Subhash Kak

Subhash Kak

Mathematically, ternary coding is more efficient than binary coding. It is little used in computation because technology for binary processing is already established and the implementation of ternary coding is more complicated, but remains relevant in algorithms that use decision trees and in communications. In this paper we present a new comparison of binary and ternary coding and their relative efficiencies are computed both for number representation and decision trees. The implications of our inability to use optimal representation through mathematics or logic are examined. Apart from considerations of representation efficiency, ternary coding appears preferable to binary coding in classification of many real-world problems of artificial intelligence (AI) and medicine. We examine the problem of identifying appropriate three classes for domain-specific applications.

Subhash Kak

This paper examines common assumptions regarding the decision-making internal environment for intelligent agents and investigates issues related to processing of memory and belief states to help obtain better understanding of the responses. In specific, we consider order effects and discuss both classical and non-classical explanations for them. We also consider implicit cognition and explore if certain inaccessible states may be best modeled as quantum states. We propose that the hypothesis that quantum states are at the basis of order effects be tested on large databases such as those related to medical treatment and drug efficacy. A problem involving a maze network is considered and comparisons made between classical and quantum decision scenarios for it.

Subhash Kak

This article explores the ideas that went into George Boole's development of an algebra for logical inference in his book The Laws of Thought. We explore in particular his wife Mary Boole's claim that he was deeply influenced by Indian logic and argue that his work was more than a framework for processing propositions. By exploring parallels between his work and Indian logic, we are able to explain several peculiarities of this work.

Subhash Kak

The paper analyzes the problem of judgments or preferences subsequent to initial analysis by autonomous agents in a hierarchical system where the higher level agents does not have access to group size information. We propose methods that reduce instances of preference reversal of the kind encountered in Simpson's paradox.

Subhash Kak

We propose that a general learning system should have three kinds of agents corresponding to sensory, short-term, and long-term memory that implicitly will facilitate context-free and context-sensitive aspects of learning. These three agents perform mututally complementary functions that capture aspects of the human cognition system. We investigate the use of CC1 and CC4 networks for use as models of short-term and sensory memory.

Subhash Kak

We examine the issue of stability of probability in reasoning about complex systems with uncertainty in structure. Normally, propositions are viewed as probability functions on an abstract random graph where it is implicitly assumed that the nodes of the graph have stable properties. But what if some of the nodes change their characteristics? This is a situation that cannot be covered by abstractions of either static or dynamic sets when these changes take place at regular intervals. We propose the use of sets with elements that change, and modular forms are proposed to account for one type of such change. An expression for the dependence of the mean on the probability of the switching elements has been determined. The system is also analyzed from the perspective of decision between different hypotheses. Such sets are likely to be of use in complex system queries and in analysis of surveys.

Subhash Kak

Data based judgments go into artificial intelligence applications but they undergo paradoxical reversal when seemingly unnecessary additional data is provided. Examples of this are Simpson's reversal and the disjunction effect where the beliefs about the data change once it is presented or aggregated differently. Sometimes the significance of the difference can be evaluated using statistical tests such as Pearson's chi-squared or Fisher's exact test, but this may not be helpful in threshold-based decision systems that operate with incomplete information. To mitigate risks in the use of algorithms in decision-making, we consider the question of modeling of beliefs. We argue that evidence supports that beliefs are not classical statistical variables and they should, in the general case, be considered as superposition states of disjoint or polar outcomes. We analyze the disjunction effect from the perspective of the belief as a quantum vector.

Prashanth Busireddygari, Subhash Kak

This paper proposes a new class of random sequences called binary primes tableau (PT) sequences that have potential applications in cryptography and communications. The PT sequence of rank p is obtained from numbers arranged in a tableau with p columns where primes are marked off until each column has at least one prime and where the column entries are added modulo 2. We also examine the dual to the PT sequences obtained by adding the rows of the tableau. It is shown that PT sequences have excellent autocorrelation properties.

Subhash Kak

Quantum decision systems are being increasingly considered for use in artificial intelligence applications. Classical and quantum nodes can be distinguished based on certain correlations in their states. This paper investigates some properties of the states obtained in a decision tree structure. How these correlations may be mapped to the decision tree is considered. Classical tree representations and approximations to quantum states are provided.

Subhash Kak

This paper reconsiders the problem of the absent-minded driver who must choose between alternatives with different payoff with imperfect recall and varying degrees of knowledge of the system. The classical absent-minded driver problem represents the case with limited information and it has bearing on the general area of communication and learning, social choice, mechanism design, auctions, theories of knowledge, belief, and rational agency. Within the framework of extensive games, this problem has applications to many artificial intelligence scenarios. It is obvious that the performance of the agent improves as information available increases. It is shown that a non-uniform assignment strategy for successive choices does better than a fixed probability strategy. We consider both classical and quantum approaches to the problem. We argue that the superior performance of quantum decisions with access to entanglement cannot be fairly compared to a classical algorithm. If the cognitive systems of agents are taken to have access to quantum resources, or have a quantum mechanical basis, then that can be leveraged into superior performance.

B. Prashanth Reddy, Subhash Kak

This paper proposes the use of the binary primes sequence to strengthen pseudorandom (PN) decimal sequences for cryptography applications. The binary primes sequence is added to the PN decimal sequence (where one can choose from many arbitrary shift values) and it is shown that the sum sequence has improved autocorrelation properties besides being computationally hard. Also, an analysis on the computational complexity is performed and it is shown that the complexity for the eavesdropper is of exponential complexity and therefore, the proposed method is an attractive procedure for cryptographic applications.

Krishnamurthy Kirthi, Subhash Kak

This paper presents a method of universal coding based on the Narayana series. The rules necessary to make such coding possible have been found and the length of the resulting code has been determined to follow the Narayana count.

Subhash Kak

A multiparty computation protocol is described in which the parties can generate different probability events that is based on the sharing of a single anonymized random number, and also perform oblivious transfer. A method to verify the correctness of the procedure, without revealing the random numbers used by the parties, is proposed.

Subhash Kak

Although random sequences can be used to generate probability events, they come with the risk of cheating in an unsupervised situation. In such cases, the oblivious transfer protocol may be used and this paper presents a variation to the DH key-exchange to serve as this protocol. A method to verify the correctness of the procedure, without revealing the random numbers used by the two parties, is also proposed.

Kam Wai Clifford Chan, Mayssaa El Rifai, Pramode K. Verma et al.

This paper presents a multi-stage, multi-photon quantum key distribution protocol based on the double-lock cryptography. It exploits the asymmetry in the detection strategies between the legitimate users and the eavesdropper. The security analysis of the protocol is presented with coherent states under the intercept-resend attack, the photon number splitting attack, and the man-in-the-middle attack. It is found that the mean photon number can be much larger than one. This complements the recent interest in multi-photon quantum communication protocols that require a pre-shared key between the legitimate users.

Krishnamurthy Kirthi, Subhash Kak

This paper investigates randomness properties of sequences derived from Fibonacci and Gopala-Hemachandra sequences modulo m for use in key distribution applications. We show that for sequences modulo a prime a binary random sequence B(n) is obtained based on whether the period is p-1 (or a divisor) or 2p+2 (or a divisor). For the more general case of arbitrary m, we use the property if the period is a multiple of 8 or not. The sequences for prime modulo have much better autocorrelation properties. These are good candidates for key distribution since the generation process is not computationally complex.

Subhash Kak

Unary coding is useful but it is redundant in its standard form. Unary coding can also be seen as spatial coding where the value of the number is determined by its place in an array. Motivated by biological finding that several neurons in the vicinity represent the same number, we propose a variant of unary numeration in its spatial form, where each number is represented by several 1s. We call this spread unary coding where the number of 1s used is the spread of the code. Spread unary coding is associated with saturation of the Hamming distance between code words.

Subhash Kak

The piggy bank idea allows one-way encryption of information that can be accessed only by authorized parties. Here we show how the piggy bank idea can be used to authenticate parties to counter man-in-the-middle (MIM) attack that can jeopardize the double-lock cryptography protocol. We call this method double-signature double lock cryptography and it can be implemented in ways that go beyond hash-based message authentication.

Subhash Kak

This paper proposes the use of sets of keys, together with corresponding identifiers, for use in wireless sensor networks and similar resource-constrained applications. A specific cryptographic scheme described in the paper is based on the use of a family of self-inverting matrices derived from the number theoretic Hilbert transform in conjunction with the Blom's scheme. In a randomized version of this scheme, the users change their published IDs at will but the parties can still reach agreement on the key by using individual scaling factors. The random protocol increases the security of the system.

Subhash Kak

This paper presents a class of random orthogonal sequences associated with the number theoretic Hilbert transform. We present a constructive procedure for finding the random sequences for different modulus values. These random sequences have autocorrelation function that is zero everywhere excepting at the origin. These sequences may be used as keys and in other cryptography applications.

Subhash Kak

Most current research on quantum cryptography requires transmission and reception of single photons that creates severe implementation challenges and limits range. This paper argues for the development of threshold quantum cryptography protocols in which the system is secure so long as the number of photons being exchanged between Alice and Bob is below a specified threshold. We speak of a (p-k-n) threshold system where if the number of photons exchanged is less than p, the system is completely secure, when it is between p and k, the system is partially secure, and when it exceeds k, the system is insecure. The BB84 protocol is (1-1-1) whereas the three-stage protocol appears to be (p-4p-n), where p is the least number of photons necessary to determine the polarization state of identically prepared photons. New quantum cryptography systems should be sought that provide greater flexibility in the choice of p and k.

Subhash Kak

This paper presents a general expression for a number-theoretic Hilbert transform (NHT). The transformations preserve the circulant nature of the discrete Hilbert transform (DHT) matrix together with alternating values in each row being zero and non-zero. Specific examples for 4-point, 6-point, and 8-point NHT are provided. The NHT transformation can be used as a primitive to create cryptographically useful scrambling transformations.

Matt Stowe, Subhash Kak

This paper examines the memory capacity of generalized neural networks. Hopfield networks trained with a variety of learning techniques are investigated for their capacity both for binary and non-binary alphabets. It is shown that the capacity can be much increased when multilevel inputs are used. New learning strategies are proposed to increase Hopfield network capacity, and the scalability of these methods is also examined in respect to size of the network. The ability to recall entire patterns from stimulation of a single neuron is examined for the increased capacity networks.

Subhash Kak

Primitive Pythagorean triples (PPT) may be put into different equivalence classes using residues with respect to primes. We show that the probability that the smaller odd number associated with the PPT triple is divisible by prime p is 2/(p+1). We have determined the autocorrelation function of the Baudhayana sequences obtained from the residue classes and we show these sequences have excellent randomness properties. We provide analytical explanation for the peak and the average off-peak values for the autocorrelation function. These sequences can be used specifically in a variety of key generation and distribution problems and, more generally, as pseudorandom sequences.

Gunjan Talati, Subhash Kak

This paper presents results on generalized public key cryptography with exponentials modulo primes and composite numbers where the mapping is not one-to-one and the uniqueness is achieved by additional side information. Such transformations may be used for oblivious transfer and generate events of specific probabilities.

Gunjan Talati, Subhash Kak

This paper presents the quartic public key transformation which can be used for public key applications if side information is also used. This extends an earlier work where the cubic transformation was similarly used. Such a transformation can be used in multiparty communications protocols.

Subhash Kak

This paper presents applications of the trope of the locked and sealed piggy-bank into which the secret can be easily inserted but from which it cannot be withdrawn without opening the box. We present a basic two-pass cryptographic scheme that can serve as template for a variety of implementations. Together with the sealed piggy-bank is sent a coded letter that lists and certifies the contents of the box. We show how this idea can help increase the security of cryptographic protocols for classical systems as well as those based on "single-state" systems. More specifically, we propose the use of a hashing digest (instead of the coded letter) to detect loss of key bits to the eavesdropper and use in communication systems where error correction is an important issue.

Subhash Kak

This paper investigates the randomness properties of a function of the divisor pairs of a natural number. This function, the antecedents of which go to very ancient times, has randomness properties that can find applications in cryptography, key distribution, and other problems of computer science. It is shown that the function is aperiodic and it has excellent autocorrelation properties.

Krishnama Raju Kanchu, Subhash Kak

Goldbach partitions can be used in creation of ellipses and circles on the number line. We extend this work and determine the count and other properties of concentric Goldbach circles for different values of n. The autocorrelation function of this sequence with respect to even and odd values suggests that it has excellent randomness properties. Cross correlation properties of ellipse and circle sequences are provided that indicate that these sequences have minimal dependencies and, therefore, they can be used in spread spectrum and other cryptographic applications.

Deepthi Cherlopalle, Subhash Kak

This paper investigates the use of the number of Goldbach triples, or the number of three prime partitions of an odd number, for use in the generation and distribution of cryptographic keys. In addition to presenting randomness properties of these triples, which turn out to be similar to that of prime partitions of even numbers, we explore the question of restricted partition sets. We propose a protocol for key distribution that is based on these numbers. Two of the three partitions of the randomly chosen number serve as cover to send the third number to the two parties that wish to communicate with each other. This third number can serve as session key and the original number of which it is a partition can be used for audit purposes.

Sayonnha Mandal, Gregory Macdonald, Mayssaa El Rifai et al.

The paper presents the implementation of a quantum cryptography protocol for secure communication between servers in the cloud. As computing power increases, classical cryptography and key management schemes based on computational complexity become increasingly susceptible to brute force and cryptanalytic attacks. Current implementations of quantum cryptography are based on the BB84 protocol, which is susceptible to siphoning attacks on the multiple photons emitted by practical laser sources. The three-stage protocol, whose implementation is described in this paper, is a departure from conventional practice and it obviates some of the known vulnerabilities of the current implementations of quantum cryptography. This paper presents an implementation of the three-stage quantum communication protocol in free-space. To the best of the authors' knowledge, this is the first implementation of a quantum protocol where multiple photons can be used for secure communication.

Krishnama Raju Kanchu, Subhash Kak

The paper studies cryptographically useful properties of the sequence of the sizes of Goldbach ellipses. We show that binary subsequences based on this sequence have useful properties. They can be used to generate keys and to provide an index-based mapping to numbers. The paper also presents a protocol for secure session keys that is based on Goldbach partitions.

Krishnama Raju Kanchu, Subhash Kak

We consider the use of Goldbach numbers as random sequences. The randomness is analyzed in terms of the autocorrelation function of the sequence of number of partitions. The distinct representations of an even number n as the sum of two primes is a local maximum for multiples of the product of the consecutive smallest primes less than the number. Specific partitions, which we call Goldbach ellipses, are examined. It is shown that such ellipse sequences also have excellent randomness property.

Subhash Kak, Yuhua Chen, Pramode Verma

This paper reports a variant of the three-stage quantum cryptography protocol which can be used in low intensity laser output regimes. The variant, which tracks the intensity of the laser beam at the intermediate stages, makes the task of the eavesdropper harder than the standard K06 protocol. The constraints on the iAQC protocol are much less than those on BB84 and in principle it can not only be used for key distribution but also for direct bitwise encryption of data. The iAQC protocol is an improvement on the K06 protocol in that it makes it harder for the eavesdropper to monitor the channel.

Sindhu Chitikela, Subhash Kak

This paper investigates the effect of permutations on blocks of a prime reciprocal sequence on its randomness. A relationship between the number of permutations used and the improvement of performance is presented. This can be used as a method for increasing the cryptographic strength of pseudorandom sequences.