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Symposium on Jainism and Mathematics - presentation : 2020
- 1. ‘Siribhūvalaya’: A Unique Scripture crafted with Modern Cryptography Constituents by Er. Anil Kumar Jain FIETE, FIE(I), SMCSI, LM-ISCA email: jain.anilk@gmail.com web-site: siri-bhoovalaya.org Virtual Symposium on Jainism and Mathematics Dec. 12 to Dec. 14, 2020
- 2. Main Objectives of this Presentation 1. To reveal that Cryptographic mechanisms devised by Acārya Kumudēndu about 1200 years ago are strikingly similar to ones used in modern Information Security applications. 2. To convey in modern mathematical expressions the underlying premise of the cryptographic Writing System (Anka-Lipi) as conceived by Acārya Kumudēndu. This is labeled as Kumudēndu Encryption System (KES) by this author. 3. To present rudiments of KES (Kumudēndu Encryption System) in modern terminology of Information Security and to corroborate its potential in present scenario. siri-bhoovalaya.org
- 3. Siribhūvalaya by Ācārya Kumudēndu • Siribhūvalaya was scripted by Ācārya Kumudēndu during 8th century AD in the state of in Karnataka, India. • Ācārya Kumudēndu devised an exclusive ‘Writing System’ for this unique creation which was based on universal script of numerals only. [Anka-Lipi] • Herein one Cakra or a page contained only integer numbers arranged in a square matrix of dimension 27. • Integer, in each of the cells of the matrix in the range of 1 to 64, represented one of the 64 Mūla Varṇas or basic phonetic alphabets as enumerated in texts of earlier Ācāryas in lineage to which Ācārya Kumudēndu belonged to. • It is claimed to be the largest scripture in the world (only one out of total of nine Khaṇḍas is available in public domain). siri-bhoovalaya.org
- 4. Siribhūvalaya manuscript at National Archives of India [One CAKRA ( 27x27 matrix) with numerals in Kannaḍa script] siri-bhoovalaya.org
- 5. siri-bhoovalaya.org Symmetric Cipher Model by Claude E. Shannon (Father of Modern Information Theory) C. E. Shannon: “Communication Theory of Secrecy Systems” : Bell Systems Technical Journal, Vol. 28, pp. 656–715, 1948. Earlier version of this research was presented in the classified report : “A Mathematical Theory of Cryptography”: Memorandum MM 45-110-02, Sept. 1, 1945, Bell Laboratories.
- 6. Kumudēndu Encryption System (KES) siri-bhoovalaya.org Schematic of Kumudēndu Encryption System .
- 7. Basic Building Blocks of KES framework siri-bhoovalaya.org KES (Kumudēndu Encryption System) framework, as deployed in Siribhūvalaya, consists of two symmetric encryption ciphers and an Integrity check mechanism : Mono Alphabetic Substitution Cipher : Here Key is identified as ‘Kumudendu Code Table’ (KCT). Here each of phonetic alphabet or Mūla Varṇa of the target script is assigned an integer numeric value in range 1 to 64. Transposition Cipher : Here key is named as ‘Bandha Traversal Path Matrix’ (BTPM). It is a matrix of 27x27 integers or a vector of 729 integers. Herein each element is a distinct integer in the range 1 to 729, and which is indicative of place of the transposed phonetic alphabets in corresponding Cakra matrix (27x27). Transposition Cipher has two variants – Cakra-Bandha and Navmānk Bandha , wherein BTPM operates over whole of 27x27 matrix or 9x9 sub-matrices of the Cakra respectively. Integrity Check : To ensure accuracy and consistency of content in Siribhūvalaya, at the end of each Adhyaya there is mention of total number of phonetic alphabets in basic decoded and derived text of the Adhyaya. This ensures that there is no addition, deletion or modification of Cakras consisting the Adhyaya.
- 8. Substitution Cipher Key (KCT) for Devanāgarī, Kannaḍa and Brāhmī scripts कु मुदेन्दु कोड सारणी में देवनागरी, कन्नड़ एवं ब्राह्मी लिपि के ध्वन्याक्षर siri-bhoovalaya.org
- 9. Mathematical Representation of Decryption Mechanism (Transposition) – [Step-1] siri-bhoovalaya.org Transformation function (matrix to vector conversion function) i.e. ‘nvfunc’ converts CAKRA matrix into NVECTOR by implementation of a method called ‘locate’ which has got two arguments viz. BTPM and an integer variable ‘p’ in the range from 1 to 729.
- 10. Mathematical Representation of Decryption Mechanism (Transposition) in Cakra 1-1-1 – [Step-2 ] siri-bhoovalaya.org Value of p 1 2 3 4 5 6 7 8 (x,y) in BTPM 1,14 27,15 26,16 25,17 24,18 23,19 22,20 21,21 cxy in CAKRA 1 58 38 1 52 1 60 2
- 11. Mathematical Representation of Decryption Mechanism (Substitution) – [Step-3] siri-bhoovalaya.org
- 12. Mathematical Representation of Decryption Mechanism (Substitution & Assimilation) – [Step-4 ] siri-bhoovalaya.org Resultant text after conjoining phonetic alphabets or Mūla Varṇas into words, phrases and verses results in following: अष्ट महाप्रातिहार्य वय्भवदिन्ि | अष्ट गुणन्गळौल् ओोम्िं || स्र्ष्ष्टगे मन्गळ पर्ायर्दितिि | अष्टम स्र्षििगेर गुवेिु ||1|| This is the first verse of first Khaṇḍa i.e. ṁagala Prābhr̥ta of Siribhūvalaya.
- 13. Robustness of KES against Brute Force Attacks siri-bhoovalaya.org Depending on the security requirements, the substitution table can be altered in size and assignment of integers. Thus even with 64 element KCT, there are possible Factorial (64) permutations resulting in a number with 89 digits. Size of Transposition key can be adjusted to thwart anticipated brute-force attacks. Preliminary estimations reveal that possible Transposition schemes of 729 elements, of 27x27 matrix as deployed in “Siribhūvalaya”, amount to Factorial (729) permutations and that results in massive number of approximately 1772 digits. That means a brute-force attacker will have to find correct Transposition Key out of so many possible ones. This will require massive computational resources which is very highly impractical in contemporary scenario. Current cryptographic standards of symmetric key encryption recommend key length of 256 bits, wherein possible key combinations are of the order 78 digits only for a brute-force attack.
- 14. KES in present Scenario siri-bhoovalaya.org KES is inherently suitable for applications on variety of platforms which use Indian Languages and Scripts. With its variable length dual key symmetric encryption scheme it can be customised for Information Security requirements of wide-range of systems and applications. Since it requires small computational overhead, KES can efficiently be deployed on systems with nominal computing resources. Even in absence of computing machines, KES can be implemented with manual operations only. That is how Acārya Kumudēndu encrypted Siribhūvalaya in 8th century and later in 20th century Pandit Yellappa Shastri was able to decrypt it manually. In addition to KES there are other Cryptographic mechanisms very creatively integrated in Siribhūvalaya. These are termed in modern terminology as Visual Cryptography (Citra Bandha) and Steganography (Antara Sāhitya).
Editor's Notes
- There are many distinctive features of this unique creation, but in this presentation focus is on the Cryptographic constituents which are strikingly similar to ones used in modern applications of Cryptography in Information Security. One can noticeably observe creative deployment of Substitution and Transposition Ciphers in Cakras of this epic. Built upon notions of Matrices these are mathematical applications of combinatorics and permutations. A generalized version of this mechanism in modern terminology has been proposed as KES (Kumudendu Encryption System) which can measure up to a versatile and robust block cipher as compared to existing ones. Besides encryption mechanisms Muni Kumudendu implemented a scheme to check Integrity of the verses in each of Adhyāyas in a smart manner. Muni Kumudendu has very creatively used Steganography for interweaving not only multi-lingual verses in base Kannada text but also deployed this technique to enclose scriptures from other sources.
- Purpose of this presentation is to highlight one of the unique characteristics of an enigmatic epic created by Jain Acārya Kumudēndu more than a century ago. Yet not fully revealed this scripture had remained out-of-sight for centuries and was rediscovered about 70 years ago by Pandit Yellappa Shastri in Bengaluru, Karnataka State of India. Acārya Kumudēndu regarded Acārya Virasen as his Guru. Subject matter of Siribhuvalaya covers wide spectrum of Jain canonical topics along with discourses on ancient sciences and scriptures of non-jain religious streams. This is very rightfully described as a scripture of scriptures (Shatron and Shatra).
- This is the image of one sheet of manuscript wherein we can identify one matrix of size 27x27 and another matrix of size 3x3 near top left corner. Elements in both of these matrix contains integer numerals in Kannada script. On minute observation one can identify partitioning of 27x27 matrix into nine sub-matrices of size 9x9 each. 3x3 matrix at the top left corner provides vital clue in construction of key for the Transposition Cipher. It is to be noted that concept of structuring and processing of data elements in matrix form is quite recent in modern mathematics, about 150 years old only. It is not yet known whether any of mathematicians prior to Acarya Kumudendu has deployed matrices so creatively.
- This slide exhibits Schematic of General Secrecy System of Claude E. Shannon which is also termed as Symmetric Key Cipher Model in Modern Cryptography. References for this model are mentioned on this slide. Kindly note that Shannon was an alumni and later a faculty member at MIT. Noticeable on this schematic are two transformation processors namely Encipherer and Decipherer which are complimentary to each other. Message source sends message to Encipherer which transforms message into a cryptogram. This transformation function has two parameters – the message M and a Key from Key source. The cryptogram is decrypted at the recipient’s end by Decipherer Transformation Processor which operates with two parameters, namely Cryptogram and the same key K from the Key Source. This model represents Symmetric Key Cryptography mechanism and is very widely deployed in modern Information Security Application. DES, 3DES, AES, IDEA are few examples of this.
- This slide brings forth the resemblance of Kumudendu Encryption System with Shannon’s Model of General Secrecy System. In this model Message source is Text in Mula Varnas, Key Source contains two keys one for Transposition Cipher and Another one for Substitution Cipher. Encipherer transforms message using Transposition and Substitution mechanisms and the Output is a CAKRA which is termed a Cryptogram by Shannon. Decipherer Transformation converts CAKRA into a string of Mula Varnas using the same Keys from Key Source.
- Slide Nos 9 to 12 describe in mathematical terms the Deciphering mechanism as applied to a CAKRA in Siribhuvalaya. Detailed description of Deciphering and Enciphering is presented in the Full Length Paper submitted for the Symposium. Here is representation of CAKRA and BTPM (Transposition Cipher Key) as 27x27 matrices respectively. Deciphering of Transposed Matrix is basically a transformation function to convert CAKRA matrix into a row-vector of 729 elements as per transpositions scheme contained in BTPM.
- This slide describes the transformation of CAKRA matrix into a row-vector - cell by cell – in accordance to place value and its location in BTPM. As an example first CAKRA of Siribhūvalaya is referred in the slide.
- This slide describes KCT - the Key for Substitution Cipher. KCT is basically an ordered row-vector of 64 elements containing Mula Varnas of the target script. KCT for Devanagari Script is taken as an example. Substitution of integers in the transposed row-vector of 729 elements results in string of Mula Varnas in target script.
- This slide represents final outcome of Deciphering. Resultant text after conjoining or assimilating phonetic alphabets into words, phrases results finally in verses. Shown in this slide is First verse of Siribhuvalaya – which the Mangalacharan and is dedicated to Eight Jina. अष्ट महाप्रातिहार्य वय्भवदिन्द | अष्ट गुणन्गळौल् ओोम्दं || स्र्ष्टिगे मन्गळ पर्यायदिनित | अष्टम जिनगेर गुवेनु || Herein, Overwhelmed with immense external grandeur of eight Mahāprātihāryas - which are attributed to Arihanta Paramēṣṭhī - Ācārya Kumudēndu envisages internal majesty of Aṣṭaguṇas i.e. eight infinite qualities which are derivatives of ‘ŌM’ and are attributed to Siddha Parameśṭhī. He pronounces that these two forms i.e. Arihanta & Siddha are enormous benefactor for the well-being of whole of the universe. Ācārya Kumudēndu thus commences Maṅgala Prābhr̥ta of Siribhūvalaya by bowing down to pay humble homage to Eighth Jina i.e. Lord Candra Prabhu or Candra Nāth.