prl: add script to compute probabilities

.github/workflows/artifacts.yml

@@ -71,6 +71,11 @@ jobs:
     steps:
     -
       uses: actions/checkout@master
+    -
+      name: Probabilities
+      run: |
+        cd prl
+        make probabilities
     -
       name: Transition graph
       run: |

README.md

@@ -320,6 +320,7 @@ cd ..
 
 | Artifact              | Paper references    | Description                            |
 |-----------------------|---------------------|----------------------------------------|
+| `probabilities`       | Sect. VII-B, Appendix D | Probabilities used for each scenario, and expected number of revocations |
 | `p-plot`              | Sect. VII-B, Fig. 6 | Plots percentiles for maximum PRL sizes under different scenarios and shares of attackers, with fixed `T_prl` and number of pseudonyms |
 | `tv-distribution`     | Sect. VII-B, Fig. 7 | Evaluates `T_eff`, heartbeat frequency, heartbeat size, and required bandwidth under different values for `T_v` |
 | `tikz-graph`          | Appendix D, Fig. 15 | Simple transition graph |
@@ -333,43 +334,49 @@ depending on your hardware.
 # go to the `prl` folder
 cd prl
 
-# Step 1: transition graph (~1-5 seconds)
+# Step 1: compute probabilities (~1-5 seconds)
+#     Expected output:
+#     - probabilities and expected number of revocations printed to standard output
+#     - "All done!" message
+make probabilities
+
+# Step 2: transition graph (~1-5 seconds)
 #     Expected output:
 #     - markov matrix and the "Done" message printed to standard output
 #     - `tikz-graph.tex` plot under `./plots`
 make tikz
 
-# Step 2: Plot series over the different probabilities (~15-20 minutes)
+# Step 3: Plot series over the different probabilities (~15-20 minutes)
 #     Expected output: 
 #     - no errors printed to standard output
 #     - several distributions cached in `./cached` and plotted in `./plots/distributions`
 #     - plots `p-plot_n800_e30.{tex,png}` under `./plots`
 make p-plot
 
-# Step 3: Plot series over the number of pseudonyms (~25-35 minutes)
+# Step 4: Plot series over the number of pseudonyms (~25-35 minutes)
 #     Expected output:
 #     - no errors printed to standard output
 #     - several distributions cached in `./cached` and plotted in `./plots/distributions`
 #     - plots `n-plot_e30.{tex,png}` under `./plots`
 make n-plot
 
-# Step 4: Plot series over the time each pseudonym stays in the list (~90-140 minutes)
+# Step 5: Plot series over the time each pseudonym stays in the list (~90-140 minutes)
 #     Expected output:
 #     - no errors printed to standard output
 #     - several distributions cached in `./cached` and plotted in `./plots/distributions`
 #     - plots `t-plot_n800.{tex,png}` under `./plots`
 make t-plot
 
-# Step 5: Generate distribution for Tv (~15-20 minutes)
+# Step 6: Generate distribution for Tv (~15-20 minutes)
 #     Expected output:
 #     - no errors printed to standard output
 #     - several distributions cached in `./cached` and plotted in `./plots/distributions`
 #     - plots `tv-distribution.{tex,png}` under `./plots`
 make tv-distribution
 
-# Step 6 (optional): Copy results somewhere safe (`plots` folder will be deleted in the next step!)
+# Step 7 (optional): Copy results somewhere safe (`plots` folder will be deleted in the next step!)
 
-# Step 7: clean up
+# Step 8: clean up
 make clean
 
 # go back to the root folder

prl/Makefile

@@ -18,6 +18,9 @@ build-docker:
 test: init_dirs
 	$(DOCKER_CMD) python3 main.py -n 3 -p 0.01 -e 2 --cache-dir=$(CACHE_DIR) --allow-cached
 
+probabilities:
+	$(DOCKER_CMD) python3 probabilities.py
+
 single: init_dirs
 	$(DOCKER_CMD) python3 main.py -n 800 -p 0.000000116323325 -e 30 --cache-dir=$(CACHE_DIR) --allow-cached
 
@@ -52,7 +55,7 @@ check:
 	fi ; \
 	fi
 
-all: init_dirs check p-plot t-plot n-plot tv-distribution tikz
+all: init_dirs check probabilities p-plot t-plot n-plot tv-distribution tikz
 	@echo Successfully created all graphs!
 
 init_dirs:

prl/README.md

@@ -41,6 +41,7 @@ you have already generated, the faster the scripts will complete.
 | Target           | Time |
 | :--------------- | :--  |
 | test             | 1 second    |
+| probabilities    | 1 second    |
 | tikz             | 1 second    |
 | single           | 3 minutes   |
 | tv-distribution  | 18 minutes  |
@@ -161,8 +162,12 @@ below for how to do this manually.
 
 ### Reproducing the data used in the first plot
 
-The appendix explains more in-depth how the probabilities were calculated.
-For the plots, the following probabilities were used:
+The appendix explains more in-depth how the probabilities were calculated. The
+probabilities that we used can be re-computed by running the `probabilities.py`
+script (via `make probabilities`). This will also print the number of expected
+revocations for each scenario, as discussed in Section VII-B. The script should
+give as output the following probabilities:
+
 ```text
 Scenario 1: vehicles get revoked at least once a day with 1% probability
 0.000000116323325
@@ -181,7 +186,7 @@ Scenario 2: vehicles get revoked at least once a day with 99% probability
 0.000196612735153
 ```
 
-Then multiple plots can be generated at the same time with the
+Then, multiple plots can be generated at the same time with the
 `generate_plots.py` script (careful, only with python up to 3.8), which is
 wrapped in the [p-plot.sh](./scripts/p-plot.sh) script.
 

prl/probabilities.py

@@ -0,0 +1,84 @@
+import numpy as np
+
+### Parameters used to compute the average number of revocations ###
+AVG_PRL_SIZE = 5 # average PRL size for Scenario 2 - 20% attackers -- computed from Markov analysis
+T_PRL = 30 # time in PRL considered
+PSEUDONYMS = 800 # number of pseudonyms considered
+
+def print_scenario(scenario, attackers):
+    assert len(scenario) == len(attackers)
+
+    for i in range(len(scenario)):
+        print(f"{int(attackers[i] * 100)}% attackers: {scenario[i]:.15f} ")
+
+# Vehicle classes: used to compute probabilities for Scenario 1 and 2
+vehicle_classes = {
+    "honest_1": {
+        "p": 0.01, # 1% probability to get revoked within the specified duration
+        "duration": 86400 # a week
+    },
+    "honest_2": {
+        "p": 0.99, # 99% probability to get revoked within the specified duration
+        "duration": 86400 # a week
+    },
+    "malicious": {
+        "p": 0.75, # 75% probability to get revoked within the specified duration
+        "duration": 1800 # 30 min
+    },
+}
+
+# Shares of attackers in the network: from 0% to 20%
+attackers = np.array([0.0, 0.01, 0.02, 0.05, 0.1, 0.2])
+
+print(f"Vehicle classes considered: {vehicle_classes}")
+print(f"Shares of attackers considered: {attackers}")
+
+print("Computing per-second revocation probabilities for each vehicle class and share of attackers..")
+probabilities = {}
+for vc in vehicle_classes:
+    # computing the per-second probability via geometric series (See Appendix D-D)
+    p =  1 - (1 - vehicle_classes[vc]["p"]) ** (1 / vehicle_classes[vc]["duration"])
+
+    probabilities[vc] = {
+        "per-second p": p,
+        "attackers": []
+    }
+    for attacker in attackers:
+        if "honest" in vc:
+            attacker_p = (1 - attacker) * p
+        else:
+            attacker_p = attacker * p
+
+        probabilities[vc]["attackers"].append(attacker_p)
+
+
+print("Computing total (honest + malicious) probabilities for each scenario..")
+
+# h1 <-> Scenario 1: 
+# honest vehicles get revoked at least once a day with 1% probability
+# malicious vehicles get revoked every 30 minutes with 75% probabiity
+h1 = np.add(probabilities["honest_1"]["attackers"], probabilities["malicious"]["attackers"])
+print(f"Probabilities Scenario 1")
+print_scenario(h1, attackers)
+
+# h2 <-> Scenario 2: 
+# honest vehicles get revoked at least once a day with 99% probability
+# malicious vehicles get revoked every 30 minutes with 75% probabiity
+h2 = np.add(probabilities["honest_2"]["attackers"], probabilities["malicious"]["attackers"])
+print(f"Probabilities Scenario 2")
+print_scenario(h2, attackers)
+
+print(f"Computing expected number of revocations over a day for n={PSEUDONYMS} and T_prl={T_PRL}..")
+
+# From the paper: E_rev = n * s * p * (1 - p_prl)
+p_prl = AVG_PRL_SIZE / PSEUDONYMS
+rev_h1 = PSEUDONYMS * 86400 * h1 * (1 - p_prl)
+rev_h2 = PSEUDONYMS * 86400 * h2 * (1 - p_prl)
+
+print(f"Expected revocations over a day Scenario 1")
+print_scenario(rev_h1, attackers)
+
+print(f"Expected revocations over a day Scenario 2")
+print_scenario(rev_h2, attackers)
+
+print("All done!")