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# Comparing Alignment and Multiple Group CFA for Analysing Political Trust in Europe During the Crisis

Lluís Coromina*a, Edurne Bartolomé Peralb

Methodology, 2020, Vol. 16(1), 21–40, https://doi.org/10.5964/meth.2791

Received: 2018-01-13. Accepted: 2019-09-14. Published (VoR): 2020-04-06.

*Corresponding author at: Department of Economics, Campus Montilivi, University of Girona, 17003, Girona, Spain. Tel. +34 972 419720, E-mail: lluis.coromina@udg.edu

This is an open access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

## Abstract

Institutional trust is in decline in many western democracies. Since the 2008 global economic and financial crisis, this increasing distrust has been closely related to trust in political institutions. Trust in institutions is one of the pillars of democracy, and its decline is one of the most evident and shared symptoms of the recession, especially in those contexts where it has been particularly acute. This article has both substantive and methodological aims. From a substantive point of view, it deals with trust in political institutions in Europe, and its decline during the recession. Differences are found among European countries, depending on the severity of the economic crisis. From a methodological point of view, the article compares two methodologies for cross-cultural analyses: the more traditional Multiple-Group Confirmatory Factor Analysis, and the newer Alignment method. Recommendations for the use of the newer method in certain circumstances are provided.

Keywords: institutional trust, political trust, measurement invariance, alignment method, multiple group confirmatory factor analysis, European social survey

Citizens´ trust in public institutions is one of the most relevant elements in a system´s legitimacy and quality of democracy. However, a great deal of literature reveals a visible decline in institutional trust and political support in established democracies. Several reasons and factors have been given for this general decline. Some of these are contextual, while others are in response to a more generalized pattern within both modern and established democracies.

No country can survive without the support of its citizens (Hofferbert & Klingemann, 1999). Therefore, political trust, among other factors, is crucial for successful democratic governance, in the sense that strong political trust also contributes to a structured and stable system of voting (Zmerli & Hooghe, 2011) and also reinforces citizens' compliance with the law and governmental demands (Rudolph & Evans, 2005).

In recent years we have witnessed a significant decline in institutional trust in a number of European democracies. This is due to the financial and economic crisis affecting Europe, and the world, since 2008. In some European countries, the decline of trust in public institutions is one of the most evident symptoms of the crisis, especially in those contexts where the crisis has been particularly severe.

The literature maintains that in order to have an effective democracy and a prosperous economy, there is a need for relevant levels of widespread social trust, accompanied by high levels of trust in state institutions (Newton, 2001; Uslaner, 2002). In Europe, we are witnessing a general decline in institutional trust due to the deep financial and economic crisis, and the citizens' perception that they are deprived of rights and resources. In this regard, individuals tend to hold institutions as the main entities responsible for their situation. This lack of trust in institutions, if persistent, has significant consequences for democratic governance, in that it could be followed in some cases by a deeper crisis of legitimacy (Easton, 1965).

The main aim of this article is twofold, as it answers both methodological and substantive questions. From a methodological point of view, the article compares two methodologies used in cross-cultural analyses in order to provide recommendations for analysing this topic, first is the more traditional Multiple-Group Confirmatory Factor Analysis (MGCFA); and the second is the newer Alignment method (Asparouhov & Muthén, 2014; Muthén & Asparouhov, 2013). From a substantive point of view, the main hypotheses are based on the level and trend of trust in different European countries during the crisis (Coromina & Bartolomé Peral, 2018).

Analysing what affects institutional trust in institutions in comparative terms, and over time is also important. A significant body of literature has focused on comparative analysis and the importance of context in studying political and institutional trust, and Zmerli and Hooghe (2011, pp. 1–11) reflect on the relevance of this approach. Central to the discussion on trust in political institutions is the need for solid methodological approaches to tackle the contextual difference, and solve the problem of equivalence. In this respect, authors such as Marien (2011) have contributed to advancing comparative analysis methodology over time and space by using Multigroup Confirmatory Factor Analysis (MGCFA) for a set of state institutions in different contexts, and in different time periods. MGCFA has proven to be a suitable method for addressing the comparative analysis of trust in political institutions. However, this method has strict requirements, and invariance and comparability can be difficult to reach if the groups differ substantially. In this article, we present a method to deal with the weaknesses of MGCFA, called Alignment method (Asparouhov & Muthén, 2014; Muthén & Asparouhov, 2013), since this method shows more flexibility than MGCFA in terms of measurement invariance.

In this paper, both methods are used on a number of European democracies hit by the financial and economic crisis to different degrees, in three time periods: 2008, 2010 and 2012. Using comparative methods enables us to assess citizens' trust in public institutions within different European economic contexts, and deepens the knowledge and understanding of politics.

The article is structured as follows: first the theoretical framework and hypotheses are presented, followed by an explanation of MGCFA and Alignment methods, focusing on the comparisons between these methods. This is followed by the results, which compare the two methods, and finally, the conclusions.

### The Relevance of Trust in Institutions

Hetherington (1998, p. 791) defines political trust as “basic evaluative orientation toward the government, founded on how well the government is operating according to people´s normative expectations”. According to Newton (2006), political trust is “the belief that those in authority and with power will not deliberately or willingly do us harm, if they can avoid it, and will look after our interests, if this is possible”.

Within this framework, trust in institutions is a crucial object of political support. It is highly relevant to democratic political systems, which need systematic public support, because if institutional trust diminishes, a system´s legitimacy can easily be called into question (Hetherington, 1998, p. 792). Institutional trust may refer to legitimate power (e.g., in the parliament), authority (e.g., in the government), or the economy (e.g., in the mode of production). It can also relate to procedures and basic practices, rather than expectations of reciprocity.

This article analyses the decline in trust in institutions and its evolution in various European societies during the years of the recent economic crisis. Decline in trust in public institutions does not affect all countries equally, nor all segments of a population. In recent years, the literature has offered several explanations for this phenomenon among western democracies (Bowler & Karp, 2004; Hetherington, 1998, 2005; Newton, 2006; Rudolph & Evans, 2005; Zmerli & Hooghe, 2011). These explanations focus on cognitive mobilization, and citizens' higher level of sophistication and political efficacy in terms of information, interest in politics and level of education (Klingemann, 1998). Some scholars blame the decline on these factors, as more sophisticated citizens tend to have higher expectations of their government; but this would not necessarily imply a fall in support for the governmental system (Easton, 1965, 1975; Newton, 2006; Zmerli & Hooghe, 2011).

Several authors point out that economic performance and economic outcomes are the main dimensions of policy performance (Criado & Herreros, 2007; Friedrichsen & Zahn, 2014; Levi & Stoker, 2000; Miller, 1974). There seems to be a general claim that poor economic performance affects institutional trust. However, this does not exactly work the other way around. Good economic performance raises trust in institutions, but this trust can also be attributed to other incumbents, who could be international leaders or previous incumbents in power (Herreros & Criado, 2008; Maravall & Przeworski, 2001).

Weatherford (1987, p. 10) contributes to the debate on the relationship between policy performance and trust in institutions by providing the necessary conditions for such a relationship. According to this author, a fitting condition would be the relevance of the policy pitfall, which can be considered as such if 1) it is not attributable to “exogenous constraints”, 2) it is not attributable to differences between the party and citizens’ ideologies, 3) persistence in time 4) a pattern of social disadvantage and social differentiation in terms of social class or income. According to these explanations, there is an important consensus of opinion in the literature, linking institutional trust with policy and economic performance.

### Hypotheses

The main claim is that individual levels of trust in political institutions have changed over time during the economic crisis. The literature explains how evaluation of performance may affect trust in institutions. Taking this into account, we expect that those citizens most affected by the crisis will evaluate the political and economic performance of their governments more poorly, and therefore, trust in their political institutions will tend to be lower (Anderson & Guillory, 1997; Citrin, 1974; Weatherford, 1987). Thus, the first hypothesis is:

• H1: Levels of trust in political institutions are expected to be significantly lower in those countries most affected by the crisis.

The second hypothesis relates to the effect of economic crisis and trust in political institutions over time. Thus, the second hypothesis is:

• H2a: It is expected that levels of trust in institutions will tend to decrease over time in all countries during the crisis.

• H2b: Levels of trust will decline more sharply in those countries most affected by the crisis.

Previous to testing the hypotheses, the methodological process of evaluating the two methods for cross-cultural comparison is carried out, taking a sample of eight European countries (Germany, Netherlands, Sweden, Norway, Greece, Ireland, Portugal, and Spain) and years of crisis (2008, 2010, and 2012).

## Methods for Cross-Cultural Comparison

Given the purpose of this study, there is a need for comparative analysis across countries and time. Relevant contributions have shown the importance of a suitable methodology to ensure equivalence. In comparative cross-cultural research, Structural Equation Modelling (SEM) is normally used to compare latent constructs; in particular, its measurement structure, known as Multiple Group Confirmatory Factor Analysis (MGCFA).

In recent years, MGCFA has proved to be the best approach to ensure comparability. However, due to its strict invariance requirements, it has proven problematic when the number of non-invariant parameters increase, which is more likely if more groups are compared (Byrne & van de Vijver, 2017). Thus, in some cases the use of MGCFA is not the best solution for substantial comparisons; in such cases a newer method known as Alignment (Asparouhov & Muthén, 2014; Muthén & Asparouhov, 2013) can be used successfully, since its invariance requirements are less strict. Both methods are discussed below.

### Multiple Group Confirmatory Factor Analysis

Multiple Group Confirmatory Factor Analysis (Ariely & Davidov, 2012; Bollen, 1989; Cieciuch, Davidov, Vecchione, Beierlein, & Schwartz, 2014; Davidov, Meuleman, Cieciuch, Schmidt, & Billiet, 2014; Davidov, Schmidt, & Billiet, 2011; Meredith, 1993; Steenkamp & Baumgartner, 1998) is generally used for cross-cultural comparison in order to test if a latent variable of interest is comparable across groups, countries and/or years, taking measurement invariance into account. In the case that invariance holds, relationships and/or latent means of the constructs can be compared across groups (countries and/or time periods).

Thus, in this paper, trust in political institutions will be measured as a latent factor with three reflective indicators, using Confirmatory Factor Analysis (Brown, 2006), which is used to estimate the measurement model, shown in a generalized form in Figure 1.

Click to enlarge

### Generalization of CFA Model

Where yi’s are three observed indicators, τi is the intercept of each of the three observed indicators, ηj is the latent variable, λij is the factor loading or slope from the j latent variable to the yi observed indicator and ei is a random measurement error for the responses for each of the three indicators. Covariances between the latent variable (ηi) and the error variance (ei) or among the error variances themselves are constrained to zero.

The estimation of each observed variables is based on the general equation:

##### 1
$y i = τ i + λ i j η j + e i$

which in this case can be broken down into the following three equations:

##### 2
$y 1 = τ 1 + λ 1 j η j + e 1$
##### 3
$y 2 = τ 2 + λ 2 j η j + e 2$
##### 4
$y 3 = τ 3 + λ 3 j η j + e 3$

where y1, y2 and y3 stand for the three indicators.

When comparison among groups is of interest, measurement invariance among the groups should be established. We can then compare the latent means and, at the same time, ensure that the latent construct has the same meaning and scaling among the groups of interest. In this context, three hierarchical levels of measurement invariance are generally tested. Configural invariance is the least strict type of invariance, which requires the same structure among groups while values for loadings and intercepts can differ among groups; thus if invariance holds, it allows the comparison of model structure. Metric invariance requires equal loadings for each group; if invariance holds, it permits relationships comparison. A more strict type of invariance than configural and metric is scalar invariance, which requires equal loading and intercepts among groups; if scalar invariance holds, it permits the comparison of latent means among groups (Davidov et al., 2014; Trusinová, 2014).

MGCFA can be used adequately when most parameters are invariant, otherwise it becomes difficult to manage (Asparouhov & Muthén, 2014; Muthén & Asparouhov, 2013) as it requires many manual adjustments to the model specification. Fit indices, expected parameter changes, or modification indices for each previous model all have to be taken into consideration for model modifications. In these cases, full or even partial invariance could not be obtained; consequently, some items or groups might be taken out of the analysis, which inhibits the comparison for all groups of interest.

The decision to take some items or groups out of the analysis must be done while taking into account both the measurement invariance and the conceptual (or substantial) validity (Billiet, 2016), since the theoretical constructs are translated into observable variables that are assumed to be valid representations of the constructs which are also determine the quality of the comparability (Billiet, 2016, p. 206).

### Alignment Method

Alignment (Asparouhov & Muthén, 2014; Muthén & Asparouhov, 2013) is an alternative method to MGCFA, and can be used to study multiple groups with substantive information (such as latent means). The main goal of Alignment method is to provide a comparison across groups, while allowing for approximate measurement invariance (Asparouhov & Muthén, 2013, 2014).

This method is based on the configural model, where the model structure is the same for all groups (see Figure 1), and estimates the latent factor means and variances without requiring the exact measurement invariance. Optimization for the Alignment method is based on maximum-likelihood, which can estimate the intercepts (τpg) parameters, factor loadings (λpg), factor means (kg), and variances (ψg) by using the assumption that the number of non-invariant measurement parameters and the amount of measurement non-invariance can be held to a minimum (Asparouhov & Muthén, 2013, p. 6). Total loss/simplicity function (F) is used, and chooses a kg and ψg, which minimizes the amount of measurement non-invariance and accumulates the total measurement non-invariance, see Equation 5:

##### 5
$F = ∑ p ∑ g 1 < g 2 w g 1, g 2 f λ p g 1,1 − λ p g 2,1 + ∑ p ∑ g 1 < g 2 w g 1, g 2 f τ p g 1,1 − τ p g 2,1$

Where the weight factor (wg1,g2) reflects the group size (Ni) and the amount of certainty in the group estimates for a particular group (bigger groups will contribute more than smaller groups to the total loss function). The weight factor is shown in Equation 6:

##### 6
$w g 1, g 2 = N g 1 N g 2$

The total loss function F is minimized at a solution where there are a few large non-invariant measurement parameters and many approximately invariant measurement parameters, rather than many medium-sized non-invariant measurement parameters (Asparouhov & Muthén, 2013, p. 8). For detailed information regarding the method, see Muthén and Asparouhov (2013), and Asparouhov and Muthén (2013, 2014).

In order to test invariance, an algorithm determines the largest invariant set of groups for each measurement parameter, where the parameter in that group is not statistically significant from the average value for that parameter across all groups in the invariant set. For each group not in the invariant set, the parameter is significantly different from that average. The first set of invariance groups is detected with a pairwise test for each pair of groups. Then the largest connected set is determined for a parameter, and this will be the starting set of groups. Next, the group is modified, by computing the average parameter following the current invariance set. The process is repeated until the invariant set stabilizes and no groups are added to, or removed from, the invariant set.

Invariance analysis is useful to determine the most invariant variables and this information can be used to improve the measurement model. Contribution to the simplicity function (F) can be isolated for each variable, reflecting its level of non-invariance. The smaller the contribution is, the more invariant the variable is. A R2 measure for each measurement parameter gives the parameter variation across groups in the configural model. This is explained by the variation in the factor means and factor variance across groups. The R2 provides a useful notion of the degree of non-invariance that can be absorbed by group-varying factor means and variances. A value close to 1 implies a high degree of invariance and a value close to 0 a low degree of invariance (Asparouhov & Muthén, 2013, p. 16, 2014, p. 500)

Another measure of non-invariance is the approximate percentage of non-invariant parameters. As a rule of thumb, a limit of 25% of non-invariant parameters may be safe for trustworthy alignment results (Muthén & Asparouhov, 2014). For higher percentages, a Monte Carlo simulation is recommended in order to assess the quality of the results (Flake & Mccoach, 2018; Muthén & Asparouhov, 2014). Monte Carlo simulations are based on the correlation between the population factor means and the estimated alignment factor means, computed over groups and averaged over replications. Correlations of at least .98 produce reliable factor means (Muthén & Asparouhov, 2014, p. 3).

### Data and Indicators Used

European Social Survey (ESS) data is used. Three different time periods of crisis (2008, 2010, and 2012) are used in eight countries. These countries were selected as they were affected by economic crisis during the period 2008-2012. Although all European countries were affected by the financial and economic crisis, there is a significant difference in the severity of the crisis in countries such as Germany, the Netherlands, Norway or Sweden (less affected by the crisis), and other countries (more affected by the crisis) such as Ireland, Greece, Portugal, and Spain. These countries had more severe austerity measures and significantly higher unemployment rates than other European countries.

Table 1 shows the sample size for each country in each time period (Coromina & Bartolomé Peral, 2018, p.62).

##### Table 1

Sample Size for Each Country and Period

Country 2008 2010 2012
Germany 2744 3015 2954
Netherlands 1775 1821 1843
Sweden 1828 1489 1843
Norway 1546 1544 1615
Greece 2066 2703 N/Aa
Ireland 1763 2542 2594
Portugal 2342 2136 2136
Spain 2554 1877 1874

Concerning the variables used in the analyses, political trust in institutions is the latent construct of interest and, according to the theory outlined in the article, the following indicators are used: a) trust in a country’s parliament, b) trust in the legal system and c) trust in politicians. Each of these three institutions represents a different type of political institution. Parliament generally represents the classic institution of state representation; the legal system accounts for effective rights and public sector performance; and politicians represent the articulation of people´s ideas and options in the political sphere through political organizations.

These indicators are obtained from the ESS questionnaire using the following question: “On a score of 0-10 how much do you personally trust each of the institutions? 0 means you do not trust an institution at all, and 10 means you have complete trust”. The institutions analysed are the following: a) the country’s parliament (namely, ‘trust in parliament’); b) the legal system (namely, ‘trust in the legal system’); and c) politicians (namely, ‘trust in politicians’).

Thus, these three variables are used in both the MGCFA and Alignment models as indicators for the latent variable ‘trust in political institutions’ (see Figure 1). Mplus 7 (Muthén & Muthén, 2012) is used for the analyses.

## Results

Firstly, invariance adjustment of both methods is shown, and secondly, factor means for the latent variable trust in political institutions in the 23 groups are calculated using both methods. Results are compared in order to draw a comparison for latent means between the traditionally used MGCFA and the newer Alignment method.

Table 2 shows each item’s intercept, loading and total contribution to the optimized simplicity function (F), and the R2 for each intercept and loading.

##### Table 2

Fit Function Contribution and R2 in Alignment Method

Intercept
Total
Fit function contribution R2 Fit function contribution R2 Fit function contribution
Trust Parliament -86.947 .590 -114.579 .926 -201.527
Trust Legal system -97.102 .459 -123.863 .929 -220.965
Trust Politicians -92.055 .147 -119.912 .942 -211.967

In accordance with the total fit function contribution, Table 2 shows that items have a similar degree of measurement invariance. Thus, they contributed similarly. However, “Trust in parliament” is the variable that contributes the least to both factor loading and intercepts, since it has the lowest total contribution to the fit function (-201.527). This result is an indicator that “Trust in parliament” shows the lowest amount of non-invariance.

R2 indicates the variation of parameters across groups in the configural model, which can be explained by the variation in the factor means and variances across groups. In this case, the factor loadings for “Trust in parliament” have the highest levels of invariance, while the degree of invariance of item intercepts is similar among them.

Table 3 shows the invariant and non-invariant parameters for the measurement item intercepts and factor loadings. Values in brackets represent the non-invariant parameters, “Trust in parliament” is the variable with more invariant parameters, with 74% of them.

##### Table 3

Intercepts/Thresholds
Trust Parliament (1) 2 3 4 5 (6) 7 8 9 10 11 12 (13) (14) (15) (16) (17) (18) 19 20 21 22 (23)
Trust Legal system (1) (2) (3) 4 5 6 7 8 9 (10) (11) (12) 13 14 15 (16) 17 18 (19) (20) 21 22 (23)
Trust Politicians 1 2 3 (4) (5) (6) 7 8 9 10 11 12 13 14 15 16 17 18 (19) (20) 21 (22) (23)
Trust Parliament 1 2 3 (4) 5 6 7 8 9 10 11 12 13 14 15 (16) 17 18 (19) 20 21 22 23
Trust Legal system 1 2 3 4 5 6 7 8 9 10 11 12 (13) 14 15 (16) (17) (18) (19) (20) 21 22 (23)
Trust Politicians 1 (2) 3 (4) 5 (6) (7) 8 9 10 11 12 13 14 15 16 17 18 (19) (20) (21) 22 (23)

Note. Non-invariant parameters between parentheses.

For the model, the percentage of non-invariant parameters is 26.1% for loadings and 37.6% for intercepts, with an average of 31.9% of non-invariant parameters. These values are slightly superior to the cut off criterion of 25% recommended by Muthén and Asparouhov (2014) for trustworthy means, and similar values to those obtained by Cieciuch, Davidov, and Schmidt (2018), where they had an average of 30% of non-invariant parameters.

In order to assess the quality of the alignment results, a Monte Carlo simulation was also carried out, with a correlation of .99 between the generated and estimated factor means, which is in agreement with that recommended by Muthén and Asparouhov (2014), showing trustworthy alignment estimations.

The model fit for the MGCFA method uses a bottom-up strategy for the test (Steenkamp & Baumgartner, 1998). The method starts with the least constrained model, then introduces more constrains to the model (metric and scalar invariance, respectively). To evaluate the model fit for each level of invariance, different fit measures are used. The first two criteria used are standardized root mean square residual (SRMR) and the root mean square error of approximation (RMSEA) measures. SRMR values of .08 or lower (Hu & Bentler, 1999) and RMSEA values of .06 or lower indicate acceptable fit (Chen, 2007). In addition, incremental fit indices, Comparative Fit Index (CFI) and Tucker-Lewis Index (TLI), are used to calculate improvements over competing models. Values higher than .90 for these two indices indicate acceptable model fit (Chen, 2007). An evaluation of these fit measures is combined with the examination of Modification Indices (MI) and Expected Parameter Change (EPC).

Table 4 shows the fit indices for each MGCFA model with different measurement invariance levels. Robust Maximum Likelihood estimator is used and missing values are dealt with using the maximum-likelihood estimation procedure that uses all available information from the variables.

##### Table 4

Fit Measures for Measurement Invariance for Trust in Political Institutions

Model Invariance χ2 df p CFI TLI RMSEA [90% CI] SRMR
a. Metric Full 439.008 44 < .001 .986 .978 .065 [.060, .071] .043
b. Metric Partial 249.750 40 < .001 .993 .987 .050 [.044, .056] .032
c. Scalar Full 5201.500 88 < .001 .820 .859 .166 [.162, .170] .110
d. Scalar Partial 391.282 60 < .001 .988 .987 .051 [.046, .056] .034

Results show that partial metric invariance (Model b in Table 4) provides a good model fit. Thus, factor loadings are comparable across countries. However, not all of them are equal due to the non-invariance of four factor loadings. Thus, relationships for trust in political institutions can be studied and interpreted across all time periods.

Full scalar invariance does not hold for all groups, which can be can be problematic when applied to large-scale and widely diverse cultural groups (Byrne & van de Vijver, 2017) due to socio-political differences, different cultural meanings attributed to policy regulations, or context-related aspects influencing different levels of identification with institutions among individuals in the different countries, etc., not merely due to the number of groups analyzed. This situation makes it difficult to compare results since cross-cultural differences might be involved. In order to obtain partial scalar invariance, 28 parameters are freed. In this case, the most non-invariant item is “Trust in legal system” (15 of 28 modifications, which required modification for intercepts in 13 groups and modification for factors loadings in 2 groups). It is followed by “Trust in Parliament” (7 of 28 modifications, which are intercepts in 6 groups and factors loadings in 1 group), and in this case, the most invariant item is “Trust in politicians” (6 out of 28 modifications, which are intercepts in 4 groups and factors loadings in 2 groups).

“Trust in legal system” is the most non-invariant item according the MGCFA and the one that more contributes to the fit function in the Alignment method, which also means the most non-invariant item. In the other side, “Trust in politicians” is the most invariant item in the MGCFA while “Trust in parliament” is the most invariant for Alignment method. Differences can also be due to the automatic adjustments in Alignment method and the required manual adjustments for the MGCFA.

### Comparative Results: MGCFA and Alignment

Table 5 shows the results of the latent means for trust in political institutions for the 23 groups (except data from Greece in 2012 which is not available), estimated using Alignment and MGCFA methods (Coromina & Bartolomé Peral, 2018, p.66). Alignment and MGCFA scores, based on the partial scalar model, shows the latent means for the different groups. Latent means are not estimated in absolute scores, but instead as an arbitrary ‘adimensional’ factor mean, reflecting average differences in the level of the latent factor across groups. Countries such as Norway, Sweden or the Netherlands have a higher level of trust in institutions, regardless of the year analysed.

##### Table 5

Results for Factor Trust in Political Institutions

Ranking Country Year Alignment score MGCFA score
1 Norway 2012 0.623 1.777
2 Norway 2010 0.507 1.561
3 Sweden 2010 0.545 1.732
4 Sweden 2012 0.375 1.387
5 Norway 2008 0.348 1.251
6 Sweden 2008 0.281 1.205
7 Netherlands 2008 0.202 0.951
8 Netherlands 2012 0.155 0.761
9 Netherlands 2010 0.143 0.815
10 Germany 2012 -0.182 0.217
11 Germany 2008 -0.296 0
12 Germany 2010 -0.431 -0.317
13 Spain 2008 -0.491 -0.242
14 Ireland 2010 -0.576 -0.456
15 Ireland 2008 -0.641 -0.388
16 Ireland 2012 -0.649 -0.55
17 Greece 2008 -0.786 -0.92
18 Spain 2010 -0.797 -1.028
19 Portugal 2008 -1.117 -1.342
20 Spain 2012 -1.236 -2.143
21 Portugal 2010 -1.365 -1.686
22 Portugal 2012 -1.450 -1.988
23 Greece 2010 -1.484 -2.511

Figure 2 summarizes the two methods, and shows the estimated latent means using both Alignment and MGCFA methods (Model d in Table 4). Both methods found that ‘Greece in 2010’ has the lowest level of trust in political institutions, while ‘Norway in 2012’ has the highest level.

Click to enlarge

### Latent Means for Alignment Method and MGCFA

Correlation between MGCFA and Alignment methods is .993. In this regard, Davidov et al., (2015) compared the Bayesian approach (an approximate measurement equivalence) with the exact MGCFA approach. They found that the scores were not comparable across countries, and discussed whether exact measurement equivalence is needed in order to conduct meaningful comparisons. The authors correlated rankings using both methods, and found that the rankings of the latent means were very similar (correlation > .97).

Taking similarities in correlations for political trust in institutions between both methods into account, additional issues must be analysed in order to decide which method is the most suitable. In this case, latent factor means from the Alignment method is used to support the hypotheses about trends in political trust in institutions between countries and over time.

### Trends in Political Trust in Institutions

Results in Figure 3 show the latent means for the Alignment method for the eight countries over time. It shows that Germany, the Netherlands, Sweden and Norway show the highest levels of trust in institutions for the different time periods, whereas Greece, Ireland, Portugal, and Spain show the lowest (Coromina & Bartolomé Peral, 2018, p.67). This suggests that those countries less affected by the economic and financial crisis show higher levels of trust in institutions, supporting H1. These results are consistent with the statement made in the theory (Criado & Herreros, 2007; Friedrichsen & Zahn, 2014) that institutional trust is sensitive to economic performance and the economic context.

Furthermore, trust in political institutions in Norway increases over time (as it does in Sweden between 2008 and 2012), before returning to the level it initially had in 2012. The Netherlands shows a stable trend, and Germany a slight decrease in 2010, with levels of trust increasing sharply in 2012. Thus, results show only partial support to H2a, which states that, the levels of trust in institutions decrease for all countries. Theory claims that for advanced democracies, processes such as citizens’ political sophistication and cognitive mobilization would generally have as a consequence a decline in institutional trust due to increasing expectations (Hetherington, 1998, 2005; Rudolph & Evans, 2005), however the results don’t reflect this claim. A possible explanation might be the positive evaluation of their institutions in a context of low economic performance and economic crisis outside their borders.

Conversely Greece, Spain, and Portugal show a clear decrease in their levels of trust in institutions between 2008 and 2012, and Ireland a slight decrease. These data reflect the fact that the countries most affected by the crisis show lower levels of institutional trust (H1), and that levels of trust continue to decrease as the economic crisis continues in these countries. Thus, these results support H2b, where the countries more affected by the crisis have a dramatic decline in trust in political institutions over time. The theoretical claim (Criado & Herreros, 2007; Friedrichsen & Zahn, 2014) about the link between economic and policy performance and institutional trust appears clearly in these results.

Click to enlarge

## Conclusions

The methodological aim is relevant when deciding which method to use under certain circumstances. In cross-cultural research, non-invariant parameters can exist not merely when the number of groups increases, but also due to the fact that especially across countries is more likely to have socio-political differences or context-specific factors influencing different levels of citizenship identification with institutions. Thus, results obtained must be compared with precaution and researchers need to be aware of such differences.

In this situation, MGCFA can become difficult to manage and more time consuming with many manual adjustments, in this article, 28 adjustments were needed. This situation might lead to the reduction of the number of groups analysed, which would mean that the comparison of factors loadings or latent means could not be possible for all groups of interest. The use of the Alignment method, less strict in the invariance requirements, enables a substantive comparison when more non-invariant parameters exist.

In any case, the decision to take items or groups out of the analysis should not only be due to issues regarding measurement, but also taking substantial or conceptual validity into consideration (Billiet, 2016), since the decision for selecting one item or another with similar measurement characteristics should be substantial. The substantive selection of items is possible in both methods. In the case of MGCFA, once items are selected, manual adjustments can also be done in order to reach partial invariance.

For those reasons, results from in any of these methods must be interpreted with caution, since researchers must be aware of the objective of the comparisons they are carrying out. Thus, these two methods can also be seen as complementary rather than competing methods, and should be used depending on the aims of the study, and the level of comparison the researcher is interested in achieving (Oberski, 2018).

However, in most structural equation modelling (SEM) programs, the Alignment method is not yet implemented, thus this article indicates how to proceed with Mplus 7 SEM program.

The substantive aim of this article is to analyse the differences in trust in political institutions across European countries and their evolution over time, specifically the most difficult years of the economic crisis, between 2008 and 2012. We have provided evidence to show that a decline in trust has taken place in nearly all the countries studied, and was more severe in those countries most affected by the crisis than in those less affected.

The findings lead to several conclusions regarding the hypotheses. Evidence provided in the data confirms the first hypotheses, H1 (Levels of trust in political institutions are expected to be significantly lower in those countries most affected by the economic crisis). From the latent means we can conclude that levels of trust in institutions are different across countries, depending on the degree to which they are affected by the crisis. The latent means clearly shows that the levels of trust in political institutions for Greece, Portugal, Spain and Ireland are lower than levels in Germany, Sweden, Norway or the Netherlands.

The second hypothesis, H2a (It is expected that levels of trust in institutions will tend to decrease over time in all countries during the crisis), is partially confirmed. The cases of Norway (where the trend is ascending), and Germany and Sweden (where the trend is combined), prove that there is no decline in trust in countries less affected by the crisis. Hypothesis H2b (Levels of trust will decline more sharply in those countries most affected by the crisis) is confirmed. A clear pattern shows that institutional trust dramatically decreases over time in those countries most affected by the crisis, particularly Portugal, Spain and Greece. This supports our claim that trust in institutions is being eroded by the economic crisis.

As discussed previously in the article, institutional trust has noticeably declined in western democracies. Our findings suggest that a situation of severe and lasting economic crisis may have serious consequences for trust in political institutions, and if it persists, may put a system’s legitimacy at risk.

## Funding

The authors have no funding to report.

## Competing Interests

The authors have declared that no competing interests exist.

## Acknowledgments

The authors have no support to report.

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text:
Home OALib Journal OALib PrePrints Submit Ranking News My Lib FAQ About Us Follow Us+
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 测绘学报 , 2011, Abstract: 基于灭点理论，提出一种利用像底点检校机载POS系统视准轴误差的新方法。首先从理论上建立了像底点与POS系统视准轴误差之间严格的数学关系，并在此基础上推导出求解视准轴误差的误差方程式，然后用一组带有POS数据的航空影像进行了试验验证。试验结果表明，所建立的利用像底点检校POS系统视准轴误差的模型是正确的，用两张以上像片上的像底点坐标即可检校出POS系统的视准轴误差，而无需布设特定的检校场和地面控制点，对城区大比例尺航空摄影时POS系统视准轴误差检校具有一定的实用价值。
 李凡,徐章艳,饶勇 计算机科学 , 2000, Abstract: In this paper,we first introduce vague set,then give the definition of intersection and union based on vague set with t-norm and t-conorm of the point value. Thus we gain some properties of intersection and union of vague set.
 计算机科学 , 2009, Abstract: The concepts of degree of truth-compatibility, degree of false-compatibility degree of truth-equality, degree of falscequality based on t-norm and t-conorm were introduced, because a vague set has the characteristic of truth-membership function and fase-membership function. Futhermore we presented the concepts of semi-vague partitions and vague partitions by using locically degree of truth-compatibility, degree of false-compatibility degree of truth-equality,degree of falscequality, and we investigated the characters of semi-vague partitions and vague partitions.
 计算机科学 , 2009, Abstract: 根据vague集具有真假隶属度的特点，首先提出了基于t-模和t-余模的真相容度、假相容度、真相等度和假相等度的概念。然后合理地利用真相容度、假相容度、真相等度和假相等度提出了半vague划分和vague划分的概念，并讨论了它们的性质。
 Arsham Borumand Saeid Le Matematiche , 2012, Abstract: By using vague sets we generalize the notion of convex sets and introduce the notion of (α, β ,T )-convex vague sets and study their properties, where T is a triangular norm on [0, 1].
 Mathematics , 2014, Abstract: The purpose of this paper is to construct topology on vague soft sets. The concept of vague soft topology is introduced and its basic properties are given.
 计算机应用研究 , 2010, Abstract: Based on the presented theory of Vague sets and the topological theory of classical sets and Fuzzy sets,this paper spreaded the relative topological theory of classical sets and Fuzzy sets through the method of analysis, presented the basic concepts of Vague topological space and Vague continuous mapping, and discussed some relative properties of them. These results further extended the scope of the research on Vague sets, and proposed the future study aspects in this field.
 Arsham Borumand Saeid Opuscula Mathematica , 2009, Abstract: In this note, by using the concept of vague sets, the notion of vague BCK/BCI-algebra is introduced. And the notions of $\alpha$-cut and vague-cut are introduced and the relationships between these notions and crisp subalgebras are studied.
 计算机科学 , 2008, Abstract: Vague set is a valid tool for processing uncertain information.The similarity measure of two uncertain patterns is important for intelligent reasoning.It is also a key problem to measure the similarity of vague values or vague sets in vague information processing systems.Many methods for similarity measure of vague sets have been proposed in recent years.However,these methods could not precisely describe the essences of the similarity between two vague sets.In this paper,some evaluation criterions for simil...
 Shokoofeh Ghorbani Journal of Discrete Mathematics , 2014, DOI: 10.1155/2014/120342 Abstract: Notions of vague filters, subpositive implicative vague filters, and Boolean vague filters of a residuated lattice are introduced and some related properties are investigated. The characterizations of (subpositive implicative, Boolean) vague filters is obtained. We prove that the set of all vague filters of a residuated lattice forms a complete lattice and we find its distributive sublattices. The relation among subpositive implicative vague filters and Boolean vague filters are obtained and it is proved that subpositive implicative vague filters are equivalent to Boolean vague filters. 1. Introduction In the classical set, there are only two possibilities for any elements: in or not in the set. Hence the values of elements in a set are only one of and . Therefore, this theory cannot handle the data with ambiguity and uncertainty. Zadeh introduced fuzzy set theory in 1965 [1] to handle such ambiguity and uncertainty by generalizing the notion of membership in a set. In a fuzzy set each element is associated with a point-value selected from the unit interval , which is termed the grade of membership in the set. This membership degree contains the evidences for both supporting and opposing . A number of generalizations of Zadeh’s fuzzy set theory are intuitionistic fuzzy theory, L-fuzzy theory, and vague theory. Gau and Buehrer proposed the concept of vague set in 1993 [2], by replacing the value of an element in a set with a subinterval of . Namely, a true membership function and a false-membership function are used to describe the boundaries of membership degree. These two boundaries form a subinterval of . The vague set theory improves description of the objective real world, becoming a promising tool to deal with inexact, uncertain, or vague knowledge. Many researchers have applied this theory to many situations, such as fuzzy control, decision-making, knowledge discovery, and fault diagnosis. Recently in [3], Jun and Park introduced the notion of vague ideal in pseudo MV-algebras and Broumand Saeid [4] introduced the notion of vague BCK/BCI-algebras. The concept of residuated lattices was introduced by Ward and Dilworth [5] as a generalization of the structure of the set of ideals of a ring. These algebras are a common structure among algebras associated with logical systems (see [6–9]). The residuated lattices have interesting algebraic and logical properties. The main example of residuated lattices related to logic is and BL-algebras. A basic logic algebra (BL-algebra for short) is an important class of logical algebras introduced by Hajek [10] in
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# When to use factoring by grouping for quadratic equation

There are several ways/methods to perform factoring. I am revising factoring at KhanAcademy, there are factoring by grouping, factoring special product and factoring difference of squares.

Although, I can work on the exercise, but I do not really understand when to apply factoring by grouping.

Suppose your quadratic factors as $$(ax+b)(cx+d)$$ Then your quadratic was $$acx^2+(ad+bc)x+bd$$ This can be factored by grouping if you can hit upon the correct way of splitting the coefficient of $x$ into two pieces, $$acx^2+(ad+bc)x+bd=(acx^2+adx)+(bcx+bd)=ax(cx+d)+b(cx+d)=(ax+b)(cx+d)$$ In other words, any quadratic that factors can, in principle, be factored by grouping, but you have to be clever/lucky to find the right grouping. E.g., given $$210x^2+421x+210$$ there are a lot of way to split $421$ into a sum of two numbers; how do you know that the way that will work is $421=225+196$? $$210x^2+421x+210=210x^2+225x+196x+210=15x(14x+15)+14(14x+15)=(15x+14)(14x+15)$$

• In this case, I would say applying quadratic formula is easier to solve all quadratic equations? – ilovetolearn Aug 17 '14 at 3:54
• I would say that if you notice the factorization, it's easier to do it that way, but if a factorization doesn't jump out at you, go for the quadratic formula. – Gerry Myerson Aug 17 '14 at 9:59

If a quadratic has the form $$ax^2+bx+c$$ then I would look at the discriminant of the quadratic formula (i.e. $b^2-4ac$) to see if it is a perfect square. If it is, then I would try to factor by grouping (or difference of squares if $b=0$). If not, then I would use the quadratic formula. By following these steps, your teacher will receive positive feedback that the brainwashing is, in fact, working.

In the real world though, we ALWAYS use the quadratic formula. Factoring by grouping is almost always completely useless in any kind of an experimental or scientific scenario. We would be very lucky, indeed, to encounter coefficients of a quadratic that would allow a grouping solution. In class we are given contrived problems like

A garden that is 4 meters wide and 6 meters long is to have a uniform border such that the area of the border is the same as the area of the garden. Find the width of the border.

to make it seem to student as though grouping may have some practical application.

If grouping is so useless, then why bother teaching it? Is it a union conspiracy to maximize the number of employed teachers? I think not. The reason why we teach factoring by grouping is to give students at least some exposure in high school to Diophantine Equations. As a continuation of Least Common Multiple, Greatest Common Divisor, Prime Factorization, and sum of digits analysis, it represents the final topic in Number Theory that students will be exposed to prior to University. As such, it is an important part of their education. I would have appreciated it, though, had my math teacher been more upfront about it.

• There are other things to do with quadratics, besides setting them equal to zero and solving for $x$. Factoring is useful because it can lead to cancellation and other simplifications, when the quadratic is part of some larger expression. – Gerry Myerson Aug 23 '14 at 14:16
• +1 good point Gerry. Perhaps I omitted others. – John Joy Aug 23 '14 at 15:55

Short answer: factoring by grouping IS factoring. All those other factoring techniques are just special cases of factoring by grouping, and once you understand how to do factoring by grouping you automatically can derive all of the special cases.

In general if you want to find the roots of a quadratic equation, this is the approach you should use:

1. Look for special factorizations of the form $x^2 \pm 2a + a^2$ or $x^2 - a^2$. If you find one of these, you're already done.
2. If the equation looks simple (or your homework assignment says you have to), use factoring by grouping.
3. Otherwise, use the quadratic formula.

What does it mean to have a "simple" equation? Basically, a quadratic equation $ax^2 + bx + c$ where the numbers $a$,$b$,$c$ are integers, or in trickier cases, simpler fractions or even radicals. $x^2 + 5x + 6$, $x^2 + \frac{3}{4}x + \frac{1}{8}$ and $x^2 + 2\sqrt{3}x + 3$ would qualify. (Actually, that last one is a special factorization, $(x+\sqrt{3})^2$.) Once you get more experience, you'll be able to see at once whether an equation is "simple" and thus worth factoring.

Some quadratics look easy but their roots are near impossible to find without the quadratic formula, for instance $x^2 + 5x + 7$. With other equations as well as equations with long decimals like $1.618x^2 + 5.1234232x + 4.329$ (the stuff that comes up more in the "real world"), it's not even worth trying. Just use the quadratic formula.

Factoring by grouping isn't restricted to quadratic polynomials. Although it becomes much more difficult, you can factor polynomials of any degree using this method.


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2 More clarification.

I will address your second question: "one has to prove that the classification diagram functor is sent under this Quillen equivalence to something weakly equivalent to the coherent nerve".

The answer is yes, this is true. First note that the simplicial category $M^\circ$ is equivalent as a simplicial category to $L^HM$, the hammock localization of M (at the weak equivalences). See here for references. So the answer to your second question (or is it a remark?) follows as a special case of this previous MO answer (which cites work of Barwick-Kan and Toen). So the classification diagram functor and the coherent nerve of $M^\circ$ are sent to equivalent things under the Quillen equivalence between quasicategories and complete Segal spaces (I prefer the term Rezk categories or Rezk $\infty$-categories).

I believe that combined with your other comments (notably using Julie Bergener's results) this is enough to answer your main question in the affirmative.

1

I will address your second question: "one has to prove that the classification diagram functor is sent under this Quillen equivalence to something weakly equivalent to the coherent nerve".

The answer is yes, this is true. First note that the simplicial category $M^\circ$ is equivalent as a simplicial category to $L^HM$, the hammock localization of M (at the weak equivalences). See here for references. So the answer to your question follows as a special case of this previous MO answer (which cites work of Barwick-Kan and Toen).


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# Registration Problem in Functional Data Analysis

##### Posted on Jan 21, 2020 (Update: Jan 29, 2020)

This post is based on the seminar, Data Acquisition, Registration and Modelling for Multi-dimensional Functional Data, given by Prof. Shi.

He illustrated with a research which is aimed at modeling patients’ recovery level after stroke via analyzing the trajectories of several bones, such as Hyoid bone, Epiglottis, head of bolus, during swallowing and other information.

The procedure of such research programme can be divided into three parts:

1. Data acquisition & preprocessing. Developed an all-in-one platform for the data acquisition, calibration, smoothing and segmentation.
2. Data registration. Introduced the GPSM method to carry out multi-dimensional functional data registration.
3. Modeling. Proposed JRC and SRC methods for simultaneous registration and classification & clustering for multi-dimensional functional data.

## Data registration using GPSM

### Landmark Registration

A point $x$ is a landmark in its neighborhood $\Omega$ if

$(\Vert D(x)\Vert > D_\mu + \lambda D_\sigma) \cap (\Vert D(x)\Vert \ge \Vert D(x')\Vert)$

for any $x’\in \Omega$.

To register query curve $w_i$ onto $w_j$ (reference curve), we need to find a warping function $h_i(t)$ such that

$w_i^*(t) = w_i[h_i(t)]\;\text{and }\; w_i^*(t_{j, landmark}) \approx w_j(t_{j,landmark})$

Warping functions usually satisfy the following conditions:

• Strict monotonicity. $h(t_1)\le h(t_2)$ for any $t_1 < t_2$
• Boundary condition. $h(0) = 0$ and $h(T)=T$
• Continuity. $h(t)$ is conditions.

### Registration: (robust) procrustes method

what is the reference curve and the query curve

## SRC

How to initialize the classification? From the training? Confused about the label updating in the joint procedure?

## Conclusion

Published in categories Seminar


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# Tag Info

## Hot answers tagged pbkdf-2

19

No, you do not need to do escaping or sanitization on data that you pass in as the user input to these functions. They accept arbitrary byte sequences, so any arbitrary byte sequence you pass is acceptable, and there should be no security risks as a consequence of it. In general, cryptographic algorithms operate on arbitrary byte sequences (possibly of ...

18

This is called Client-Independent Update, according to the Catena paper. It is desirable to be able to compute a new password hash (with some higher security parameter) from the old one (with the old and weaker security parameter), without having to involve user interaction, i.e., without having to know the password. We call this feature a client-...

17

SHA-512 is a cryptographically secure hash, PBKDF2 is what we call a Password Based Key Derivation Function. If the resulting secret isn't used as key but as hash value it's also called a password hash. Password hashes differ from secure hashes in the sense that they contain a salt and a work factor / iteration count. Both cryptographic hashes and password ...

14

Password hashes need first pre-image resistance and should not cause many collisions among typical passwords (preserve the entropy). This collision "attack" violates neither requirement and causes no practical security issues. While this issue can find trivial collisions, they're not between commonly chosen passwords. A SHA-1 hash (and thus the shorter of ...

13

Yes, it is. PBKDF2 derives a DK, a "derived key", which is indistinguishable from random. This is mainly because function within PBKDF2 is HMAC, and HMAC is a PRF. Let's see the definition from Wikipedia: In cryptography, a pseudorandom function family, abbreviated PRF, is a collection of efficiently-computable functions which emulate a random oracle in ...

13

Assume you have an IND-CCA secure cryptosystem $E$ that runs a password through a slow KDF and implicitly handles salts and random IVs, a human-chosen password $p$, and messages $m_1$ through $m_n$ to encrypt. Is $E_p(m_1+m_2+\cdots+m_n)$ or $E_p(m_1)+E_p(m_2)+\cdots+E_p(m_n)$ better for this? Each invocation of $E$ is slow due to it running a KDF on $p$, ...

12

To paraphrase my answer to an earlier question, PBKDF2 is a generic high-level algorithm that internally calls a pseudorandom function (PRF) to process its input. The PBKDF2 spec does not mandate any particular PRF, so implementors are free to choose any PRF they want (as long as it meets the definition of a secure PRF, and can accept the input PBKDF2 gives ...

11

Use gpg --s2k-mode 3 --s2k-count N, where N is the number of iterations you want to use. The manual page says the default is 65536, and you can use any number between 1024 and 65011712. If you like to tweak the defaults, I suggest making this number as large as you can bear it, without introducing noticeable slowdown (e.g., ideal would be to make the ...

10

So in general, isn't this equivalent to what Bcrypt and PBKDF2 do in terms of password storage security? PBKDF2, yes, pretty much. The only real difference is that salt/password are used the other way around, with the password mixed in at every step. Bcrypt, however, is different. In your case an attacker only needs a small amount of memory compared to ...

10

TL;DR: You want to use Argon2d here. Even though Argon2 was standardized only somewhat recently, it is the result of the Password-Hashing Competition (2013-2015) and was a late re-design of Argon which also picked up ideas from a few other finalists. Since then there have been attacks on it, which caused the scheme to be tweaked to counter them better, this ...

9

That particular usage of SHA-1 uses HMAC, and then iterates that as part of PBKDF2 (which is actually defined for any PRF, not just HMAC-SHA1). As of this date (2017-05-18) HMAC-SHA1 is unbroken in terms of collisions and other attacks, so PBKDF2-HMAC-SHA1 is still considered safe. The HMAC construction, along with the many iterations in PBKDF2, protects ...

8

PBKDF2 uses the password as the PRF key. From the RFC: The first argument to the pseudorandom function PRF serves as HMAC's "key," and the second serves as HMAC's "text." In the case of PBKDF2, the "key" is thus the password and the "text" is the salt.

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PBKDF2 (as defined by RFC 2898) is a function of the form $$DK = \text{PBKDF2}(\text{PRF}, Password, Salt, c, dkLen)$$ In most practical use cases, the $\text{PRF}$ is $\text{HMAC}$ instantiated with a Merkle-Damgård hash function such as $\text{SHA-1}$. The time to compute $\text{PBKDF2}$ is roughly linear with the iteration parameter $c$, all other ...

8

I think you are overstating the complexity of PBKDF2, and also, not matching it feature-wise with your alternative. Let's dispatch the latter first: as gammatester's comment mentioned, PBKDF2 supports variable output size. If you built that into your proposal, it would become more complex. Once we control for that, PBKDF2 is hardly more complex than your ...

8

You're right. Unless the software is doing something very peculiar, which should be very apparent when reading the code and documentation, and I can't imagine what it could be, the salt generation and the application of PBKDF2 are unnecessary. If you have 32 bytes from a cryptographically secure random generator, you can use these 32 bytes as an AES-256 key....

7

Hash algorithm strength is important, but it is not so important in key derivation functions. It is unlikely that even if SHA-1 is broken that it would influence the security of PBKDF2. You are better off using SHA-1, and increase the iteration count up to a level that is tweaked for your specific configuration. If you must, you could use Bouncy Castle to ...

7

So, here's my question: is there a point where the salt size doesn't matter anymore in terms of security and where it might even decrease it? The purpose of a salt is to prevent the attacker from targeting multiple users' passwords with the same try or caching common passwords' hashes in a table. You need enough salts that each user has a unique salt. After ...

7

Firstly, How much time will it take to crack PBKDF2 while using a 9 character password? and how do I calculate the cost? I'm not specifying any specific system or platform. If a brute force attack is made using the best ever super computer around how much time will it take to crack it? Unless the underlying PRF is broken, brute force and dictionary attacks ...

7

There's no practical limit to password or salt length you can use with PBKDF2. Theoretical limit, however, is determined by the hash function used by PBKDF2: under the hood, it uses $HMAC(Password, Salt || Counter)$, which in turn will translates to a series of hash function calls: $K_0 = H(Password)$ (assuming password is longer than hash block) $HMAC =... 7 TL;DR: Yes, you can use PBKDF2 as a stream cipher. However, you should not use it for that and for its intended purpose (i.e. password-based key derivation) at the same time. Instead, if you need to do both, call it twice. PBKDF2 is a password-based key derivation function — or, rather, a scheme for constructing such a function out of a variable-key-... 7 These hash functions are also used for the key derivation function. Both are roughly equally secure for a KDF in the sense of collision resistance etc., but SHA512 offers a bit more security as countermeasure for custom hardware attacks, since the memory requirements are a higher than for SHA256 and custom hardware to crack the passwords like FPGAs is more ... 7 All the parameters you've mentioned can be public. Furthermore, only the salt, IV and ciphertext cannot simply be guessed. The IV is generally easy to retrieve once the key is known, so the making the IV secret doesn't make sense. In CBC mode and many other modes you may just not get the first plaintext block if the IV is completely absent, but the rest of ... 7 There's no point in using either an ASIC or a GPU to calculate a single password hash. That's true whether you use PBKDF2 or scrypt or Argon2 or whatever. What massively parallel devices like GPUs or ASICs are good for is hashing millions or billions of passwords at the same time. That's useful if you're mining a cryptocurrency or trying to crack a hash ... 6 In a scenario such as yours, where there is only one password/passphrase, but it is used as key material for the encryption of multiple CBC encrypted files, you will (as you noted yourself) obviously not make it any harder for an attacker to compute your password, should you use a salt. However, using a salt would mean that the encryption of each file is ... 6 Of course you can - but as to whether or not it's a practical or advisable idea, I don't think so. It's not really prudent to implement crypto systems/protocols and assume that they'll be fine in 10 years. Cryptography is a dynamic field that changes rapidly; algorithms get broken, hardware improves, governments try to undermine the field, and attacks only ... 6 You have two algorithms,$A$and$B$, that claim to compute two (essentially) injective, hard-to-invert and costly-to-compute functions$F$and$G$. Your fear is that either of the algorithms instead compute functions$F'$and$G'\$, that may be neither injective, hard-to-invert, nor costly-to-compute. This may happen because of programming mistakes or (these ...

6

The threat model of password storage is that of server compromision, where the attacker gain access to the database and server code. The attacker can then run the code to test password candidates, possibly making modifications, porting to faster platform, etc. The attacker will not bother computing the fake hash and fake salt. So this scheme is twice as ...

6

Is there anything wrong with swapping SHA2 for Keccak for use in PBKDF2? There are 2 main issues. Performance and implementation. Performance The performance issue is from an attacker-defender scenario, where the attacker is able to provide FPGA or ASIC resources. In this scenario, Keccak is several times faster than SHA-2. I will use a Blake comparison ...

6

No, you should not use a password directly as an HMAC key. However, it is fine to use HMAC as part of a key derivation function, which generates keys from a password. However, do not mistake the output as naturally having higher entropy than whatever you put in. "password" has essentially 0-bits of entropy, and running it through a KDF will not magically ...

6

The only downside to SHA-2 over SHA-1 is that SHA-2 is a little slower. In this case, that is of no practical consequence whatsoever because you're trying to make something slow to compute, and it is only a matter of raising the number of iterations to make it cost a certain number of bit operations in serial. However, please consider using the modern ...

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text:
The Jaccard overlap score (or Jaccard index) between two sets $A$ and $B$ is the size of their intersection divided by the size of their union. In image segmentation, the set is usually represented by a discrete indicator, i.e., a boolean-valued matrix where one indicates the object (foreground) and zero indicates non-object (background). In this case, when $A,B \in \{0,1\}^{m \times n}$, the Jaccard overlap is

$J(A,B) = \frac{\sum_{ij} a_{ij} \land b_{ij}}{\sum_{ij} a_{ij} \lor b_{ij}}$

where $\land$ and $\lor$ denote “logical and” and “logical or”, respectively. This overlap score is between zero and one, where $J=0$ indicates no overlap between $A$ and $B$ and $J=1$ indicates perfect overlap. Usually, one of the matrices is considered to be the proposed, or approximate, segmentation, while the other is considered to be the exact, or “ground truth”, segmentation of some object. So, the score $J(A,B)$ gives a measure of the quality of the proposed segmentation against the ground truth segmentation (the former usually provided by an algorithm and the latter usually provided by an expert).

The Jaccard overlap score is ubiquitous because it is simple to understand and implement, and provides a reasonable way to measure segmentation quality. Further, it continues to be popular because it is popular (i.e., it provides a standard metric to compare different methods in literature). Here, I show a particular aspect of the score: it is sensitive to the size of the ground truth segmentation. Let me make this precise.

Let $A,B \in \{0,1\}^{m \times n}$, and consider $B$ to be the “ground truth” segmentation. Now, let’s suppose that matrix $A$ is formed by removing $p$ points from $B$ and by adding $q$ points to $A$ that are not in $B$. More precisely,

1. Set $A=B$.
2. Choose randomly a subset $P \subset \{ (i,j) \, | \, b_{ij}=1 \}$ where $|P|=p$. Set $a_{ij}=0$ for all $(i,j)$ in $P$.
3. Choose randomly a subset $Q \subset \{ (i,j) \, | \, b_{ij}=0 \}$ where $|Q|=q$. Set $a_{ij}=1$ for all $(i,j)$ in $Q$.

It’s not too hard to see that this covers all proposed segmentations, $A$. Now, by 2., we have $|A \cap B| = |B| - p$, and by 3., we have $|A \cup B| = |B| + q$. This gives

\begin{align*} J(A,B) &= \frac{|B| - p}{|B| + q} \\ &= 1 - \frac{p+q}{|B| + q} \end{align*}

Note that when $p=q=0$ (i.e., $A=B$) we have that $J=1$ as expected. Otherwise, we miss the perfect score of 1 by a factor of $\frac{p+q}{|B| + q}$. This factor is increasing in both $p$ and $q$, if everything else is held fixed, which is expected: if we increase the number mismatched points ($p$ or $q$), we increase the factor $\frac{p+q}{|B| + q}$, and therefore decrease the overlap $J(A,B)$.

However, notice that the factor $\frac{p+q}{|B| + q}$ also depends on $|B|$, which is the size of the ground truth segmentation. Notice further that $\frac{p+q}{|B| + q}$ is decreasing in $|B|$, which means that for two segmentations where $|B_1| < |B_2|$ with $p$ and $q$ held fixed, then $\frac{p+q}{|B_1| + q} > \frac{p+q}{|B_2| + q}$.

In other words, even though $A_1$ and $A_2$ have the same number of pixel-wise errors ($p+q$), if $B_1$ is smaller than $B_2$, then we will have $J(A_1, B_1) < J(A_2, B_2)$.

This is something to be aware of while using this score. If you are segmenting objects over a large range of sizes, you might observe higher variability in the Jaccard overlap score for smaller objects because as shown above, it takes less pixel-wise errors to produce larger changes in overlap score for smaller objects.

#### Extras: Foreground vs background disagreements and the effect on the Jaccard score

Here’s something additional to ponder. Given the choice, which is better in terms of the Jaccard overlap score: (a) setting a point in $A$ which is zero in $A$ and one in $B$ to one (i.e., reducing $p$ to $p-1$), or (b) setting a point in $A$ that is one in $A$ and zero in $B$ to zero (i.e., reducing $q$ to $q-1$)?

The answer is (a), because the inequality,

$\frac{|B| - (p-1)}{|B| + q} > \frac{|B| - p}{|B| + (q-1)}$

follows from

1. $p+q > 1$ (which is true under the assumption that $p,q \geq 1$)
2. $|B| \geq p$ (which is true because $p$ can be at most $|B|$)
3. $|B|+q > 1$ (which is true if $|B| \geq 1$)

This means if there’s a point where $a_{ij}=0$ and $b_{ij}=1$ and if there is separate point where $a_{st}=1$ and $b_{st}=0$, then we produce a larger increase in the Jaccard overlap by “flipping the bit” where $a_{ij}$ to 1 rather than setting $a_{st}$ to 0.

This is a somewhat of a surprising result. For example, consider the situation where $a_{ij}=1$ everywhere except $a_{st}=0$ at a single point where also $b_{st}=1$. The above says that we produce a larger increase in Jaccard overlap by setting $a_{st}=1$, rather than by switching any of the places in $A$ that equal to one to a zero, despite the fact that there are probably many more places where zeros must be switched to ones.

How much do we have to reduce $q$ to catch up with the overlap increase that results from reducing $p$ by one? If you fiddle with the inequalities, you find that we must reduce $q$ by at least $\frac{|B|+q}{|B|-(p-1)}$ (which is greater than 1), in order to produce the same increase in Jaccard overlap score as reducing $p$ by one.

For example, suppose the image is $101^2$ pixels, and $B$ indicates a square of size $21^2$ pixels. Consider the situation above where $A$ is ones everywhere except in a single place where $B$ is zero. In this case, $|B| = 21^2$, $q = 101^2 - 21^2$, and $p=1$. Then, we have to reduce $q$ by at least 24 (i.e., set 24 of the bits in $A$ to zero where $B$ is also zero) in order to produce the same increase in overlap score that results from simply from reducing $p$ to zero (i.e., setting $A$ to one in the only place it is zero).

The situation is less dramatic in a more realistic situation. For example, instead of $A$ being nearly all ones, suppose it is a square of ones centered on $B$ that is a single pixel too large, i.e., a square of ones size $23^2$. Suppose a single pixel on the interior of $A$ is zero when it should be one. Now, in this case, $|B|=21^2$, $p=1$, but now $q = 4\cdot 21 + 4$. So then $\frac{|B|+q}{|B|-(p-1)} \approx 1.2$, which says that we have to reduce $q$ by at least 2 to see the same increase that reducing $p$ by one produces.