Algebraic Structures and its Applications in Cryptography Dr. - PowerPoint PPT Presentation

Algebraic Structures and its Applications in Cryptography Dr. Sucheta Chakrabarti Scientist - G Scientific Analysis Group DRDO Delhi E-mail – suchetadrdo@hotmail.com IC-W 2020 29/8/2020

Outline of the Presentation • Secure Communication & Cryptography • Role of probability and entropy in secure communication from information theoretic approach • Commonly used Algebraic Structures in Cryptography • New Direction of (in) Cryptography based on Non-commutative / Non- associative Algebraic Structures • Quaigroups • Quasigroup – Based Transformations and its cryptographic applications 29/8/2020 IC-W 2020

Secure Communications Over Open Channels Aim :  To Protect Information  To coordinate operations ( command and control )  To carry out online business transaction ( E- commerce ) Service required for secure communication – • Data confidentiality : It ensures the privacy of data i.e only the authorized person can only access the information • Data Integrity : It ensures the protection from any unauthorized alteration i.e. no insertion, deletion or modification has been done in the information by Non-legitimate party .It provides the assurance that the data is present in its original form as it was sent by the sender. 29/8/2020 IC-W 2020

• Data availability : This means that the data is always available for access whenever required • Authentication : This ensures that the communication is being held among the right individuals. • Non-repudiation : According to this, the sender or the receiver cannot deny being responsible for the data being transmitted . 29/8/2020 IC-W 2020

Fundamental building block of security is Cryptography 1949 is the turning point for cryptography – it turns to scientific based on mathematical grounds by the research article Communication Theory of secrecy system - C.E. Shannon 29/8/2020 IC-W 2020

• Security needs continuous improvement / up gradation against adversary capabilities viz. (i) computational • Computationally unbounded – Unconditional security ( Info. theoretical or perfect secrecy ) • Computationally bounded – Computational security & Provable security ( the cryptographic primitive reduced to certain problem which is proved to be (well known )hard problem . It implies breaking of the primitive computationally infeasible ) (ii) other capabilities - • Active - can corrupt parties, inject / modify messages • Passive / eavesdropper – only listens (intercepts) messages • Other resources i.e. ability to decrypt some messages. • Security is based on Arbitrary Adversary Principle (AAP ) – i.e it assume restrictions on adversary capabilities , but not that the adversary is using specific strategies or attacks • Secure electronic identities and information protection are key for digital evolution 29/8/2020 IC-W 2020

In the Modern digital world Cryptography ( Crypto-primitives / algorithms ) deals with information security & secure communications over insecure channels. Mainly deals with Confidentiality , Authenticity , Integrity & Non-repudiation It needs set of elements and specific operations that are applied to the elements of the set is called Algebraic Structures 29/8/2020 IC-W 2020

Basic Components of Cryptography  Functions • one – one • one-way • trapdoor one way • encryption / decryption 29/8/2020 IC-W 2020

Encryption/Decryption function has to satisfy the following condition : For E ∈ E and 𝑙 𝑓 ≡ 𝑓 ∈ 𝒧 , 𝐹 𝑓 : ℳ → 𝒟 is a 1-1 mapping & so there exists a corresponding D ∈ D and 𝑙 𝑒 ≡ 𝑒 ∈ 𝒧 such that 𝐸 𝑒 : 𝒟 → ℳ and 𝐸 𝑒 𝐹 𝑓 𝑛 = 𝑛 𝑔𝑝𝑠 𝑏𝑚𝑚 𝑛 ∈ ℳ In other words Cryptographic Algorithms - consist of ℳ , 𝒟 , 𝒧 and set 𝐹 𝑓 , 𝑓 ∈ 𝒧 of encryption transformations and corresponding set 𝐸 𝑒 , 𝑒 ∈ 𝒧 of decryption transformations with the property that for −1 i.e each 𝑓 ∈ 𝒧 there exists a unique , 𝑒 ∈ 𝒧 s.t 𝐸 𝑒 ≡ 𝐹 𝑓 𝐸 𝑒 𝐹 𝑓 𝑛 = 𝑛 𝑔𝑝𝑠 𝑏𝑚𝑚 𝑛 ∈ ℳ 29/8/2020 IC-W 2020

Domain & Codomain of Encryption / Decryption Functions • Alphabet set - A • Message space - ℳ • Crypt space - 𝒟 • Key space - 𝒧 Set of encryption and decryption functions are denoted by E & D respectively 29/8/2020 IC-W 2020

Cryptosystems Three Sets : Message / Plaintext – ℳ Ciphertext - 𝒟 - 𝒧 Keys 𝐿𝐻, 𝐹, 𝐸 Three randomized algorithms : 𝐿𝐻: 𝑇 ∗ → 𝒧 Key generation Algo Encryption Algo 𝐹: 𝒧 × ℳ → 𝒟 Decryption Algo 𝐸 ∶ 𝒧 × 𝒟 → ℳ For any k ey 𝑙 ∈ 𝒧 and 𝑛 ∈ ℳ holds 𝐸 𝑙 𝐹 𝑙 𝑛 = 𝑛 So a cryptosystem consists of five tuples which represent as ℳ, 𝒟 , 𝒧 , 𝐹, 𝐸 29/8/2020 IC-W 2020

Probability & Entropy Concepts for Secure Communication  The concept of entropy has evolved in probability theory to create information theoretical model for secure communication .  In 1947-48 by classic work of C. Shannon gives birth of Information theory , a new branch in applied probability theory to handle practical problem of communication.  Security generally expressed in terms of probability and amount of information (entropy)  Here we will discuss some important concepts of discrete probabilities Probability Space : 𝕐, 𝑄𝑠 , where • 𝕐 − the sample space which is a finite set of possible outcomes ( events) • 𝑄𝑠 – a function from 𝒬 𝕐 → 0,1 such that 𝑗 𝑄𝑠 𝕐 = 1, 𝑗𝑗 𝑄𝑠 Φ = 0, 𝑗𝑗𝑗 𝑄𝑠 𝑌 ∪ 𝑍 = 𝑄𝑠 𝑌 + 𝑄𝑠 𝑍 if 𝑌 ∩ 𝑍 = Φ ( iv ) 𝑄𝑠 𝑌 ∩ 𝑍 = 𝑄𝑠 𝑌 𝑄𝑠 𝑍 if 𝑌 ∩ 𝑍 = Φ 𝑄𝑠 is called a probability distribution , a probability measure or just a probability 𝑄𝑠 of X ∈ 𝒬 𝕐 determined by 𝑄𝑠 𝑦 ∀ 𝑦 ∈ 𝑌 29/8/2020 IC-W 2020

Joint Probabilities : Two probability spaces viz . 𝕐, 𝑄𝑠 𝕑, 𝑄𝑠 1 2 It can create joint probability space 𝕐 × 𝕑, 𝑄𝑠 where 𝑄𝑠 define as follows : 𝑄𝑠 𝑦, 𝑧 = 𝑄𝑠 𝑦 𝑄𝑠 𝑧 1 2 Conditional Probability • 𝑄𝑠 𝑌 𝑍 = 𝑄𝑠 𝑌 ∩ 𝑍 /𝑄𝑠 𝑍 - only defined if 𝑄𝑠 𝑍 > 0 • 𝑌 and 𝑍 are independent if 𝑄𝑠 𝑌 = 𝑦|𝑍 = 𝑧 = Pr 𝑌 = 𝑦 or 𝑄𝑠 𝑦| 𝑧 = 𝑄𝑠 𝑦 & also 𝑄𝑠 𝑌 = 𝑦 ∩ 𝑍 = 𝑧 = Pr 𝑌 = 𝑦 Pr 𝑍 = 𝑧 ∀𝑦, 𝑧 𝑄𝑠 𝑌 𝑄𝑠(𝑍|𝑌) Bayes Theorem : 𝑄𝑠 𝑌|𝑍 = 𝑄𝑠 𝑍 29/8/2020 IC-W 2020

Random Variables • A random variable 𝑌 is a function from underlying set of probability space ( all possible outcomes 𝕐 ) to some set of values ( some set of 𝒬 𝕐 ) • Given a probability space and a random variable 𝑌 , the probability that the random variable 𝑌 takes value 𝑦 is 𝑄𝑠 𝑥 𝑌 𝑥 = 𝑦 29/8/2020 IC-W 2020

Application to Cryptography for security analysis Plaintext Distribution : • 𝑌 discrete random variable over the plaintext set ℳ • Sender choose 𝑦 from ℳ based on some probability distribution - Let Pr 𝑌 = 𝑦 be the probability that 𝑦 is chosen - This probability may depend on the language Key Distribution: Sender & Receiver agree on a key 𝑙 chosen from a key set 𝒧 • 𝐿 discrete random variable over 𝒧 • Pr 𝐿 = 𝑙 ,the probability that 𝑙𝑓𝑧 𝑗𝑡 𝑙 Note that here Probability space ( Plaintext , Key) 29/8/2020 IC-W 2020

Ciphertext Probability Distribution 𝑍 is a discrete random variable over the set 𝒟 The probability of obtaining a particular ciphertext 𝑧 depends on the probability of Plaintext and key - 𝑄𝑠 𝑧 = σ 𝑦,𝑙|𝑓 𝑙 𝑦 =𝑧 𝑄𝑠 𝑦 𝑄𝑠(𝑙) = σ 𝑙 𝑄𝑠 𝑙 𝑄𝑠(𝑒 𝑙 ( y))  Attacker Aims to determine the plaintext 𝑦 • Attacker’s does not know /observe ciphertext 𝑧 o Probability (a pri riori i probabil ilit ity ) that the plaintext is 𝑦 : 𝑄𝑠 𝑌 = 𝑦 ≡ Pr(𝑦) o It depends on plaintext distribution i.e language characteristics • Attacker’ s knows / observes ciphertext 𝑧 o Probability ( a posterio iori i probabil ilit ity)that the plaintext is 𝑦– 𝑄𝑠 𝑌 = 𝑦|𝑍 = 𝑧 ≡ 𝑄𝑠 𝑦|𝑧 Computation of attacker’s a a posterio ior (c (condit itio ional) l) probabil ilit itie ies • Apply Bayes theorem 29/8/2020 IC-W 2020

𝑄𝑠 𝑌 = 𝑦|𝑍 = 𝑧 ≡ 𝑄 𝑠 𝑦|𝑧 𝑄𝑠 𝑦 ×𝑄𝑠 𝑧|𝑦 = 𝑄𝑠 𝑧 Here 𝑄𝑠 𝑦 - Probability of the plaintext 𝑄𝑠 𝑧 - Probability of this ciphertext – It ind nduced by probabil ilit ity of f plain intext an and key distr trib ibutio ions 𝑄𝑠 𝑧 = ෍ 𝑄𝑠 𝑦 𝑄𝑠 𝑙 𝑦,𝑙|𝑓 𝑙 𝑦 =𝑧 𝑄𝑠 𝑧|𝑦 - probability that the 𝑧 is obtained for a given 𝑦 depends on the keys which provide such a mapping from plaintext domain (Message space ) to ciphertext domain (Cipher space) - 𝑄𝑠 𝑧|𝑦 = ෍ 𝑄𝑠 𝑙 𝑙|𝑓 𝑙 𝑦 =𝑧 𝑝𝑠𝑒 𝑙 𝑧 =𝑦 29/8/2020 IC-W 2020