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### Generalization of a few results in Integer Partitions [article]

Manosij Ghosh Dastidar, Sourav Sen Gupta
2011 arXiv   pre-print
In this paper, we generalize a few important results in Integer Partitions; namely the results known as Stanley's theorem and Elder's theorem, and the congruence results proposed by Ramanujan for the partition function. We generalize the results of Stanley and Elder from a fixed integer to an array of subsequent integers, and propose an analogue of Ramanujan's congruence relations for the 'number of parts' function instead of the partition function. We also deduce the generating function for
more » ... ing function for the 'number of parts', and relate the technical results with their graphical interpretations through a novel use of the Ferrer's diagrams.

### Extension of Stanley's Theorem for Partitions [article]

Manosij Ghosh Dastidar, Sourav Sen Gupta
2010 arXiv   pre-print
In this paper we present an extension of Stanley's theorem related to partitions of positive integers. Stanley's theorem states a relation between "the sum of the numbers of distinct members in the partitions of a positive integer n" and "the total number of 1's that occur in the partitions of n". Our generalization states a similar relation between "the sum of the numbers of distinct members in the partitions of n" and the total number of 2's or 3's or any general k that occur in the
more » ... ur in the partitions of n and the subsequent integers. We also apply this result to obtain an array of interesting corollaries, including alternate proofs and analogues of some of the very well-known results in the theory of partitions. We extend Ramanujan's results on congruence behavior of the 'number of partition' function p(n) to get analogous results for the 'number of occurrences of an element k in partitions of n'. Moreover, we present an alternate proof of Ramanujan's results in this paper.

### Attack on Broadcast RC4 Revisited [chapter]

Subhamoy Maitra, Goutam Paul, Sourav Sen Gupta
2011 Lecture Notes in Computer Science
In this paper, contrary to the claim of Mantin and Shamir (FSE 2001), we prove that there exist biases in the initial bytes (3 to 255) of the RC4 keystream towards zero. These biases immediately provide distinguishers for RC4. Additionally, the attack on broadcast RC4 to recover the second byte of the plaintext can be extended to recover the bytes 3 to 255 of the plaintext given Ω(N 3 ) many ciphertexts. Further, we also study the non-randomness of index j for the first two rounds of PRGA, and
more » ... ounds of PRGA, and identify a strong bias of j2 towards 4. This in turn provides us with certain state information from the second keystream byte. Input: Secret Key K. Output: S-Box S generated by K. Initialize counter: j = 0; Initialize the counters: i = j = 0; ABT AB with A, B bytes and T a string of bytes of small length), and the state recovery attack presented by Maximov and Khovratovich [6] . However, the major portion of the literature in RC4 cryptanalysis involves results related to initial keystream bytes of PRGA [2, 5] (also see the references therein). To get rid of these problems, one may throw away some initial bytes of RC4 PRGA as suggested in [3, 7] . But it may not be easy to modify the actual implementations immediately by throwing away some initial keystream bytes, since RC4 is already in use in many commercial applications. Thus the cryptanalytic results related to the initial bytes are still of importance. Moreover, these results are always of theoretical significance in terms of studying one of the most popular stream ciphers. The trend continues, including the most recent biases in this direction [8] that relates the initial keystream bytes, state variables and secret key of RC4. Recently, another paper [9] accepted at Eurocrypt 2011 exploited the known biases of RC4 (mostly involving the initial bytes) to provide distinguishers against WEP and WPA. Using related idea, this paper also proposes the best key recovery attack against WPA till date. Notation. Let S r , i r , j r , z r denote the state, index i, index j, and the keystream byte respectively, after r (≥ 1) rounds of PRGA have been performed. Let S 0 denote the state just before the PRGA starts, i.e., right after the KSA ends. Further, let p r,x denote the probability Pr(S r [x] = x), after r rounds of PRGA, where r ≥ 1 and 0 ≤ x ≤ N − 1. Motivation and Contribution. In FSE 2001, Mantin and Shamir [5] pub-

### Coronaviruses: origin and evolution

Sourav Sen, Kavita Bala Anand, Santosh Karade, R.M. Gupta
2020 Medical Journal Armed Forces India
The world is currently face to face with a pandemic which is spreading rapidly across the globe caused by SARS-CoV-2, a strain of Coronaviruses (CoVs) belonging to subgenus Sarbecovirus of genus Betacoronavirus. World Health Organisation (WHO) on 11 Feb 20 named this disease caused by SARS-CoV-2 as Covid-19. This pandemic is spreading rapidly and more than 20,00,000 cases have occurred globally. The human Coronaviruses discovered in 1960s were considered potentially harmless endemic viruses
more » ... endemic viruses with seasonal distribution before late 2002. The CoVs are found in a large number of domestic and wild animals and birds. The first pandemic caused by Coronavirus caused by SARS-CoV was recognized in the late 2002 in Guangdong Province and resulted in widespread morbidity and mortality. This was followed by MERS-CoV which began in 2012 in the Arabian peninsula with multiple outbreaks related to it in various parts of the globe. Various studies have suggested how these viruses made their entry from their natural reservoir bats via intermediate host like civets and camels in case of SARS-CoV and MERS-CoV respectively. The intermediate host of the SARS-CoV-2 still needs to be established. The SARS-CoV-2 has 96.2% similarity to the bat Severe Acute Respiratory Syndrome related-Coronavirus (SARSr-CoV RaTG13). SARS-CoV-2 has been found to be more distant in relation to and MERS-CoV (50%). At the whole genome sequence level pangolin CoV and SARSr-CoV RaTG13 show 91.02% and 96.2% similarity with SARS-CoV-2 but the S1 subunit of spike protein of pangolin CoV is more closely related to SARS-CoV-2 than SARSr-CoV RaTG13. The genetic analysis of the currently circulating strains of the pandemic have shown 99.98-100% similarity in their genomes implying a recent shift to humans. The animal source of SARS-CoV-2 needs to be identified to implement control measures in the present pandemic. Also, how the virus moves interspecies will help predict and prevent future pandemics. .e lsev ie r. co m/ lo cate/ mj afi m e d i c a l j o u r n a l a r m e d f o r c e s i n d i a 7 6 ( 2 0 2 0 ) 1 3 6 e1 4 1

### Designing integrated accelerator for stream ciphers with structural similarities

Sourav Sen Gupta, Anupam Chattopadhyay, Ayesha Khalid
2012 Cryptography and Communications
Till date, the basic idea for implementing stream ciphers has been confined to individual standalone designs. In this paper, we introduce the notion of integrated implementation of multiple stream ciphers within a single architecture, where the goal is to achieve area and throughput efficiency by exploiting the structural similarities of the ciphers at an algorithmic level. We present two case studies to support our idea. First, we propose the merger of SNOW 3G and ZUC stream ciphers, which
more » ... ciphers, which constitute a part of the 3GPP LTE-Advanced security suite. We propose HiPAcc-LTE, a high performance integrated design that combines the two ciphers in hardware, based on their structural similarities. The integrated architecture reduces the area overhead significantly compared to two distinct cores, and also provides almost double throughput in terms of keystream generation, compared with the state-of-the-art implementations of the individual ciphers. As our second case study, we present IntAcc-RCHC, an integrated accelerator for the stream ciphers RC4 and HC-128. We show that the integrated accelerator achieves a slight reduction in area without any loss in throughput compared to our standalone implementations. We also achieve at least 1.5 times better throughput compared to general purpose processors. Long term vision of this hardware integration approach for cryptographic primitives is to build a flexible core supporting multiple designs having similar algorithmic structures.

### Proof of Empirical RC4 Biases and New Key Correlations [chapter]

Sourav Sen Gupta, Subhamoy Maitra, Goutam Paul, Santanu Sarkar
2012 Lecture Notes in Computer Science
In SAC 2010, Sepehrdad, Vaudenay and Vuagnoux have reported some empirical biases between the secret key, the internal state variables and the keystream bytes of RC4, by searching over a space of all linear correlations between the quantities involved. In this paper, for the first time, we give theoretical proofs for all such significant empirical biases. Our analysis not only builds a framework to justify the origin of these biases, it also brings out several new conditional biases of high
more » ... biases of high order. We establish that certain conditional biases reported earlier are potentially non-causal in nature as they are correlated with a third event with much higher probability. This gives rise to the discovery of new keylength-dependent biases of RC4, some as high as 50/N . The new biases in turn result in successful keylength prediction from the initial keystream bytes of the cipher.

### On Double Danielewski Surfaces and the Cancellation Problem

Neena Gupta, Sourav Sen
2019 Journal of Algebra
We study a two-dimensional family of affine surfaces which are counter-examples to the Cancellation Problem. We describe the Makar-Limanov invariant of these surfaces, determine their isomorphism classes and characterize the automorphisms of these surfaces.

### Factoring RSA Modulus Using Prime Reconstruction from Random Known Bits [chapter]

Subhamoy Maitra, Santanu Sarkar, Sourav Sen Gupta
2010 Lecture Notes in Computer Science
This paper discusses the factorization of the RSA modulus N (i.e., N = pq, where p, q are primes of same bit size) by reconstructing the primes from randomly known bits. The reconstruction method is a modified brute-force search exploiting the known bits to prune wrong branches of the search tree, thereby reducing the total search space towards possible factorization. Here we revisit the work of Heninger and Shacham in Crypto 2009 and provide a combinatorial model for the search where some
more » ... rch where some random bits of the primes are known. This shows how one can factorize N given the knowledge of random bits in the least significant halves of the primes. We also explain a lattice based strategy in this direction. More importantly, we study how N can be factored given the knowledge of some blocks of bits in the most significant halves of the primes. We present improved theoretical result and experimental evidences in this direction.

### Diarrheagenic Pathogens in Polymicrobial Infections

Brianna Lindsay, T. Ramamurthy, Sourav Sen Gupta, Yoshifumi Takeda, Krishnan Rajendran, G. Balakrish Nair, O. Colin Stine
2011 Emerging Infectious Diseases

### Some Observations on Modulo 5 Congruences for 2-Color Partitions [article]

Suparno Ghoshal, Sourav Sen Gupta
2018 arXiv   pre-print
The 2-color partitions may be considered as an extension of regular partitions of a natural number n, with p_k(n) defined as the number of 2-colored partitions of n where one of the 2 colors appears only in parts that are multiples of k. In this paper, we record the complete characterization of the modulo 5 congruence relation p_k(25n + 24 - k) ≡ 0 5 for k ∈{1, 2, ..., 24}, in connection with the 2-color partition function p_k(n), providing references to existing results for k ∈{1, 2, 3, 4, 7,
more » ... k ∈{1, 2, 3, 4, 7, 8, 17}, simple proofs for k ∈{5, 10, 15, 20} for the sake of completeness, and counter-examples in all the remaining cases. We also propose an alternative proof in the case of k = 4, without using the Rogers-Ramanujan ratio, thereby making the proof considerably simpler compared to the proof by Ahmed, Baruah and Ghosh Dastidar (JNT 2015).

### Diarrheagenic Pathogens in Polymicrobial Infections

Brianna Lindsay, T. Ramamurthy, Sourav Sen Gupta, Yoshifumi Takeda, Krishnan Rajendran, G. Balakrish Nair, O. Colin Stine
2011 Emerging Infectious Diseases
During systematic active surveillance of the causes of diarrhea in patients admitted to the Infectious Diseases and Beliaghata General Hospital in Kolkata, India, we looked for 26 known gastrointestinal pathogens in fecal samples from 2,748 patients. Samples from about one-third (29%) of the patients contained multiple pathogens. Polymicrobial infections frequently contained Vibrio cholerae O1 and rotavirus. When these agents were present, some coinfecting agents were found signifi cantly less
more » ... ignifi cantly less often (p = 10 -5 to 10 -33 ), some were detected signifi cantly more often (p = 10 -5 to 10 -26 ), and others were detected equally as often as when V. cholerae O1 or rotavirus was absent. When data were stratifi ed by patient age and season, many nonrandom associations remained statistically signifi cant. The causes and effects of these nonrandom associations remain unknown.

### One Byte per Clock: A Novel RC4 Hardware [chapter]

Sourav Sen Gupta, Koushik Sinha, Subhamoy Maitra, Bhabani P. Sinha
2010 Lecture Notes in Computer Science
RC4, the widely used stream cipher, is well known for its simplicity and ease of implementation in software. In case of a special purpose hardware designed for RC4, the best known implementation till date is 1 byte per 3 clock cycles. In this paper, we take a fresh look at the hardware implementation of RC4 and propose a novel architecture which generates 1 keystream byte per clock cycle. Our strategy considers generation of two consecutive keystream bytes by unwrapping the RC4 cycles. The same
more » ... C4 cycles. The same architecture is customized to perform the key scheduling algorithm at a rate of 1 round per clock.

### Proving TLS-attack related open biases of RC4

Santanu Sarkar, Sourav Sen Gupta, Goutam Paul, Subhamoy Maitra
2014 Designs, Codes and Cryptography
= 0 has been done by Maitra, Paul and Sen Gupta [11] in 2011.  ...  In [22, Section 2] , Sen Gupta et al. presented a family of biases in RC4 that are dependent on the length of the secret key.  ...

### Partial Key Exposure Attack on RSA – Improvements for Limited Lattice Dimensions [chapter]

Santanu Sarkar, Sourav Sen Gupta, Subhamoy Maitra
2010 Lecture Notes in Computer Science
Consider the RSA public key cryptosystem with the parameters N = pq, q < p < 2q, public encryption exponent e and private decryption exponent d. In this paper, cryptanalysis of RSA is studied given that some amount of the Most Significant Bits (MSBs) of d is exposed. In Eurocrypt 2005, a lattice based attack on this problem was proposed by Ernst, Jochemsz, May and de Weger. In this paper, we present a variant of their method which provides better experimental results depending on practical
more » ... g on practical lattice parameters and the values of d. We also propose a sublattice structure that improves the experimental results significantly for smaller decryption exponents.

### The Role of Vibrio cholerae Genotyping in Africa

Rituparna De, Jayeeta Banerjee Ghosh, Sourav Sen Gupta, Yoshifumi Takeda, G. Balakrish Nair
2013 Journal of Infectious Diseases
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