Discussion on: Daily Challenge #42 - Caesar Cipher

-module( caesar ).
-compile( export_all ).
-export( [ caesar/2 ] ).

-include_lib("eunit/include/eunit.hrl").

shift_char( Char, Key ) when Char >= $a, Char =< $z ->
    $a + ( Char - $a + Key ) rem 26;
shift_char( Char, Key ) when Char >= $A, Char =< $Z ->
    $A + ( Char - $A + Key ) rem 26;
shift_char( Char, _ ) ->
    Char.

caesar( Message, Key ) when is_integer( Key ), Key > 0, Key < 26 ->
    lists:map( fun( C ) -> shift_char( C, Key ) end, Message ).

shift_char_test_() -> [
    ?_assertEqual( $b, shift_char( $a, 1 ) ),
    ?_assertEqual( $f, shift_char( $a, 5 ) ),
    ?_assertEqual( $a, shift_char( $z, 1 ) ),
    ?_assertEqual( $z, shift_char( $r, 8 ) ),
    ?_assertEqual( $b, shift_char( $r, 10 ) ),
    ?_assertEqual( $M, shift_char( $K, 2 ) ),
    ?_assertEqual( $D, shift_char( $W, 7 ) )
].

casear_test_() -> [
    ?_assertEqual( "b", caesar( "a", 1 ) ),
    ?_assertEqual( "e", caesar( "a", 4 ) ),
    ?_assertEqual( "a", caesar( "z", 1 ) ),
    ?_assertEqual( "efgfoe uif fbtu xbmm pg uif dbtumf", caesar( "defend the east wall of the castle", 1 ) ),
    ?_assertEqual( "dwwdfn iurp wkh zrrgv dw gdzq", caesar( "attack from the woods at dawn", 3 ) ),
    ?_assertEqual( "Dro aesmu, lbygx pyh TEWZON yfob dro vkji nyq!", caesar( "The quick, brown fox JUMPED over the lazy dog!", 10 ) )
].

% calculate the frequencies of the letters in the message
freqs( Message ) ->
    freqs( string:lowercase( Message ), #{}, 0 ).
freqs( [ C | Rest ], Freqs, Count ) when C >= $a, C =< $z ->
    NewFreqs = maps:put( C, maps:get( C, Freqs, 0 ) + 1, Freqs ),
    freqs( Rest, NewFreqs, Count + 1 );
freqs( [ _ | Rest ], Freqs, Count ) ->
    freqs( Rest, Freqs, Count );
freqs( [], Freqs, Count ) ->
    maps:map( fun( _C, Freq ) -> 100 * Freq / Count end, Freqs ).

% calculate the distance between the frequency of a letter in the message
% and the frequency of the same letter in the English language
freq_distance( C, Freq ) ->
    EnglishFreq = maps:get( C, english_frequencies() ),
    math:pow( EnglishFreq - Freq, 2 ).

% calculate the distance between frequencies in the message and frequencies in the English language
distance( Freqs ) ->
    lists:foldr(
        fun( { _, Dist }, Acc ) -> Acc + Dist end,
        0,
        maps:to_list(
            maps:map( fun freq_distance/2, Freqs )
        )
    ).

% try all keys and sort them starting with the decrypted message that has the least distance
% with the frequencies of the English language
crack( Message ) ->
    Keys = lists:seq( 1, 25 ),
    Candidates = lists:map( fun( Key ) -> { 26 - Key, caesar( Message, Key ) } end, Keys ),
    Freqs = lists:map( fun( { Key, Msg } ) -> { Key, Msg, freqs( Msg ) } end, Candidates ),
    Distances = lists:map( fun( { Key, Msg, Frq } ) -> { Key, Msg, distance( Frq ) } end, Freqs ),
    _BestKeys = lists:sort( fun( { _, _, Dist1 }, { _, _, Dist2 } ) -> ( Dist1 - Dist2 ) =< 0 end, Distances ).

% frequencies of the letters in the English language, from
% http://pi.math.cornell.edu/~mec/2003-2004/cryptography/subs/frequencies.html
english_frequencies() ->
    _Frequencies = #{
        $e => 12.02, $t => 9.10, $a => 8.12, $o => 7.68, $i => 7.31, $n => 6.95, $s => 6.28,
        $r => 6.02, $h => 5.92, $d => 4.32, $l => 3.98, $u => 2.88, $c => 2.71, $m => 2.61,
        $f => 2.30, $y => 2.11, $w => 2.09, $g => 2.03, $p => 1.82, $b => 1.49, $v => 1.11,
        $k => 0.69, $x => 0.17, $q => 0.11, $j => 0.10, $z => 0.07
    }.

% erl
1> c(caesar).
{ok,caesar}

2> caesar:test().
  All 13 tests passed.
ok

3> caesar:caesar( "attack from the woods at dawn", 3 ).
"dwwdfn iurp wkh zrrgv dw gdzq"

4> caesar:crack("efgfoe uif fbtu xbmm pg uif dbtumf" ).
[{1,"defend the east wall of the castle",232.3872551020408},
 {12,"stutcs iwt tphi lpaa du iwt rphiat",339.90096938775514},
 {13,"rstsbr hvs sogh kozz ct hvs qoghzs",514.9585122448979},
 ...

5> caesar:crack("dwwdfn iurp wkh zrrgv dw gdzq").
[{3,"attack from the woods at dawn",309.13214444444446},
 {18,"leelnv qczx esp hzzod le olhy",409.1672777777778},
 {25,"exxego jvsq xli asshw ex hear",415.19877777777776},

6> caesar:crack("Dro aesmu, lbygx pyh TEWZON yfob dro vkji nyq!").
[{10,"The quick, brown fox JUMPED over the lazy dog!",
  142.00382345679012},
 {20,"Jxu gkysa, rhemd ven ZKCFUT eluh jxu bqpo tew!",
  281.83945679012345},
 {21,"Iwt fjxrz, qgdlc udm YJBETS dktg iwt apon sdv!",
  307.0069345679012},