. Retrieval Number:100.1/ijcns.B144205021125 DOI: 10.54105/ijcns.B1442.05021125 Journal Website: www.ijcns.latticescipub.com For i=1 to t si = w1i-1 ⊕ w3i 3. W2, W3 -> S. If we recode the secret by using the second and third shadows, the calculations are a little more complicated. For i=1 to t a. yi = w2i ⊕ w3i b. xi = yi ⊕ xi-1 c. si = xi ⊕ w2i III. PROPERTIES OF THE SCHEME The proposed SSS is - Ideal: The size of each share equals the size of the secret. - Perfect secrecy: No information is leaked from a single share. - Efficient: Uses only XOR and operates byte-by-byte without matrices. - Redundant: Any two out of three shares suffice to reconstruct the original secret. IV. EXPERIMENTAL RESULTS The proposed algorithm was implemented in the FORTRAN programming language. To verify correctness, an image file was encoded and decoded. Since all bytes, including format metadata, were encrypted, shadows cannot be visualized. (Figure 2). Figure 2: Example of an image file encoded and decoded by the proposed SSS. A performance test was conducted on a directory containing 1,977 files, totalling 8.6 GB. Results: - Sharing time: 27.9 minutes (including I/O); 4.6 minutes (pure computation) - Recovery time: 33.8 minutes (including I/O); 0.8 minutes (pure computation) - System: Intel® Core™ i7-10510U laptop For comparison, a basic Shamir’s scheme took: - Sharing: 35.3 minutes (6.4 pure) - Recovery: 34.6 minutes (1.8 pure) V. CONCLUSION We presented a simple and efficient (3,2) XOR-based secret sharing scheme operating at the byte level. The algorithm offers high speed and minimal memory usage, making it suitable for secure storage, transfer, and real-time applications. DECLARATION STATEMENT I must verify the accuracy of the following information as the article's author. ▪ Conflicts of Interest/ Competing Interests: Based on my understanding, this article has no conflicts of interest. ▪ Funding Support: This article has not been funded by any organizations or agencies. This independence ensures that the research is conducted with objectivity and without any external influence. ▪ Ethical Approval and Consent to Participate: The content of this article does not necessitate ethical approval or consent to participate with supporting documentation. ▪ Data Access Statement and Material Availability: The adequate resources of this article are publicly accessible. ▪ Author’s Contributions: The authorship of this article is contributed solely by the author. REFERENCES 1. Chen, L., Laing, T.M., Martin, K.M. (2016). Efficient, XOR-Based, Ideal (t,n)−threshold Schemes. In: Foresti, S., Persiano, G. (eds) Cryptology and Network Security. CANS 2016. Lecture Notes in Computer Science(), vol 10052. Springer, Cham. DOI: https://doi.org/10.1007/978-3-319-48965-0_28 2. Wang H, Jiang Z, Qian Q, Wang H. An efficient XOR-based visual cryptography scheme with lossless reconfigurable algorithms. International Journal of Distributed Sensor Networks. 2022;18(4). DOI: http://doi.org/10.1177/15501329221084223 3. Shima K., Doi H. Hierarchical Secret-Sharing Scheme Based on XOR Operations, Journal of Information Processing, September 2024, 32:719-730, DOI: http://doi.org/10.2197/ipsjjip.32.719 AUTHOR’S PROFILE Sergei Kulikov, PhD, Head, Department of Biostatistics, National Medical Research Center of Hematology, Ministry of Health of the Russian Federation, Moscow. Education: Moscow Physical Technical Institute of Science, Russia, Diploma in Applied Mathematical Statistics in 1975. Institute of Control Science, USSR Academy of Sciences, Moscow, PhD in Machine Learning, 1986. Research specialities and interests: Biostatistics & Statistical Epidemiology, Population Haematology, Data Science Analytics. Special Skills: SAS programming and analytics. Committee: Member of the Boarding Committee of the Russian Haematology Society, Member of the Biostatistics working party of the European Leukaemia Network. Current h-index = 20 Appendix A. Fortran Implementation of the XOR-Based Secret Sharing Scheme The following FORTRAN program demonstrates an implementation of a simple (3,2) XOR-based secret sharing scheme. It reads binary files from a specified source directory, splits each file into three shadow shares using a bytewise XOR operation, and saves each share into a separate output directory. The implementation ensures that any two of the three shares can reconstruct the original data, while a single share reveals no information. This version includes inline comments for clarity and was used to verify the algorithm's performance and correctness during testing. program main implicit none ! Constants and array declarations integer, parameter :: NR = 1000 character, dimension(NR) :: IX = "?" character, dimension(NR) :: IXa = "!", IXb = "!", IXc = "!" integer inbite, outbite, sha1, sha2, sha3 Indian Journal of Cryptography and Network Security (IJCNS) ISSN: 2582-9238 (Online), Volume-5 Issue-2, November 2025 14 Published By: Lattice Science Publication (LSP) © Copyright: All rights reserved. Retrieval Number:100.1/ijcns.B144205021125 DOI: 10.54105/ijcns.B1442.05021125 Journal Website: www.ijcns.latticescipub.com integer, dimension(NR) :: CX = 999 integer i, j, k, ios, lest, icr, ipos, endc, ifile integer, parameter :: Nf = 3 integer, parameter :: NN = 16 integer:: re_i, r, kk, ind, reason, NstationFiles, iStatio integer :: lendir, randJ, r1, r2, r3, n, r0, cot integer: a(NN) real: f(NN) real: ra, randn, t_start, t_stop, t_sum integer: rr(130) ! Directory and file name buffers CHARACTER (40): fromdir CHARACTER (40): todir1, todir2, todir3 CHARACTER (100): filename0, filename1, filename2, filename3 character (LEN=100), dimension (:), allocatable: station File Names ! Read input/output directories from file OPEN (unit=15, file="config.txt", status='old') read (15, *) fromdir read (15, *) todir1 read (15, *) todir2 read (15, *) todir3 close (15) ! Copy input files to each of the three output directories call system ('CHCP 1251 >nul') call system ('xcopy ' //TRIM (fromdir)//' '//TRIM (todir1)//' /H /Y /C /R /S /I /E /T ') call system('xcopy ' //TRIM(fromdir)//' '//TRIM(todir2)//' /H /Y /C /R /S /I /E /T ') call system('xcopy ' //TRIM(fromdir)//' '//TRIM(todir3)//' /H /Y /C /R /S /I /E /T ') ! Get list of files from source directory call system ('dir '//TRIM (fromdir)//' /a: -d/s/b/l > fileContents.txt') lendir = len(trim(fromdir)) + 1 open (31, file='fileContents.txt', action="read") i = 0 do read (31, '(a)', iostat=reason) ra if (reason /= 0) exit i = i + 1 end do NstationFiles = i allocate(stationFileNames(NstationFiles)) rewind (31) ! Process each file individually t_sum = 0 do ifile = 1, NstationFiles read (31, '(a)') stationFileNames(ifile) filename0 = stationFileNames(ifile) filename1 = trim(todir1) // stationFileNames(ifile) (lendir:) filename2 = trim(todir2) // stationFileNames(ifile) (lendir:) filename3 = trim(todir3) // stationFileNames(ifile)(lendir:) OPEN (unit=1, file=filename0, status='old', ACCESS='stream', form='unformatted') OPEN (unit=2, file=filename1, status='new', ACCESS='stream') OPEN (unit=3, file=filename2, status='new', ACCESS='stream') OPEN (unit=4, file=filename3, status='new', ACCESS='stream') do i = 1, 90000 IX = "-" CX = 999 icr = 0 ipos = 0 endc = NR ! Read raw bytes into IX [] do j = 1, NR read (1, iostat=ios) IX(j) if (ios /= 0) exit end do endc = j - 1 call cpu_time(t_start) r0 = 111! Initial random seed do j = 1, endc ! Encode each byte of the secret using two random keys inbite = ichar(IX(j)) + 1 call random_number(randn) r = FLOOR (255 * randn) r1 = r r2 = xor(inbite, r0) r3 = xor(inbite, r) r0 = r ! Store shadows IXa(j) = char(r1) IXb(j) = char(r2) IXc(j) = char(r3) end do call cpu_time(t_stop) t_sum = t_sum + (t_stop - t_start) ! Write the three shadows to files write (2) (IXa(j), j=1, endc) write (3) (IXb(j), j=1, endc) write (4) (IXc(j), j=1, endc) if (ios < 0) exit end do close (1) close (2) close (3) close (4) end do close (31) write (*, *) "Total processing time (seconds):", t_sum 9999 write (*,*) "<<<<<<- End of program" end program main Simplest (3,2) Threshold Secret Sharing Schemes Based on EXCLUSIVE-OR Function 15 Published By: Lattice Science Publication (LSP) © Copyright: All rights reserved. Retrieval Number:100.1/ijcns.B144205021125 DOI: 10.54105/ijcns.B1442.05021125 Journal Website: www.ijcns.latticescipub.com Appendix B. Fortran Implementation of the Secret Recovery Procedure This appendix provides the FORTRAN implementation for recovering the original data from two of the three shadow shares produced by the XOR-based secret sharing scheme. Depending on the selected rule, the program reads different combinations of shadow files. It applies XOR operations, optionally using sequential shifts and an initial random seed, to reconstruct the original binary file. The method ensures correct recovery when any two shares are available. ! A fortran95 program for G95 ! By WQY program main implicit none integer re_i, i, j, k, icr, ios, endc, xi, yi, icode, ai, zi integer r1, r2, r3, rr, fi1, fi2, inbite, outbite, ifile integer nstationfiles, reason, rule integer lendir0, lendir1, lendir2, lendir3 integer, parameter :: NR = 1000 integer outcode(256,256), vvv(1000) character, dimension(NR) :: IXa="!", IXb="!", IXout character(40) :: fromdir, todir1, todir2, todir3 character(100) :: filename1, filename2, filename3, filename0, ra character(LEN=100), dimension(:), allocatable :: stationFileNames real :: t_start, t_stop, t_sum ! Read input/output directories and rule from config open (15, file=" config.txt", status='old') read (15, *) fromdir read (15, *) todir1 read (15, *) todir2 read (15, *) todir3 read (15, *) rule close (15) ! Copy directory structure call system ('CHCP 1251 >nul') call system ('xcopy "' // trim(todir1) // '" "' // trim(fromdir) // '" /H /Y /C /R /S /I /E /T') ! Get list of files to process call system ('dir "' // trim(todir1) // '" /a: -d/s/b/l > fileContents.txt') lendir0 = len(trim(fromdir)) + 1 lendir1 = len(trim(todir1)) + 1 lendir2 = len(trim(todir2)) + 1 lendir3 = len(trim(todir3)) + 1 open (31, file='fileContents.txt', action="read") i = 0 do read(31,'(a)', iostat=reason) ra if (reason /= 0) exit i = i + 1 end do nstationfiles = i allocate(stationFileNames(nstationfiles)) rewind(31) t_sum = 0 do ifile = 1, nstationfiles read(31,'(a)') stationFileNames(ifile) filename0 = trim(fromdir) // stationFileNames(ifile)(lendir1:) filename1 = trim(todir1) // stationFileNames(ifile)(lendir1:) filename2 = trim(todir2) // stationFileNames(ifile)(lendir2:) filename3 = trim(todir3) // stationFileNames(ifile)(lendir3:) open(1, file=filename0, status='new', access='stream') open(2, file=filename1, status='old', access='stream', form='unformatted') open(3, file=filename2, status='old', access='stream', form='unformatted') open(4, file=filename3, status='old', access='stream', form='unformatted') ! Determine which shadows to use based on the rule if (rule == 1) then fi1 = 2 fi2 = 3 else if (rule == 2) then fi1 = 2 fi2 = 4 else if (rule == 3) then fi1 = 3 fi2 = 4 end if do i = 1, 10000 endc = NR do j = 1, NR read(fi1, iostat=ios) IXa(j) if (ios < 0) exit read (fi2, iostat=ios) IXb(j) end do endc = j - 1 call cpu_time(t_start) ! Apply selected recovery logic if (rule == 1) then do j = 1, endc if (j == 1) then xi = 111 else xi = ichar(IXa(j-1)) end if yi = ichar(IXb(j)) icode = xor(xi, yi) IXout(j) = char(icode) end do else if (rule == 2) then do j = 1, endc xi = ichar(IXa(j)) yi = ichar(IXb(j)) icode = xor(xi, yi) IXout(j) = char(icode) end do else if (rule == 3) then do j = 1, endc xi = ichar(IXa(j)) Indian Journal of Cryptography and Network Security (IJCNS) ISSN: 2582-9238 (Online), Volume-5 Issue-2, November 2025 16 Published By: Lattice Science Publication (LSP) © Copyright: All rights reserved. Retrieval Number:100.1/ijcns.B144205021125 DOI: 10.54105/ijcns.B1442.05021125 Journal Website: www.ijcns.latticescipub.com yi = ichar(IXb(j)) r1 = xor(xi, yi) if (j == 1) then r2 = xor(r1, 111) else r2 = xor(r1, rr) end if rr = r2 icode = xor(yi, rr) IXout(j) = char(icode) end do end if call cpu_time(t_stop) t_sum = t_sum + (t_stop - t_start) write(1) (IXout(j), j=1, endc) end do close(1) close(2) close(3) close(4) end do write(*,*) "Sum time = ", t_sum write(*,*) "End program." end program main