Question
This is a project in which a pseudo-random noise sequence is used to encrypt a one-word message. In order to decrypt the message, a probabilistic approach is used in which advantage is made of the fact that binary-coding, using ASCII, creates a situation in which letters of the alphabet have 8-bit ASCII codes in which the first bit is a 0. For this problem, a 7 letter English-language codeword is first encrypted using a substitution cipher in which each letter is shifted by a fixed number of places in the alphabet. (For example, applying this algorithm to the word "dog" would yield "eph" if each letter was shifted by 1.) For the problem here, the shift can be anything between 1 and 25, with the same shift applied to all letters in the codeword. After this, the encrypted version of the seven letter word is changed to binary using ASCII encoding. It is then further encrypted by being XOR'd, at the binary level, by a pseudo-random sequence (PRN) created using a 4th order linear recursive sequence. This encryption yields, in hexadecimal, the following sequence: FA41086EF9153F.

1. Your first task is to show that the XOR process, when applied to binary bits, yields an output bit stream in which the probability of a 1 is equal to .5, as is the probability of a 0, if either of the inputs is random (i.e., pl = p0 = 0.5). 2.
2. Your second task is to use the fourth order recursion provided below to generate a pseudo-random sequence that repeats every 24 = 15 bits. Sequences like this are used in spread-spectrum systems such as GPS. Use the recursion x sub 4 = X sub 1 XOR x sub 0, x sub 5 = x sub 2 XOR x sub 1, etc, for the bit stream x0 x1x2x3x4x5etc..
3. Decode the message to recover: the original codeword, the number of alphabet letters by which the original codeword was shifted before the ASCII message was encrypted by the pseudo-random noise sequence the recursion, starting with the correct choice of initial bits
4. Explain how you used probability models to determine the initial four bits of the recursion, and to determine the number of letter positions by which the original codeword was shifted before its binary encryption. Hint: if the initial four bits of the recursion are 1111, the next seven bits are 0001001; ASCII characters are represented from most significant bit (msb) to Isb, left to right.
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Task 1)

When XORing two binary bits, there are 4 possibilities based on the random input bits, as follows:

0 XOR 0 =0
1 XOR 1 =0
0 XOR 1 =1
1 XOR 0 = 1

It can be noticed the probability of obtaining a 0 is equal with 2/4= 0.5.

It can also be noticed the probability of obtaining a 1 in the output stream is equal with 2/4= 0.5.

In conclusion, the probability of a 1 or of a 0 in bistream equals 0.5, as required.
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      • Illuminate\Cache\Events\CacheHit (112μs)
      • Illuminate\Cache\Events\CacheHit (115μs)
      • Illuminate\Cache\Events\CacheHit (136μs)
      • Illuminate\Cache\Events\CacheHit (104μs)
      • Illuminate\Cache\Events\CacheHit (118μs)
      • Illuminate\Cache\Events\CacheHit (96μs)
      • Illuminate\Cache\Events\CacheHit (109μs)
      • Illuminate\Cache\Events\CacheHit (96μs)
      • Illuminate\Cache\Events\CacheHit (108μs)
      • Illuminate\Cache\Events\CacheHit (94μs)
      • Illuminate\Cache\Events\CacheHit (117μs)
      • Illuminate\Cache\Events\CacheHit (95μs)
      • Illuminate\Cache\Events\CacheHit (130μs)
      • Illuminate\Cache\Events\CacheHit (88μs)
      • Illuminate\Cache\Events\CacheHit (161μs)
      • Illuminate\Cache\Events\CacheHit (113μs)
      • Illuminate\Cache\Events\CacheHit (116μs)
      • Illuminate\Cache\Events\CacheHit (512μs)
      • Illuminate\Cache\Events\CacheHit (179μs)
      • Illuminate\Cache\Events\CacheHit (111μs)
      • Illuminate\Cache\Events\CacheHit (122μs)
      • Illuminate\Cache\Events\CacheHit (100μs)
      • Illuminate\Cache\Events\CacheMissed (188μs)
      • Illuminate\Database\Events\StatementPrepared (741μs)
      • Illuminate\Database\Events\QueryExecuted (786μs)
      • eloquent.retrieved: App\Models\SubjectCat (62μs)
      • Illuminate\Cache\Events\KeyWritten (246μs)
      • Illuminate\Cache\Events\CacheHit (149μs)
      • Illuminate\Cache\Events\CacheHit (147μs)
      • Illuminate\Cache\Events\CacheHit (109μs)
      • Illuminate\Cache\Events\CacheHit (132μs)
      • Illuminate\Cache\Events\CacheHit (99μs)
      • Illuminate\Cache\Events\CacheHit (112μs)
      • Illuminate\Cache\Events\CacheHit (150μs)
      • Illuminate\Cache\Events\CacheHit (145μs)
      • Illuminate\Cache\Events\CacheHit (141μs)
      • Illuminate\Cache\Events\CacheHit (136μs)
      • Illuminate\Cache\Events\CacheHit (186μs)
      • Illuminate\Cache\Events\CacheHit (137μs)
      • Illuminate\Cache\Events\CacheHit (99μs)
      • Illuminate\Cache\Events\CacheHit (111μs)
      • Illuminate\Cache\Events\CacheHit (95μs)
      • Illuminate\Cache\Events\CacheHit (108μs)
      • Illuminate\Cache\Events\CacheHit (95μs)
      • creating: site.layouts.app (323μs)
      • composing: site.layouts.app (16μs)
      • creating: components.canonical (359μs)
      • composing: components.canonical (91μs)
      • creating: components.open-graph (137μs)
      • composing: components.open-graph (48μs)
      • creating: site.headers.header (404μs)
      • composing: site.headers.header (237μs)
      • Illuminate\Cache\Events\CacheHit (1.56ms)
      • creating: components.footer (115μs)
      • composing: components.footer (67μs)
      • Illuminate\Cache\Events\CacheHit (575μs)
      • Illuminate\Cache\Events\CacheMissed (238μs)
      • Illuminate\Database\Events\StatementPrepared (691μs)
      • Illuminate\Database\Events\QueryExecuted (840μs)
      • eloquent.retrieved: App\Models\SubjectCat (61μs)
      • Illuminate\Cache\Events\KeyWritten (260μs)
      • Illuminate\Cache\Events\CacheHit (227μs)
      • Illuminate\Cache\Events\CacheHit (144μs)
      • Illuminate\Cache\Events\CacheHit (115μs)
      • Illuminate\Cache\Events\CacheHit (107μs)
      • Illuminate\Cache\Events\CacheHit (110μs)
      • Illuminate\Cache\Events\CacheHit (181μs)
      • Illuminate\Cache\Events\CacheHit (299μs)
      • Illuminate\Cache\Events\CacheHit (216μs)
      • Illuminate\Cache\Events\CacheHit (130μs)
      • Illuminate\Cache\Events\CacheHit (113μs)
      • Illuminate\Cache\Events\CacheHit (183μs)
      • Illuminate\Cache\Events\CacheHit (143μs)
      • creating: components.forms.contact-us (202μs)
      • composing: components.forms.contact-us (105μs)
      • creating: components.forms.get-started (110μs)
      • composing: components.forms.get-started (43μs)
      • creating: components.forms.free-tool-download (153μs)
      • composing: components.forms.free-tool-download (48μs)
      • creating: components.forms.claim-free-worksheet (80μs)
      • composing: components.forms.claim-free-worksheet (103μs)
      • creating: components.forms.tutor-subscription-waitlist (266μs)
      • composing: components.forms.tutor-subscription-waitlist (49μs)
      • creating: components.forms.tutor-subscription-join (86μs)
      • composing: components.forms.tutor-subscription-join (40μs)
      • creating: components.forms.tutor-support (77μs)
      • composing: components.forms.tutor-support (38μs)
      • 311 x Illuminate\Cache\Events\CacheHit (15.56%)
        58.92ms
        20 x Illuminate\Database\Events\StatementPrepared (9.81%)
        37.16ms
        20 x Illuminate\Database\Events\QueryExecuted (7.74%)
        29.31ms
        11 x Illuminate\Cache\Events\CacheMissed (2.18%)
        8.25ms
        1 x Illuminate\Foundation\Events\LocaleUpdated (1.05%)
        3.97ms
        11 x Illuminate\Cache\Events\KeyWritten (0.9%)
        3.40ms
        1 x Illuminate\Routing\Events\Routing (0.4%)
        1.50ms
        12 x eloquent.retrieved: App\Models\SubjectCat (0.2%)
        759μs
        1 x Illuminate\Database\Events\ConnectionEstablished (0.19%)
        732μs
        1 x Illuminate\Routing\Events\RouteMatched (0.17%)
        637μs
        1 x creating: components.breadcrumbs (0.13%)
        491μs
        1 x eloquent.booting: App\Models\SubjectCat (0.11%)
        422μs
        1 x creating: site.headers.header (0.11%)
        404μs
        1 x creating: components.canonical (0.09%)
        359μs
        1 x creating: site.layouts.app (0.09%)
        323μs
        1 x creating: homework.show (0.07%)
        270μs
        1 x creating: components.forms.tutor-subscription-waitlist (0.07%)
        266μs
        1 x composing: site.headers.header (0.06%)
        237μs
        7 x eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibrary (0.06%)
        219μs
        1 x creating: components.forms.contact-us (0.05%)
        202μs
        1 x eloquent.booting: App\Models\HomeworkLibrary\HomeworkLibrary (0.05%)
        195μs
        2 x eloquent.retrieved: App\Models\Subject (0.05%)
        193μs
        1 x composing: homework.show (0.05%)
        190μs
        11 x eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibraryFile (0.05%)
        176μs
        1 x creating: components.forms.free-tool-download (0.04%)
        153μs
        1 x eloquent.booted: App\Models\HomeworkLibrary\HomeworkLibrary (0.04%)
        140μs
        1 x creating: components.open-graph (0.04%)
        137μs
        1 x composing: components.breadcrumbs (0.03%)
        125μs
        1 x creating: components.footer (0.03%)
        115μs
        1 x creating: components.forms.get-started (0.03%)
        110μs
        1 x eloquent.booting: App\Models\HomeworkLibrary\HomeworkLibraryFile (0.03%)
        108μs
        1 x composing: components.forms.contact-us (0.03%)
        105μs
        1 x composing: components.forms.claim-free-worksheet (0.03%)
        103μs
        1 x eloquent.booting: App\Models\Subject (0.02%)
        92μs
        1 x composing: components.canonical (0.02%)
        91μs
        1 x creating: components.forms.tutor-subscription-join (0.02%)
        86μs
        1 x creating: components.forms.claim-free-worksheet (0.02%)
        80μs
        1 x creating: components.forms.tutor-support (0.02%)
        77μs
        1 x composing: components.footer (0.02%)
        67μs
        1 x composing: components.forms.tutor-subscription-waitlist (0.01%)
        49μs
        1 x composing: components.forms.free-tool-download (0.01%)
        48μs
        1 x composing: components.open-graph (0.01%)
        48μs
        1 x composing: components.forms.get-started (0.01%)
        43μs
        1 x eloquent.booted: App\Models\Subject (0.01%)
        43μs
        1 x eloquent.booted: App\Models\SubjectCat (0.01%)
        42μs
        1 x composing: components.forms.tutor-subscription-join (0.01%)
        40μs
        1 x eloquent.booted: App\Models\HomeworkLibrary\HomeworkLibraryFile (0.01%)
        40μs
        1 x composing: components.forms.tutor-support (0.01%)
        38μs
        1 x composing: site.layouts.app (0%)
        16μs
      14 templates were rendered
      • 1x homework.showshow.blade.phpblade
      • 1x components.breadcrumbsbreadcrumbs.blade.phpblade
      • 1x site.layouts.appapp.blade.phpblade
      • 1x components.canonicalcanonical.blade.phpblade
      • 1x components.open-graphopen-graph.blade.phpblade
      • 1x site.headers.headerheader.blade.phpblade
      • 1x components.footerfooter.blade.phpblade
      • 1x components.forms.contact-uscontact-us.blade.phpblade
      • 1x components.forms.get-startedget-started.blade.phpblade
      • 1x components.forms.free-tool-downloadfree-tool-download.blade.phpblade
      • 1x components.forms.claim-free-worksheetclaim-free-worksheet.blade.phpblade
      • 1x components.forms.tutor-subscription-waitlisttutor-subscription-waitlist.blade.phpblade
      • 1x components.forms.tutor-subscription-jointutor-subscription-join.blade.phpblade
      • 1x components.forms.tutor-supporttutor-support.blade.phpblade
      uri
      GET college-homework-library/{category}/{subject}/{id}
      middleware
      web, utm.parameters
      controller
      App\Http\Controllers\HomeworkLibraryController@show
      namespace
      where
      as
      homework.show
      file
      app/Http/Controllers/HomeworkLibraryController.php:79-176
      20 statements were executed, 5 of which were duplicates, 15 unique. Show only duplicated51.3ms
      • Connection Establishedtwenty4_siteHomeworkLibraryController.php#91
        Backtrace
        • 13. app/Http/Controllers/HomeworkLibraryController.php:91
        • 14. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 15. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 16. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
        • 17. vendor/laravel/framework/src/Illuminate/Routing/Route.php:205
      • select * from `solutionslibrary` where `status` = 'published' and `price` > 0 and `solutionslibrary`.`id` = '31497' limit 1
        22.83mstwenty4_siteHomeworkLibraryController.php#97
        Bindings
        • 0: published
        • 1: 0
        • 2: 31497
        Backtrace
        • 16. app/Http/Controllers/HomeworkLibraryController.php:97
        • 17. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 18. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 19. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
        • 20. vendor/laravel/framework/src/Illuminate/Routing/Route.php:205
      • select * from `subjects` where `subjects`.`id` in (103)
        2.47mstwenty4_siteHomeworkLibraryController.php#97
        Backtrace
        • 21. app/Http/Controllers/HomeworkLibraryController.php:97
        • 22. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 23. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 24. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
        • 25. vendor/laravel/framework/src/Illuminate/Routing/Route.php:205
      • select * from `solutionslibrary_files` where `solutionslibrary_files`.`solutionlib_id` in (31497)
        1.71mstwenty4_siteHomeworkLibraryController.php#97
        Backtrace
        • 21. app/Http/Controllers/HomeworkLibraryController.php:97
        • 22. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 23. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 24. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
        • 25. vendor/laravel/framework/src/Illuminate/Routing/Route.php:205
      • select * from `subject_cats` where `subject_cats`.`id` = 3 limit 1
        1.7mstwenty4_siteHomeworkLibrary.php#201
        Bindings
        • 0: 3
        Backtrace
        • 20. app/Models/HomeworkLibrary/HomeworkLibrary.php:201
        • 26. app/Http/Controllers/HomeworkLibraryController.php:105
        • 27. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 28. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 29. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
      • select * from `solutionslibrary` where `id` <> 31497 and `subject` = 103 and `status` = 'published' and `price` > 0 order by RAND() limit 6
        4.67mstwenty4_siteHomeworkLibraryRepository.php#30
        Bindings
        • 0: 31497
        • 1: 103
        • 2: published
        • 3: 0
        Backtrace
        • 14. app/Repositories/HomeworkLibraryRepository.php:30
        • 15. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 16. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 18. app/Repositories/HomeworkLibraryRepository.php:39
        • 19. app/Http/Controllers/HomeworkLibraryController.php:139
      • select * from `subjects` where `subjects`.`id` in (103)
        1.27mstwenty4_siteHomeworkLibraryRepository.php#30
        Backtrace
        • 19. app/Repositories/HomeworkLibraryRepository.php:30
        • 20. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 21. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 23. app/Repositories/HomeworkLibraryRepository.php:39
        • 24. app/Http/Controllers/HomeworkLibraryController.php:139
      • select * from `solutionslibrary_files` where `solutionslibrary_files`.`solutionlib_id` = 31497 and `solutionslibrary_files`.`solutionlib_id` is not null and `publish` = 'question' order by `order` asc, `id` asc
        960μstwenty4_siteHomeworkLibrary.php#260
        Bindings
        • 0: 31497
        • 1: question
        Backtrace
        • 15. app/Models/HomeworkLibrary/HomeworkLibrary.php:260
        • 16. app/Transformers/HomeworkLibrary/HomeworkLibraryTransformer.php:58
        • 19. vendor/league/fractal/src/TransformerAbstract.php:128
        • 20. vendor/league/fractal/src/TransformerAbstract.php:107
        • 21. vendor/league/fractal/src/Scope.php:383
      • select * from `solutionslibrary_files` where `solutionslibrary_files`.`solutionlib_id` = 31497 and `solutionslibrary_files`.`solutionlib_id` is not null and `publish` = 'teaser' order by `order` asc, `id` asc
        800μstwenty4_siteHomeworkLibrary.php#260
        Bindings
        • 0: 31497
        • 1: teaser
        Backtrace
        • 15. app/Models/HomeworkLibrary/HomeworkLibrary.php:260
        • 16. app/Transformers/HomeworkLibrary/HomeworkLibraryTransformer.php:69
        • 19. vendor/league/fractal/src/TransformerAbstract.php:128
        • 20. vendor/league/fractal/src/TransformerAbstract.php:107
        • 21. vendor/league/fractal/src/Scope.php:383
      • select * from `solutionslibrary_files` where `solutionslibrary_files`.`solutionlib_id` = 31497 and `solutionslibrary_files`.`solutionlib_id` is not null and `publish` = 'solution' order by `order` asc, `id` asc
        860μstwenty4_siteHomeworkLibrary.php#260
        Bindings
        • 0: 31497
        • 1: solution
        Backtrace
        • 15. app/Models/HomeworkLibrary/HomeworkLibrary.php:260
        • 16. app/Transformers/HomeworkLibrary/HomeworkLibraryTransformer.php:80
        • 19. vendor/league/fractal/src/TransformerAbstract.php:128
        • 20. vendor/league/fractal/src/TransformerAbstract.php:107
        • 21. vendor/league/fractal/src/Scope.php:383
      • select * from `subject_cats` where `subject_cats`.`id` = 3 limit 1
        1.21mstwenty4_siteHomeworkLibrary.php#201
        Bindings
        • 0: 3
        Backtrace
        • 20. app/Models/HomeworkLibrary/HomeworkLibrary.php:201
        • 32. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
        • 33. vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58
        • 34. vendor/livewire/livewire/src/ComponentConcerns/RendersLivewireComponents.php:69
        • 35. vendor/laravel/framework/src/Illuminate/View/Engines/CompilerEngine.php:70
      • select * from `subject_cats` where `subject_cats`.`id` = 1 limit 1
        1.15mstwenty4_siteSubject.php#100
        Bindings
        • 0: 1
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 3 limit 1
        3.29mstwenty4_siteSubject.php#100
        Bindings
        • 0: 3
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 4 limit 1
        990μstwenty4_siteSubject.php#100
        Bindings
        • 0: 4
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 10 limit 1
        1.01mstwenty4_siteSubject.php#100
        Bindings
        • 0: 10
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 33 limit 1
        1.15mstwenty4_siteSubject.php#100
        Bindings
        • 0: 33
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 11 limit 1
        990μstwenty4_siteSubject.php#100
        Bindings
        • 0: 11
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 5 limit 1
        1.05mstwenty4_siteSubject.php#100
        Bindings
        • 0: 5
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 34 limit 1
        1.1mstwenty4_siteSubject.php#100
        Bindings
        • 0: 34
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 31 limit 1
        1.05mstwenty4_siteSubject.php#100
        Bindings
        • 0: 31
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 36 limit 1
        1.04mstwenty4_siteSubject.php#100
        Bindings
        • 0: 36
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 28. view::components.footer:170
      App\Models\SubjectCat
      12SubjectCat.php
      App\Models\HomeworkLibrary\HomeworkLibraryFile
      11HomeworkLibraryFile.php
      App\Models\HomeworkLibrary\HomeworkLibrary
      7HomeworkLibrary.php
      App\Models\Subject
      2Subject.php
          _token
          druQC4A9qEM5pUw4PJUGN2d0OQJDtHTuKdAHERkG
          utm_source
          direct
          redirectUrl
          /college-homework-library/Computer-Science/Cryptography/31497
          _previous
          array:1 [ "url" => "https://staging.dev.24houranswers.com/college-homework-library/Computer-Scienc...
          _flash
          array:2 [ "old" => [] "new" => [] ]
          PHPDEBUGBAR_STACK_DATA
          []
          path_info
          /college-homework-library/Computer-Science/Cryptography/31497
          status_code
          200
          
          status_text
          OK
          format
          html
          content_type
          text/html; charset=UTF-8
          request_query
          []
          
          request_request
          []
          
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