
Areas of Interest Quantum and Classical
Information & Computation, Geometry, Information Geometry and Stochastic
Processes Stochastic Calculus,
Computational Mathematics, Physics and Earth Sciences Intrusion Prevention &
Detection, Cyberwarfare, Internet of Things Anonymity,
Cryptography & Cryptanalysis 

1.
Quantum and Classical Information and Computation
1.1
Quantum
Distributed Systems and Networks
Research in this area is progressing in a theoretical setting, exploring the significance of various models and how they relate to experimental results.
J. Spring, Quantum Distributed Systems, 13^{th} Biennial International Quantum Structures Association, Leicester, 11^{th} – 15^{th} July, 2016
1.2
Quantum
Geometry
Research in this area has centred on non – Euclidean representations of quantum states. In particular Hyperbolic representations.
1.3
Cyber
Security
Research in this area involves the application of game theory to networks and in traffic behaviour. Stochastic models are also considered.
A
Selection of Publications
J. Abegunde, H. Xiao, and J. Spring, A Dynamic Game with Adaptive Strategies for IEEE 802.15.4 and IoT, IEEE Trustcom (2016)
J. Abegunde, H. Xiao, and J. Spring, Resilient TitForTat (RTFT)  A Game Solution For Wireless Misbehaviour, IEEE Xplore, (2015)
J. Abegunde, H. Xiao, and J. Spring, A Resilient MAC Protocol for Wireless Networks, IMA Conference on Game Theory and its Applications, (2014)
A Selection
of Presentations
Paul Wernick, Bruce
Christianson and Joseph Spring, Simulating Perceptions of Security, Security
Protocols, 25^{th} International Workshop, Cambridge, UK, (2017),
J. Spring, Cyber
Security: Aspects of Classical and Quantum Cryptography, British Council
Computer Science and IT Tour, Pune and Chennai, India, 23^{rd} – 27^{th}
January, (2017)
1.4
Cryptography
The theory of divisors has played a remarkably productive role in classical cryptography, particularly in the application of function fields from algebraic geometry. Asymmetric (public) key systems based upon the Discrete Logarithm Problem [DLP] for finite fields, the Elliptic Curve DLP and more recently the Hyperelliptic Curve DLP have each depended heavily upon mathematical structures reliant upon the theory of divisors as indeed have systems based upon the Integer Factorisation Problem.
With the advent of Shors algorithms, and the Hidden Subgroup Problem the continuing successful application of finite fields in asymmetric key cryptography has been put into question. Recent workshops within the classical community however, have cited various schemes that may be resistant to such quantum attacks (for example the DiffieLamportMerkle Signature system, the NTRU encryption system, the McEliece encryption system, and the HFE signature system). Additionally, applications of Galois theory to classical protocols are not restricted solely to their use in asymmetric key cryptography but may be found for example, in error detection and symmetric (private) key cryptography. In quantum applications such as coding, cryptography, tomography and computing an increasing role is being played by concepts drawn from algebraic geometry.
Finite fields in a quantum setting have appeared in various papers with discussions for example, on
· the complexity of performing Galois Field arithmetic in quantum black boxes (referred to by Shor in his factorisation algorithms for the IFP and DLP)
· phase space (defined over finite fields as the scalar field),
· maximally unbiased bases
· Galois rings
· Galois operators.
We focus on an alternative and (we believe) a more fundamental approach to the development and application of such primitives.
A
selection of Publications
JOSEPH SPRING, Polynomials in a Multipartite Setting,
(to appear)
JOSEPH SPRING, Quantum
Primitives, Quantum Communication, Measurement and Computing 9, AIP,
(2009)
JOSEPH SPRING, Entanglement and Irreducibility, Quantum Communication, Measurement and Computing 8, AIP, (2007)
1.5
Information
Geometry
We are currently exploring applications of information geometry within our main research areas
1.6
Random
Matrices, Lie Algebras, Hopf Algebras (aka Quantum
Groups), Free Probability and Quantum Information Theory
More to follow.
2. Quantum Probability
Classical probability theory for martingales (stochastic integration) is traditionally based upon a probability space (Ω, F, m) consisting of a sample space Ω, a sigma algebra of subspaces of the sample space and a probability measure m together with a filtration { F_{t}}_{t }_{≥0}consisting of complete s  subrings of F. A conditional expectation operator E ( .  F_{t }) is also employed mapping functions integrable with respect to F to functions integrable with respect to F_{t}. A martingale is then a process X_{t }such that " t' ≥ t, E (X_{t'} F_{t}) = X_{t}. Reformulating the theory in terms of the Hilbert space L^{2}(Ω, F , m) with associated filtration L^{∞}(Ω, F_{t }, m) (a von Neumann algebra filtration) and conditional expectation (projection) from L^{2}(Ω, F , m) to L^{2}(Ω, F_{t }, m) one may develop a quantum analogue of the classical construction based for example on Irving Segals probability gage space (H, A, m) consisting of a Hilbert space H, a von Neumann algebra A, a filtration of von Neumann algebras { A _{z}}_{ z} and gage m. We consider noncommutative constructions in Clifford and QuasiFree settings resulting in orthogonal, centred L^{2 }martingales, obeying isometry properties as stochastic integrals.
Representation theorems for such martingales have been established and research is actively being pursued in this and related areas of probability theory.
A Selection of Publications
W. J. SPRING, A Quantum Stochastic Calculus (2012), University of Hertfordshire
W. J. SPRING, Multidimensional Partial Orderings and Quantum Stochastic Processes, Quantum Communication, Measurement and Computing 10, AIP, (2011)
W. J. SPRING, Multiparameter Quantum Stochastic Processes, QPRT 30, Quantum Probability and White Noise Analysis, (2010)
W. J. SPRING, Quantum Stochastic Processes, Quantum Communication, Measurement and Computing 9, AIP, (2009)
W. J. SPRING, Multiparameter Quantum Processes over the Clifford Sheet, International Journal of Pure and Applied Mathematics, (Invited), 49, (No. 3), (2008), 451457
W. J. SPRING, QuasiFree Stochastic Integrals and Martingale Representation, (CCR), Quantum Probability and White Noise Analysis, 23, (2008).
W. J. SPRING, Martingale Representation in the Clifford and QuasiFree Sheet, Quantum Communication, Measurement and Computing 8, AIP, (2007)
W. J. SPRING, The ItoClifford WongZakai Integrals and Martingale Representation, Foundations of Probability and Physics 4, AIP, (2007)
W. J. SPRING AND I. F. WILDE., QuasiFree Fermion Planar Quantum Stochastic Integrals, QPPQ, Quantum Probability and White Noise Analysis, 15, (2003).
W. J. SPRING AND I. F. WILDE, QuasiFree Quantum Stochastic Integrals in the Plane, Rep. Math. Phys. 49 (2002), 6376
W. J. SPRING AND
3.
Voting Schemes
3.1
Classical
and Quantum Voting
Various properties have emerged as being desirable from the literature regarding classical secret ballot voting schemes. Amongst these is the concept of resilience which involves the properties of universal verifiability, privacy, and robustness. A universally verifiable election scheme is a scheme deemed open to scrutiny by all interested parties. Compliance with this property ensures that ballots are carried out correctly and that subsequent tallies are fairly assessed. From a scheme satisfying the privacy property an honest participant is assured that their vote remains confidential, provided the number of attackers does not grow too large. With the property of robustness an election scheme has the capacity to recover from faults again, provided the number of parties involved does not grow too large. Schemes satisfying these three properties are said to be resilient.
The concept of a receiptfree election scheme has also emerged as a desirable property particularly as a counter to the risk of vote buying/coercion. Receiptfree election schemes ensure that voters cannot prove, to other parties, the particular vote cast within the scheme. Further desirable properties, are to be found in the literature.
Voting protocols performed within a classical setting are in general grouped according to their use of: homomorphism, MIX nets and blind signatures.
Homomorphic election schemes involve the use of a (*_{P }, *_{C}) homomorphic probabilistic encryption scheme E. These consist of a plaintext space V, a ciphertext space C (each of which form a group structure (V,) and also (C,') under appropriate binary operations and ( ' ) together with a family of homomorphic encryption schemes { E _{i} } such that each E_{i}_{ }maps V to C by c = E _{i}_{ }(v). The homomorphic property [CGS:1997] may be defined as follows: let c_{j} = E_{i_j}(v_{j}) and c__{k} = E_{_{i_k}}(v__{k}) for some i_j, i_k positive natural numbers. Then there exists a natural number i such that c_{j}' c_{k} = E_{i} (v_{j}_{ }v_{k} ). Homomorphic election schemes are felt to be important in that, for example, they allow one to derive tallies without observing specific votes, the specific votes not being decrypted. Further such schemes lead to resiliant election schemes [CFSY: 1996, CGS: 1997].
MIX net schemes, first introduced by David Chaum, have found applications in scenarios involving anonymity, elections and payments. The development of classical MIX net schemes to achieve in particular privacy initially led to ciphertext whose size was proportional to the number of MIX servers involved in the scheme. This problem was resolved by Park, Itoh and Kurosawa, resulting in ciphertext whose length was independent of the number of MIX servers. Sako and Kilian, produced a general MIX net scheme satisfying verifiability but failing with regard to robustness. The first resilient MIX net scheme was produced by Ogata, Kurosawa, Sako and Takatani.
A MIX net election scheme involves the use of shuffle machine agents referred to as MIX servers, which take as input, a ciphertext vector, these could be for example, encrypted votes, (c_{1}, c_{2}, ... , c_{n} ) submitted by for example voters, (v_{1}, v_{2}, ... , v_{n} ) and produces as output a permuted vector (in which the components are shuffled) of corresponding output (for example, decrypted votes) such that the source of each ciphertext (vote) remains hidden. In a kMIX scheme the output is then passed to the next shuffle agent, the process is repeated with another permutation, and then passed to the next MIX, until each of the agents has processed the data.
The resilience properties of privacy, verifiability and robustness may be presented in terms of tprivacy, tverifiability and trobustness, where it is understood that t refers to the number of malicious MIX servers that the scheme can withstand given at most n  2 malicious sources. A scheme satisfying the above three tproperties is said to be tresilient.
Blind signature schemes, also introduced by David Chaum, have been developed with applications in anonymity, election and payment schemes. The basic concept involves obtaining a signature to authenticate a ciphertext, for example an encrypted vote, without the signer being able to observe the plaintext vote itself. Verification regarding the signature is supported by such schemes whilst maintaining privacy regarding the actual plaintext. The signer is thus denied the ability to link a particular plaintext with its corresponding blind signature. Variations upon such schemes are to be found with for example Fair Blind Signatures in which the possibility of, for example, blackmail is discussed.
Various anonymity schemes within a quantum setting are considered.
References
[CGS] R. Cramer, R. Gennaro and B. Schoenmakers, Eurocrypt '97 Proceedings, LNCS, 1233, 103 (1997).
[CFSY] R. Cramer, M.
Franklin, B. Schoenmakers and M. Yung, Eurocrypt '96 Proceedings, LNCS, 1070, 72 (1996).
A
Selection of Publications
J. A.VACCARO, J. SPRING AND A. CHEFLES, Quantum Protocols for Anonymous Voting and Surveying, Phys Rev A, 75, 1 (2007).
J. A.VACCARO, J. SPRING AND A. CHEFLES, Quantum Protocols for Anonymous Voting and Surveying, quantph/0504161.
Note: The first publication on quantum voting.
4.
Computational Mathematics and Physics
I am interested in the application of algorithms to modelling concepts involving mathematics and physics. From a security perspective: techniques involving intrusion detection. From a physics perspective: Modelling Earth Systems. From a mathematics perspective: I am interested in exploring applications involving Representation Theory and Algebraic Geometry.
Seminars and Lectures
In Computing I have delivered seminars in Cryptography and Computer and Internet Security. I have been the driving force in establishing Computer Security Groups, created to support and encourage undergraduate and postgraduate research in security related areas. Leading these groups has involved delivering seminars, inviting: guest speakers, colleagues and students, to deliver presentations to the group, for the benefit of the group. In computing I have lectured on a wide range of courses summarised in the lecture links.
In Mathematics I have delivered seminars on Tensor Products, Fock Space, Measure Theory, von Neumann Algebras, and Quantum Probability and Quantum Information Theory. Research has involved Quantum Stochastic Models, Commutative and NonCommutative Integration Theory, Martingales, Operator Theory, C* Algebras, von Neumann Algebras and Quantum Statistical Mechanics. Papers have been published jointly with colleagues in Quantum Stochastic Calculus, and Quantum Voting Protocols, Quantum Information Theory and in applications of Game Theory to Security. I have taught mathematics to a wide range of students and levels e.g. Quantum Computing, Operator Theory, Gyrovectors and Hyperbolic Geometry, Algebraic Geometry, Finite Field Theory, Statistics, Mathematical Models and Methods, Quantitative Methods, Logic, Functions, Group Theory, Linear Algebra, ALevel (Pure and Applied), GCSE, Access, and Numeracy.
Supervision
I have supervised students for PhD, MPhil, MSc (Res), MSc. and BSc.(Hons). Currently, supervising students in areas including: Quantum Distributed Systems, Cyber Security, Intrusion Detection, Applications of Game Theory to Secure Systems, Security and VoIP, Security and the Internet of Things, Cryptography, Cryptographic algorithms, and the representation of Quantum states via Hyperbolic Geometry.