Rozważmy masz funkcji skrótu $\mathcal{H}$ , który trwa ciągi długości $2n$ i powrót ciągi o długości $n$ i ma tę właściwość, piękny, że jest odporna na zderzenia , czyli trudno jest znaleźć dwa różne ciągi $s \neq s'$ z tego samego skrótu $\mathcal{H}(s) = \mathcal{H}(s')$ .

Chciałbyś teraz zbudować nową funkcję skrótu $\mathcal{H'}$ która pobiera ciągi o dowolnej długości i odwzorowuje je na ciągi o długości $n$ , zachowując jednocześnie odporność na kolizje.

Na szczęście już w 1979 r. Opublikowano metodę znaną obecnie jako konstrukcja Merkle – Damgård, która osiąga właśnie to.

Zadaniem tego wyzwania będzie wdrożenie tego algorytmu, dlatego najpierw przejrzymy formalny opis konstrukcji Merkle – Damgård, a następnie przejrzymy krok po kroku przykład, który powinien wykazać, że podejście jest prostsze niż może się na początku pojawić.

Biorąc pod uwagę liczbę całkowitą $n > 0$ , funkcję skrótu $\mathcal{H}$ jak opisano powyżej i ciąg wejściowy $s$ o dowolnej długości, nowa funkcja skrótu $\mathcal{H'}$ wykonuje następujące czynności:

• Ustaw $l = |s|$, długość $s$ i podział $s$ na kawałki o długości $n$ , w razie potrzeby wypełniając ostatni fragment końcowymi zerami. Daje to $m = \lceil \frac{l}{n} \rceil$wiele kawałków oznaczonych jako$c_1, c_2, \dots, c_m$.
• Dodaj początkowy i końcowy fragment $c_0$ i $c_{m+1}$ , gdzie $c_0$ jest łańcuchem składającym się z $n$ zer, a $c_{m+1}$ jest $n$ binarnie, wypełnione zerami wiodącymi do długości $n$ .
• Teraz iteracyjnie zastosuj $\mathcal{H}$ do bieżącego fragmentu $c_i$ dołączonego do poprzedniego wyniku $r_{i-1}$ : $r_i = \mathcal{H}(r_{i-1}c_i)$ , gdzie $r_0 = c_0$ . (Ten krok może być bardziej wyraźny po zapoznaniu się z poniższym przykładem).
• Wydajność $\mathcal{H'}$ jest wynikiem końcowym $r_{m+1}$ .

Zadanie

Napisz program lub funkcję, która przyjmuje na wejściu dodatnią liczbę całkowitą $n$ , funkcję skrótu $\mathcal{H}$ jako czarną skrzynkę i niepuste ciągi $s$ i zwraca ten sam wynik co $\mathcal{H'}$ na tych samych wejściach.

To jest golf golfowy , więc wygrywa najkrótsza odpowiedź w każdym języku.

Przykład

Powiedzmy, że $n = 5$ , więc nasza podana funkcja skrótu $\mathcal{H}$ przyjmuje łańcuchy o długości 10 i zwraca łańcuchy o długości 5.

• Biorąc pod uwagę $s = \texttt{"Programming Puzzles"}$ , otrzymujemy następujące fragmenty: $s_1 = \texttt{"Progr"}$ , $s_2 = \texttt{"ammin"}$ , $s_3 = \texttt{"g Puz"}$ i $s_4 = \texttt{"zles0"}$ . Zauważ, że $s_4$ trzeba było uzupełnić do długości 5 jednym końcowym zerem.
• $c_0 = \texttt{"00000"}$ to tylko ciąg pięciu zer, a$c_5 = \texttt{"00101"}$ to pięć cyfr binarnych ($\texttt{101}$ ), wypełnionych dwoma zerami wiodącymi.
• Teraz fragmenty są łączone z $\mathcal{H}$ :
  $r_0 = c_0 = \texttt{"00000"}$
  $r_1 = \mathcal{H}(r_0c_1) = \mathcal{H}(\texttt{"00000Progr"})$
  $r_2 = \mathcal{H}(r_1c_2) = \mathcal{H}(\mathcal{H}(\texttt{"00000Progr"})\texttt{"ammin"})$ $r_3 = \mathcal{H}(r_2c_3) = \mathcal{H}(\mathcal{H}(\mathcal{H}(\texttt{"00000Progr"})\texttt{"ammin"})\texttt{"g Puz"})$
  $r_4 = \mathcal{H}(r_3c_4) = \mathcal{H}(\mathcal{H}(\mathcal{H}(\mathcal{H}(\texttt{"00000Progr"})\texttt{"ammin"})\texttt{"g Puz"})\texttt{"zles0"})$
  $r_5 = \mathcal{H}(r_4c_5) = \mathcal{H}(\mathcal{H}(\mathcal{H}(\mathcal{H}(\mathcal{H}(\texttt{"00000Progr"})\texttt{"ammin"})\texttt{"g Puz"})\texttt{"zles0"})\texttt{"00101"})$
• $r_5$ jest naszym wyjściem.

Zobaczmy, jak wyglądałoby to wyjście w zależności od niektórych opcji ¹ dla $\mathcal{H}$ :

• Jeśli $\mathcal{H}(\texttt{"0123456789"}) = \texttt{"13579"}$ , tj. $\mathcal{H}$ zwraca tylko co drugi znak, otrzymujemy:
  $r_1 = \mathcal{H}(\texttt{"00000Progr"}) = \texttt{"00Por"}$
  $r_2 = \mathcal{H}(\texttt{"00Porammin"}) = \texttt{"0oamn"}$
  $r_3 = \mathcal{H}(\texttt{"0oamng Puz"}) = \texttt{"omgPz"}$
  $r_4 = \mathcal{H}(\texttt{"omgPzzles0"}) = \texttt{"mPze0"}$
  $r_5 = \mathcal{H}(\texttt{"mPze000101"}) = \texttt{"Pe011"}$
  So $\texttt{"Pe011"}$ needs to be the output if such a $\mathcal{H}$ is given as black box function.
• If $\mathcal{H}$ simply returns the first 5 chars of its input, the output of $\mathcal{H'}$ is $\texttt{"00000"}$. Similarly if $\mathcal{H}$ returns the last 5 chars, the output is $\texttt{"00101"}$.
• If $\mathcal{H}$ multiplies the character codes of its input and returns the first five digits of this number, e.g. $\mathcal{H}(\texttt{"PPCG123456"}) = \texttt{"56613"}$, then $\mathcal{H}'(\texttt{"Programming Puzzles"}) = \texttt{"91579"}$.

^{1 For simplicity, those $\mathcal{H}$ are actually not collision resistant, though this does not matter for testing your submission.}

Haskell, 91 90 86 bytes

• -1 byte thanks to Laikoni
• -4 bytes thanks to xnor

n!h|let a='0'<$[1..n];c?""=c;c?z=h(c++take n(z++a))?drop n z=h.(++mapM(:"1")a!!n).(a?)

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Explanation

a='0'<$[1..n]

Just assigns the string "00...0" ('0' $n$ times) to a

c?""=c
c?z=h(c++take n(z++a))?drop n z

The function ? implements the recursive application of h: c is the hash we have obtained so far (length $n$), z is the rest of the string. If z is empty then we simply return c, otherwise we take the first $n$ characters of z (possibly padding with zeros from a), prepend c and apply h. This gives the new hash, and then we call ? recursively on this hash and the remaining characters of z.

n!h=h.(++mapM(:"1")a!!n).(a?)

The function ! is the one actually solving the challenge. It takes n, h and s (implicit) as inputs. We compute a?s, and all we have to do is append n in binary and apply h once more. mapM(:"1")a!!n returns the binary representation of $n$.

R, 159 154 bytes

function(n,H,s,`?`=paste0,`*`=strrep,`/`=Reduce,`+`=nchar,S=0*n?s?0*-(+s%%-n)?"?"/n%/%2^(n:1-1)%%2)(function(x,y)H(x?y))/substring(S,s<-seq(,+S,n),s--n-1)

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Yuck! Answering string challenges in R is never pretty, but this is horrible. This is an instructive answer on how not to write "normal" R code...

Thanks to nwellnhof for fixing a bug, at a cost of 0 bytes!

Thanks to J.Doe for swapping the operator aliasing to change the precedence, good for -4 bytes.

The explanation below is for the previous version of the code, but the principles remain the same.

function(n,H,s,               # harmless-looking function arguments with horrible default arguments 
                              # to prevent the use of {} and save two bytes
                              # then come the default arguments,
                              # replacing operators as aliases for commonly used functions:
 `+`=paste0,                  # paste0 with binary +
 `*`=strrep,                  # strrep for binary *
 `/`=Reduce,                  # Reduce with binary /
 `?`=nchar,                   # nchar with unary ?
 S=                           # final default argument S, the padded string:
  0*n+                        # rep 0 n times
  s+                          # the original string
  0*-((?s)%%-n)+              # 0 padding as a multiple of n
  "+"/n%/%2^(n:1-1)%%2)       # n as an n-bit number
                              # finally, the function body:
 (function(x,y)H(x+y)) /      # Reduce/Fold (/) by H operating on x + y
  substring(S,seq(1,?S,n),seq(n,?S,n))  # operating on the n-length substrings of S

C (gcc), 251 bytes

#define P sprintf(R,
b(_){_=_>1?10*b(_/2)+_%2:_;}f(H,n,x)void(*H)(char*);char*x;{char R[2*n+1],c[n+1],*X=x;P"%0*d",n,0);while(strlen(x)>n){strncpy(c,x,n);x+=n;strcat(R,c);H(R);}P"%s%s%0*d",R,x,n-strlen(x),0);H(R);P"%s%0*d",R,n,b(n));H(R);strcpy(X,R);}

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Not as clean as the bash solution, and highly improvable.

The function is f taking H as a function that replaces its string input with that string's hash, n as in the description, and x the input string and output buffer.

Description:

#define P sprintf(R,     // Replace P with sprintf(R, leading to unbalanced parenthesis
                         // This is replaced and expanded for the rest of the description
b(_){                    // Define b(x). It will return the integer binary expansion of _
                         // e.g. 5 -> 101 (still as integer)
  _=_>1?                 // If _ is greater than 1
    10*b(_/2)+_%2        // return 10*binary expansion of _/2 + last binary digit
    :_;}                 // otherwise just _
f(H,n,x)                 // Define f(H,n,x)
  void(*H)(char*);       // H is a function taking a string
  char*x; {              // x is a string
  char R[2*n+1],c[n+1],  // Declare R as a 2n-length string and c as a n-length string
  *X=x;                  // save x so we can overwrite it later
  sprintf(R,"%0*d",n,0); // print 'n' 0's into R
  while(strlen(x)>n){    // while x has at least n characters
    strncpy(c,x,n);x+=n; // 'move' the first n characters of x into c
    strcat(R,c);         // concatenate c and R
    H(R);}               // Hash R
  sprintf(R,"%s%s%0*d"   // set R to two strings concatenated followed by some zeroes
    R,x,                 // the two strings being R and (what's left of) x
    n-strlen(x),0);      // and n-len(x) zeroes
  H(R);                  // Hash R
  sprintf(R,"%s%*d",R,n, // append to R the decimal number, 0 padded to width n
    b(n));               // The binary expansion of n as a decimal number
  H(R);strcpy(X,R);}     // Hash R and copy it into where x used to be

Ruby, 78 bytes

->n,s,g{(([?0*n]*2*s).chop.scan(/.{#{n}}/)+["%0#{n}b"%n]).reduce{|s,x|g[s+x]}}

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How it works:

([?0*n]*2*s).chop    # Padding: add leading and trailing 
                     # zeros, then remove the last one
.scan(/.{#{n}}/)     # Split the string into chunks
                     # of length n
+["%0#{n}b"%n]       # Add the trailing block
.reduce{|s,x|g[s+x]} # Apply the hashing function
                     # repeatedly

Jelly, 23 bytes

0Ṿ;s;BṾ€ṚWƲ}z”0ZU0¦;Ç¥/

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Accepts $\mathcal H$ at the line above it, $s$ as its left argument, and $n$ as its right argument.

Bash, 127-ε bytes

Z=`printf %0*d $1` R=$Z
while IFS= read -rn$1 c;do R=$R$c$Z;R=`H<<<${R::2*$1}`;done
H< <(printf $R%0*d $1 `bc <<<"obase=2;$1"`)

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This works as a program/function/script/snippet. H must be resolveable to a program or function that will perform the hashing. N is the argument. Example call:

$ H() {
>   sed 's/.\(.\)/\1/g'
> }
$ ./wherever_you_put_the_script.sh 5 <<< "Programming Puzzles"  # if you add a shebang
Pe011

Description:

Z=`printf %0*d $1`

This creates a string of $1 zeroes. This works by calling printf and telling it to print an integer padded to extra argument width. That extra argument we pass is $1, the argument to the program/function/script which stores n.

R=$Z

This merely copies Z, our zero string, to R, our result string, in preparation for the hashing loop.

while IFS= read -rn$1 c; do

This loops over the input every $1 (n) characters loading the read characters into c. If the input ends then c merely ends up too short. The r option ensures that any special characters in the input don't get bash-interpreted. This is the -ε in the title - that r isn't strictly necessary, but makes the function more accurately match the input.

R=$R$c$Z

This concatenates the n characters read from input to R along with zeroes for padding (too many zeroes for now).

R=`H<<<${R::2*$1}`;done

This uses a here string as input to the hash function. The contents ${R::2*$1} are a somewhat esoteric bash parameter substitution which reads: R, starting from 0, only 2n characters.

Here the loop ends and we finish with:

H< <(printf $R%0*d $1 `bc <<<"obase=2;$1"`)

Here the same format string trick is used to 0 pad the number. bc is used to convert it to binary by setting the output base (obase) to 2. The result is passed to the hash function/program whose output is not captured and thus is shown to the user.

Pyth, 24 bytes

Since Pyth doesn't allow H to be used for a function name, I use y instead.

uy+GH+c.[E=`ZQQ.[ZQ.BQ*Z

Try it online! Example is with the "every second character" version of H.

Perl 6, 79 68 bytes

{reduce &^h o&[~],comb 0 x$^n~$^s~$n.fmt("%.{$n-$s.comb%-$n}b"): $n}

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Explanation

{
  reduce         # Reduce with
    &^h o&[~],   # composition of string concat and hash function
    comb         # Split string
      0 x$^n     # Zero repeated n times
      ~$^s       # Append input string s
      ~$n.fmt("  # Append n formatted
        %.       # with leading zeroes,
        {$n             # field width n for final chunk
         -$s.comb%-$n}  # -(len(s)%-n) for padding,
        b")      # as binary number
      :          # Method call with colon syntax
      $n         # Split into substrings of length n
}

Clean, 143 bytes

import StdEnv
r=['0':r]
$n h s=foldl(\a b=h(a++b))(r%(1,n))([(s++r)%(i,i+n-1)\\i<-[0,n..length s]]++[['0'+toChar((n>>(n-p))rem 2)\\p<-[1..n]]])

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Python 2, 126 113 bytes

lambda n,H,s:reduce(lambda x,y:H(x+y),re.findall('.'*n,'0'*n+s+'0'*(n-len(s)%n))+[bin(n)[2:].zfill(n)])
import re

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-13 thanks to Triggernometry.

Yeah, this is an abomination, why can't I just use a built-in to split a string into chunks...? :-(

Python 2, 106 102 bytes

For once, the function outgolfs the lambda. -4 bytes for simple syntax manipulation, thanks to Jo King.

def f(n,H,s):
 x='0'*n;s+='0'*(n-len(s)%n)+bin(n)[2:].zfill(n)
 while s:x=H(x+s[:n]);s=s[n:]
 return x

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Japt, 27 bytes

òV ú'0 pV¤ùTV)rÈ+Y gOvW}VçT

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I haven't found any capability for Japt to take functions directly as an input, so this takes a string which is interpreted as Japt code and expects it to define a function. Specifically, OvW takes the third input and interprets it as Japt, then g calls it. Replacing that with OxW allows input as a Javascript function instead, or if the function were (somehow) already stored in W it could just be W and save 2 bytes. The link above has the worked example of $\mathcal{H}$ that takes characters at odd indexes, while this one is the "multiply char-codes and take the 5 highest digits" example.

Due to the way Japt takes inputs, $s$ will be U, $n$ will be V, and $\mathcal{H}$ will be W

Explanation:

òV                             Split U into segments of length V
   ú'0                         Right-pad the short segment with "0" to the same length as the others
       p     )                 Add an extra element:
        V¤                       V as a base-2 string
          ùTV                    Left-pad with "0" until it is V digits long
              r                Reduce...
                        VçT          ...Starting with "0" repeated V times...
               È       }                                                  ...By applying:
                +Y               Combine with the previous result
                   gOvW          And run W as Japt code

GolfScript, 47 bytes

~:π'0'*:s\+s+π/);[sπ2base{48+}%+0π->]+{+1$~}*\;

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oK, 41 bytes

{(x#48)(y@,)/(0N,x)#z,,/$((x+x!-#z)#2)\x}

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{                                       } /x is n, y is H, z is s.
                          (x+x!-#z)       /number of padding 0's needed + x
                         (         #2)\x  /binary(x) with this length
                      ,/$                 /to string
                    z,                    /append to z
             (0N,x)#                      /split into groups of length x
       (y@,)/                             /foldl of y(concat(left, right))...
 (x#48)                                   /...with "0"*x as the first left string