Average-case complexity of the Euclidean algorithm with a fixed polynomial over a finite field

We analyse the behaviour of the Euclidean algorithm applied to pairs (g,f) of univariate nonconstant polynomials over a finite field $\mathbb{F}_{q}$ of q elements when the highest degree polynomial g is fixed. Considering all the elements f of fixed degree, we establish asymptotically optimal bounds in terms of q for the number of elements f that are relatively prime with g and for the average degree of $\gcd(g,f)$ . We also exhibit asymptotically optimal bounds for the average-case complexity of the Euclidean algorithm applied to pairs (g,f) as above.

SIGACT News Complexity Theory Column 108

Warmest thanks to Rafael Pass and Muthu Venkitasubramaniam for this issue's guest column, "Average-Case Complexity Through the Lens of Interactive Puzzles." When I mentioned to them that my introduction would have a section on Alan Selman's passing, they immediately wrote back that they were very sorry to hear of Alan's passing, and mentioned (as you will see discussed in the second page of their article), "The main problem that we are addressing actually goes back to a paper of Even, Selman, and Yacobi from 1984: "The Complexity of Promise Problems with Applications to Public-Key Cryptography'." It is beautiful, and a tribute to the lasting influence of Alan's research, that in the 2020s his work from many decades earlier is helping shape the field's dialogue.

Guest Column

We review a study of average-case complexity through the lens of interactive puzzles- interactive games between a computationally bounded Challenger and computationally-bounded Solver/Attacker. Most notably, we use this treatment to review a recent result showing that if NP is hard-on-the-average, then there exists a sampleable distribution over only true statements of an NP language, for which no probabilistic polynomial time algorithm can find witnesses. We also discuss connections to the problem of whether average-case hardness in NP implies averagecase hardness in TFNP, or the existence of cryptographic one-way functions.

A pragmatic mixed-methods review of changing “case-complexity” of referrals to an intensive support service

Purpose “Case-complexity” is a widely used but under-explored concept across health and social care. A region’s Intensive Support Teams (ISTs) had been reporting an increase in “case-complexity”, but had not tested this hypothesis against data. This study aims to investigate this question through a pragmatic mixed-methods approach as part of a wider service evaluation. Design/methodology/approach Health of the Nation Outcome Scales for People with Learning Disabilities (HoNOS-LD) scores were used (n = 1,766) to estimate average “case-complexity” of referrals over an eight-year sample period. Two focus groups for IST staff (n = 18) explored why “case-complexity” appears to be increasing. Participant perspectives were subjected to thematic analysis. Findings Average HoNOS-LD scores have steadily increased over the sample period, suggestive of increasing “case-complexity”. Focus groups identified three broad themes to potentially explain the increased complexity: effects of Transforming Care; people’s changing and unchanging support systems; and issues related to mild and borderline intellectual disability. Many perspectives are grounded in or supported by evidence. Research limitations/implications Implications and limitations of findings are discussed, including areas for further consideration and research. The well-designed “short-cut” is promoted as a strategy for busy professionals in need of practice-based evidence but with limited research time and resources. Originality/value The findings and discussion will be of value to anyone involved in the design, commissioning and delivery of mental health and challenging behaviour services to people with intellectual and developmental disabilities (IDD) under Transforming Care. Study methodology is easily replicable to build broader picture about “case-complexity” among UK’s IDD population.

Statistical Properties of Lambda Terms

We present a quantitative, statistical analysis of random lambda terms in the De Bruijn notation. Following an analytic approach using multivariate generating functions, we investigate the distribution of various combinatorial parameters of random open and closed lambda terms, including the number of redexes, head abstractions, free variables or the De Bruijn index value profile. Moreover, we conduct an average-case complexity analysis of finding the leftmost-outermost redex in random lambda terms showing that it is on average constant. The main technical ingredient of our analysis is a novel method of dealing with combinatorial parameters inside certain infinite, algebraic systems of multivariate generating functions. Finally, we briefly discuss the random generation of lambda terms following a given skewed parameter distribution and provide empirical results regarding a series of more involved combinatorial parameters such as the number of open subterms and binding abstractions in closed lambda terms.