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{{Short description|Way to detect errors in decimal codes}}
The '''Verhoeff algorithm'''<ref name="Verhoeff_1969" /> is a [[checksum]] for [[error detection]] first published by Dutch mathematician [[Jacobus Verhoeff]] in 1969.<ref name=Kirtland_2001 /><ref name=Salomon_2005 />  It was the first decimal [[check digit]] algorithm which detects all single-digit errors, and all transposition errors involving two adjacent digits,<ref name=Haunsperger_2006 /> which was at the time thought impossible with such a code.

The method was independently discovered by H. Peter Gumm in 1985, this time including a formal proof and an extension to any base.<ref>{{cite journal |last1=Gumm |first1=H. |title=A new class of check-digit methods for arbitrary number systems (Corresp.) |journal=IEEE Transactions on Information Theory |date=January 1985 |volume=31 |issue=1 |pages=102–105 |doi=10.1109/TIT.1985.1056991 |url=https://www.researchgate.net/publication/3084126}}</ref>

==Goals==
Verhoeff had the goal of finding a decimal code—one where the check digit is a single decimal digit—which detected all single-digit errors and all transpositions of adjacent digits. At the time, supposed proofs of the nonexistence<ref name=Sisson_1958 /> of these codes made base-11 codes popular, for example in the [[International Standard Book Number#Check digits|ISBN check digit]].

His goals were also practical, and he based the evaluation of different codes on live data from the Dutch postal system, using a weighted points system for different kinds of error.  The analysis broke the errors down into a number of categories: first, by how many digits are in error; for those with two digits in error, there are ''transpositions'' (''ab'' → ''ba''), ''twins'' (''aa'' → 'bb'), ''jump transpositions'' (''abc'' → ''cba''), ''phonetic'' (''1a'' → ''a0''), and ''jump twins'' (''aba'' → ''cbc'').  Additionally there are omitted and added digits.  Although the frequencies of some of these kinds of errors might be small, some codes might be immune to them in addition to the primary goals of detecting all singles and transpositions.

The phonetic errors in particular showed linguistic effects, because in Dutch, numbers are typically read in pairs; and also while 50 sounds similar to 15 in Dutch, 80 doesn't sound like 18.

Taking six-digit numbers as an example, Verhoeff reported the following classification of the errors:.
{| class="wikitable"
|- valign="middle"
! style="background:#ffc" | Digits in error
! style="background:#ffc" | Classification
! style="background:#ffc" | Count
! style="background:#ffc" | Frequency
|-
|style="background:#ffc" |1||style="background:#ffc" |Transcription||align="right"|9,574||align="right"|79.05%
|-
|style="background:#ffc" rowspan=8 |2||style="background:#ffc" |Transpositions||align="right"|1,237||align="right"|10.21%
|-
|style="background:#ffc" |Twins||align="right"|67||align="right"|0.55%
|-
|style="background:#ffc" |Phonetic||align="right"|59||align="right"|0.49%
|-
|style="background:#ffc" |Other adjacent||align="right"|232||align="right"|1.92%
|-
|style="background:#ffc" |Jump transpositions||align="right"|99||align="right"|0.82%
|-
|style="background:#ffc" |Jump Twins||align="right"|35||align="right"|0.29%
|-
|style="background:#ffc" |Other jump errors||align="right"|43||align="right"|0.36%
|-
|style="background:#ffc" |Other||align="right"|98||align="right"|0.81%
|-
|colspan=2 style="background:#ffc" |3||align="right"|169||align="right"|1.40%
|-
|colspan=2 style="background:#ffc" |4||align="right"|118||align="right"|0.97%
|-
|colspan=2 style="background:#ffc" |5||align="right"|219||align="right"|1.81%
|-
|colspan=2 style="background:#ffc" |6||align="right"|162||align="right"|1.34%
|-
|colspan=2 style="background:#ffc" |'''Total''' || align="right"|'''12,112'''
|
|}

==Description==
The general idea of the algorithm is to represent each of the digits (0 through 9) as elements of the [[dihedral group]] <math>D_5</math>. That is, map digits to <math>D_5</math>, manipulate these, then map back into digits. Let this mapping be <math>m: [0, 9] \to D_5
</math>

<math>m = \begin{pmatrix}
0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9\\
e & r & r^2 & r^3 & r^4 & s & rs & r^2s & r^3s & r^4s\end{pmatrix}
</math>

Let the nth digit be <math>a_n
</math> and let the number of digits be <math>k
</math>.

For example given the code 248 then <math>k
</math> is 3 and <math>a_3 = m(8) = r^{3}s
</math>. 

Now define the permutation <math>f: D_5 \to D_5
</math>

<math>f = \begin{pmatrix}
e & r & r^2 & r^3 & r^4 & s & rs & r^2s & r^3s & r^4s\\
r & s & r^2s & rs & r^2 & r^3s & r^3 & e & r^4s & r^4\end{pmatrix}
</math>

For example <math>f(r^3) = rs
</math>. Another example is <math>f^2(r^3) = r^3
</math> since <math>f(f(r^3)) = f(rs) = r^3
</math>

Using multiplicative notation for the group operation of <math>D_5</math>, the check digit is then simply a value <math>c
</math> such that

<math> f(a_1) \cdot f^2(a_2) \cdot \ldots \cdot f^k(a_k) \cdot f^{k+1}(c) = e
</math>

<math>c
</math> is explicitly given by inverse permutation

<math>c = f^{-1-k}\left(\prod_{n=1}^k f^n(a_n)^{-1}\right)
</math>

For example the check digit for 248 is 5. To verify this, use the mapping to <math>D_5</math> and insert into the LHS of the previous equation

<math> f(r^2) \cdot f^2(r^4) \cdot f^3(r^3s) \cdot f^4(s) = e
</math>

To evaluate this permutation quickly use that

<math> f^4(s) = f^3(r^3s) = f^2(r^4) = f(r^2) = r^2 s
</math>

to get that

<math> r^2 s \cdot r^2 s \cdot r^2 s \cdot r^2 s = e
</math>

This is the same reflection being iteratively multiplied. Use that reflections are their own inverse.<ref name="Gallian_2010" />

<math> (r^2 s \cdot r^2 s) \cdot (r^2 s \cdot r^2 s) = e^2 = e 
</math>

In practice the algorithm is implemented using simple [[lookup table]]s without needing to understand how to generate those tables from the underlying group and permutation theory. This is more properly considered a family of algorithms, as other permutations work too. Verhoeff's notes that the particular permutation, given above, is special as it has the property of detecting 95.3% of the phonetic errors.<ref>{{harvnb|Verhoeff|1969|p=95}}</ref>

The strengths of the algorithm are that it detects all transliteration and transposition errors, and additionally most twin, twin jump, jump transposition and phonetic errors.

The main weakness of the Verhoeff algorithm is its complexity. The calculations required cannot easily be expressed as a formula in say <math>{\displaystyle \mathbb {Z} /10\mathbb {Z} }
</math>. Lookup tables are required for easy calculation. A similar code is the [[Damm algorithm]], which has similar qualities.

== Table-based algorithm ==
The Verhoeff algorithm can be implemented using three tables:
a multiplication table ''d'', an inverse table ''inv'', and a permutation table ''p''.

{{col-begin}}
{{col-break}}
{| class="wikitable" style="text-align:center"
|- valign="middle"
! rowspan=2 colspan=2 style="background:#ffc" | {{tmath|d(j,k)}}<ref>{{harvnb|Verhoeff|1969|p=83}}</ref>
! colspan=10  | ''k''
|- valign="middle"
! style="background:#ffc; width:1em" |  0
! style="background:#ffc; width:1em" |  1
! style="background:#ffc; width:1em" |  2
! style="background:#ffc; width:1em" |  3
! style="background:#ffc; width:1em" |  4
! style="background:#ffc; width:1em" |  5
! style="background:#ffc; width:1em" |  6
! style="background:#ffc; width:1em" |  7
! style="background:#ffc; width:1em" |  8
! style="background:#ffc; width:1em" |  9
|- valign="middle"
! rowspan=10 style="width:1em" |  ''j''
! style="background:#ffc; width:1em" |  0
| 0 || 1|| 2 || 3|| 4 || 5|| 6 || 7|| 8 || 9
|- valign="bottom"
! style="background:#ffc" | 1
| 1|| 2 || 3|| 4 || 0|| 6 || 7|| 8 || 9|| 5
|- valign="bottom"
! style="background:#ffc" | 2
| 2|| 3 || 4|| 0 || 1|| 7 || 8|| 9 || 5|| 6
|- valign="bottom"
! style="background:#ffc" | 3
| 3|| 4 || 0|| 1 || 2|| 8 || 9|| 5 || 6|| 7
|- valign="bottom"
! style="background:#ffc" | 4
| 4|| 0 || 1|| 2 || 3|| 9 || 5|| 6 || 7|| 8
|- valign="bottom"
! style="background:#ffc" | 5
| 5|| 9 || 8|| 7 || 6|| 0 || 4|| 3 || 2|| 1
|- valign="bottom"
! style="background:#ffc" | 6
| 6|| 5 || 9|| 8 || 7|| 1 || 0|| 4 || 3|| 2
|- valign="bottom"
! style="background:#ffc" | 7
| 7|| 6 || 5|| 9 || 8|| 2 || 1|| 0 || 4|| 3
|- valign="bottom"
! style="background:#ffc" | 8
| 8|| 7 || 6|| 5 || 9|| 3 || 2|| 1 || 0|| 4
|- valign="bottom"
! style="background:#ffc" | 9
| 9|| 8 || 7|| 6 || 5|| 4 || 3|| 2 || 1|| 0
|}
{{col-break|gap=1em}}
{| class="wikitable" style="text-align:center"
|- valign="middle"
! style="background:#ffc; width:1em; height:3.5em;" |  ''j''
! style="width:1em; height:3.5em;" |  {{tmath|inv(j)}}
|-
! style="background:#ffc; width:1em" |  0
| 0
|- valign="bottom"
! style="background:#ffc" | 1
| 4
|- valign="bottom"
! style="background:#ffc" | 2
| 3
|- valign="bottom"
! style="background:#ffc" | 3
| 2
|- valign="bottom"
! style="background:#ffc" | 4
| 1
|- valign="bottom"
! style="background:#ffc" | 5
| 5
|- valign="bottom"
! style="background:#ffc" | 6
| 6
|- valign="bottom"
! style="background:#ffc" | 7
| 7
|- valign="bottom"
! style="background:#ffc" | 8
| 8
|- valign="bottom"
! style="background:#ffc" | 9
| 9
|}
{{col-break|gap=1em}}
{| class="wikitable" style="text-align:center"
|- valign="middle"
! rowspan=2 colspan=2 style="background:#ffc" | {{tmath|p(pos,num)}}
! colspan=10 | ''num''
|- valign="middle"
! style="background:#ffc; width:1em" |  0
! style="background:#ffc; width:1em" |  1
! style="background:#ffc; width:1em" |  2
! style="background:#ffc; width:1em" |  3
! style="background:#ffc; width:1em" |  4
! style="background:#ffc; width:1em" |  5
! style="background:#ffc; width:1em" |  6
! style="background:#ffc; width:1em" |  7
! style="background:#ffc; width:1em" |  8
! style="background:#ffc; width:1em" |  9
|- valign="middle"
! rowspan=8 {{verth|va=middle| {{math| ''pos'' (mod 8)}} }}
! style="background:#ffc; width:1em" |  0
| 0 || 1|| 2 || 3|| 4 || 5|| 6 || 7|| 8 || 9
|- valign="bottom"
! style="background:#ffc" | 1
| 1|| 5 || 7|| 6 || 2|| 8 || 3|| 0 || 9|| 4
|- valign="bottom"
! style="background:#ffc" | 2
| 5|| 8 || 0|| 3 || 7|| 9 || 6|| 1 || 4|| 2
|- valign="bottom"
! style="background:#ffc" | 3
| 8|| 9 || 1|| 6 || 0|| 4 || 3|| 5 || 2|| 7
|- valign="bottom"
! style="background:#ffc" | 4
| 9|| 4 || 5|| 3 || 1|| 2 || 6|| 8 || 7|| 0
|- valign="bottom"
! style="background:#ffc" | 5
| 4|| 2 || 8|| 6 || 5|| 7 || 3|| 9 || 0|| 1
|- valign="bottom"
! style="background:#ffc" | 6
| 2|| 7 || 9|| 3 || 8|| 0 || 6|| 4 || 1|| 5
|- valign="bottom"
! style="background:#ffc" | 7
| 7|| 0 || 4|| 6 || 9|| 1 || 3|| 2 || 5|| 8
|}
{{col-end}}

The first table, '''''d''''', is based on multiplication in the dihedral group D<sub>5</sub>.<ref name=Gallian_2010 /> and is simply the [[Cayley table]] of the group.  Note that this group is not [[commutative]], that is, for some values of ''j'' and ''k'', ''d''(''j'',''k'') ≠ ''d''(''k'', ''j'').

The inverse table '''''inv''''' represents the multiplicative inverse of a digit, that is, the value that satisfies ''d''(''j'', ''inv''(''j'')) = 0.

The permutation table '''''p''''' applies a [[permutation]] to each digit based on its position in the number. This is actually a single permutation {{nowrap|(1 5 8 9 4 2 7 0)(3 6)}} applied iteratively; i.e. ''p''(''i''+''j'',''n'') = ''p''(''i'', ''p''(''j'',''n'')).

The Verhoeff checksum calculation is performed as follows:

# Create an array ''n'' out of the individual digits of the number, taken from right to left (rightmost digit is ''n''<sub>0</sub>, etc.).
# Initialize the checksum ''c'' to zero.
# For each index ''i'' of the array ''n,'' starting at zero, replace ''c'' with {{tmath|d(c, p(i \bmod 8, n_i))}}.

The original number is valid if and only if {{tmath|1=c = 0}}.

To generate a check digit, append a {{val|0}}, perform the calculation: the correct check digit is {{tmath|inv(c)}}.

== Examples ==
{{col-begin}}
{{col-2}}
'''Generate''' a check digit for ''236'':
{| class="wikitable" style="text-align:center"
|-
! style="background:#ffc" | ''i''
! style="background:#ffc" | ''n<sub>i</sub>''
! style="background:#ffc" | ''p''(''i,n<sub>i</sub>'')
! style="background:#ffc" | ''c''
|-
|0||0||0||0
|-
|1||6||3||3
|-
|2||3||3||1
|-
|3||2||1||2
|}
''c'' is 2, so the check digit is ''inv''(2), which is 3.
{{col-2}}
'''Validate''' the check digit ''2363'':
{| class="wikitable" style="text-align:center"
|-
! style="background:#ffc" |  ''i''
! style="background:#ffc" |  ''n<sub>i</sub>''
! style="background:#ffc" |  ''p''(''i,n<sub>i</sub>'')
! style="background:#ffc" | ''c''
|-
|0||3||3||3
|-
|1||6||3||1
|-
|2||3||3||4
|-
|3||2||1||0
|}
''c'' is zero, so the check is correct.
{{col-end}}

== See also ==
* [[Luhn algorithm]], earlier (1960) check digit algorithm

== References ==
{{reflist|refs=
<ref name="Verhoeff_1969">
{{cite book
| title=Error Detecting Decimal Codes (Tract 29)
| last=Verhoeff |first=J.
| publisher=The Mathematical Centre, Amsterdam
| year=1969
| doi=10.1002/zamm.19710510323 |bibcode=1971ZaMM...51..240N
}}
</ref>
<ref name=Kirtland_2001>{{cite book | title=Identification Numbers and Check Digit Schemes |last=Kirtland |first=Joseph | author-link=Joseph Kirtland |publisher=Mathematical Association of America | year=2001 | page=153 | isbn=0-88385-720-0 |chapter=5. Group Theory and the Verhoeff Check Digit Scheme | chapter-url=https://books.google.com/books?id=npTxORxmLosC&pg=PA121 }}</ref>

<ref name=Salomon_2005>{{cite book | title=Coding for Data and Computer Communications | last=Salomon |first=David | publisher=Springer | year=2005 | isbn=0-387-21245-0 |pages=56–58 |chapter=§2.11 The Verhoeff Check Digit Method | chapter-url=https://books.google.com/books?id=A88kvYwIVu0C&pg=PA57  }}</ref>

<ref name=Haunsperger_2006>{{cite book | title=The Edge of the Universe: Celebrating Ten Years of Math Horizons | editor1-last=Haunsperger | editor1-first=Deanna | editor2-last=Kennedy | editor2-first=Stephen | publisher=Mathematical Association of America | year=2006 | isbn=978-0-88385-555-3 | lccn=2005937266 | url=https://books.google.com/books?id=jiaIeCUpoFwC&pg=PA39 | page=38 }}</ref>

<ref name=Sisson_1958>{{cite journal |first=Roger L. |last=Sisson |title=An improved decimal redundancy check |journal=Communications of the ACM |volume=1 |issue=5 |pages=10–12 |date=May 1958 |doi=10.1145/368819.368854 |url=|doi-access=free }}</ref>

<ref name=Gallian_2010>{{cite book | title=Contemporary Abstract Algebra | edition=7th | last=Gallian |first=Joseph A. | publisher=Brooks/Cole | year=2010 | isbn=978-0-547-16509-7 | lccn=2008940386 | page=[https://archive.org/details/contemporaryabst00gall_1/page/111 111] | url=https://archive.org/details/contemporaryabst00gall_1 | url-access=registration | quote=verhoeff check digit. | access-date=August 26, 2011}}</ref>

}}

==External links==
{{Wikibooks|Algorithm_Implementation/Checksums/Verhoeff_Algorithm}}
* [http://www.cs.utsa.edu/~wagner/laws/verhoeff.html Detailed description of the Verhoeff algorithm]

{{DEFAULTSORT:Verhoeff Algorithm}}
[[Category:Modular arithmetic]]
[[Category:Checksum algorithms]]
[[Category:Error detection and correction]]
[[Category:1969 introductions]]