We do not labour the advantages of FPE. For certain applications it is clearly nice
that the ciphertext has exactly the same form as the plaintext. Clearly the
ciphertext can be stored and displayed in exactly the same format as the
plaintext, and no special provision needs to be made for it. In legacy database
applications a field for data which now needs to be encrypted, will not require any
However getting FPE right is quite tricky, as the size of the plaintext can often be
quite small. For each of the methods considered the authors have gone to some length
to ensure the security of their solutions.
Each mode is based on a block cipher, and this can be chosen at will. Here we
have used the Advanced Encryption Standard (AES) as the basis for all of
our implementations. In the “real world” we imagine that most would do
The FFX mode from Bellare, Rogaway and Spies is probably the most
complex, and clearly a lot of thought has gone into it. It has been through
several iterations, the latest being FFX[Radix]. It is patented. Test vectors are
The BPS mode from Brier, Peyrin and Stern is unpatented. It comes in for some
criticism from Bellare et al. for the way in which it deals with longer strings – in this
case it is not a PRP – a pseudo-random permutation. However as they themselves
point out long strings may not be appropriate for these FPE modes anyway, and BPS
using the AES can for example handle strings of up to 56 decimal digits without
invoking its longer strings mechanism, so in practice this may not be an issue. There
are no test vectors provided.
The VFPE is a method invented by the Visa corporation, and is patented. It is
the simplest mode considered here. There are no test vectors provided.
We assume that all of the proposals are equally secure (although this may not be
the case – but such comparison is beyond the scope of this note).
2 General Implementation Issues
The absence of test vectors in two out of three cases is to be severely deprecated.
Implementation from the provided descriptions is difficult enough even when one has
test vectors to ensure correctness at each step. Without them it is impossible to be
100% sure that the implementation is correct. One common issue is how to present
input for AES encryption. If a number is to be encrypted, should it be presented to
the block cipher in big or little endian form? For example the AES accepts input as a
block of 16 bytes, B, B etc. If the number 1 is to be encrypted should the AES
input have B=1 or B=1? It would help considerably if authors were explicit
3 Format Preserving Encryption
One of the challenges to be faced by the designer of an FPE mode is that the
input strings can be quite small. Now the reason that a block cipher has a
relatively large block size of typically 64 or 128 bits is to prevent a code book
attack. Recall that a block cipher can be visualised as a massive fixed (and
reversible) look-up table whereby the output ciphertext is a pseudo-random
permutation of the input plaintext. There is a completely different look-up table for
each possible key. Reductio ad absurdam – consider now a 1-bit block cipher
with a 128-bit key. Clearly a 0 will consistently encipher as a 0 or a 1 for a
fixed key. With one known bit of input and output the complete two-entry
look-up table can be established and the system is broken – an attacker
simply looks up each bit of ciphertext in the look-up table and ignores the
Therefore to work with a small block size a new idea is needed. One common idea
is to make the block cipher “tweakable”. Each time a tweakable block cipher is
tweaked, it in effect becomes a completely different block cipher. So if each encryption
with the same key uses a different tweak, the result is a completely different
ciphertext, and code book attacks are frustrated.
All of the modes considered here are tweakable, although VFPE achieves
the same functionality by changing the counter mode initialisation vector
(IV). As the strings to be encrypted get longer this tweakability becomes less
4 FFX mode
Although test vectors are provided, it was a struggle to interpret them. The first test
vector appears to have been transcribed incorrectly. For example the provided P
vector has only 15 entries – it should have 16. The input of length 10 is
0123456789. This converts to binary (in big endian form) as the two bytes 231,223.
However test vector 2 has the same input, which this time converts to binary as
221,213. The latter appears correct to us, so we quickly gave up on test vector
Test vector 2 worked smoothly. Test vector 3 included a tweak. However the tweak
is handled as an ascii string of characters rather than as base 10 numbers as is the
case for the input. Test Vector 4 is the only one to have an uneven input size. This is
a case that needs to be handled carefully, as this situation can easily arise in practise.
In fact the decryption algorithm as described in the paper is rather misleading. When
the input is split A is the smaller part and B the bigger part – bigger by one.
Then inside the loop A is assigned to B. This leaves the most significant
element of B unassigned. Then A is updated. At this stage the most significant
element of B should be updated from the most significant element of the new
Test vector 5 uses base 36, and the input is represented using the digits 0-9
followed by A-Z. These need to be carefully converted from ascii to base 36 numbers.
With its larger input and tweak, this test requires an extra invocation of the AES
algorithm when internally calculating the CBC-MAC.
Our implementation optimises as suggested by the authors by encrypting the
CBC-MAC IV, the vector P, just once across all of the rounds. The authors appear
concerned that “When the radix is not a power of two, the mod at line 38 may
require significant attention”. We were puzzled by this remark, as we found this not
to be an issue.
5 BPS mode
This mode is quite similar to FFX in that it also uses a Fesitel structure. Its advent
seems to have motivated some of the recent changes to FFX. It was quite
straightforward to implement, the only major issue being the lack of test vectors.
There appears to be a small typo in the second last line of Algorithm 3, where
There is the same problem in algorithm 4.
The CBC mechanism for handling longer strings may raise doubts, but it is
certainly convenient and requires less memory than FFX.
6 VFPE mode
This mode was probably the simplest to implement. It has a parameter
which is basically the number of digits to the chosen base
encrypted from each invocation of the AES. This should be a little less that the maximum possible,
as for larger
more ciphertext blocks might need to be rejected as they may introduce a bias. Blocks of
digits are generated using the AES in counter mode, which are then added modulo
to each digit
in the plaintext. The same random digits are subtracted to restore the plaintext on decryption.
For base and
using the AES,
appears to be a good choice.
The only tricky part is determining the threshold for acceptability of AES ciphertexts. They
must be less than .
On the face of it, it might appear to require multi-precision
arithmetic to calculate this threshold. The trick is to convert
division by .
Then the threshold is simply this number with its last
digits set to zero. Note that the AES ciphertext must be converted to base
anyway for use in the encryption process, and comparison with the adjusted threshold
is now trivial.
The counter used in the AES counter mode must be initialised externally. If the plaintext is
digits, as many invocations of the AES as are needed are carried out, incrementing
the counter for each fresh block.
Note that VFPE is “malleable”, as it alone is based on a stream-cipher mode and
does not use the Feistel structure of the other two modes. This makes it open to an
active known plaintext substitution attack, and so some extra mechanism (typically a
MAC) must be deployed to prevent this.
The timings will be roughly preportional to the number of AES encryptions required,
and this in turn will be proportional to the number of digits in the plaintext and the
number base. For FFX and BPS we assume a 64-bit tweak. We first count the number
of AES encryptions in Table 1.
Table 1: Number of AES calls vs Number of decimal digits
As can be seen FFX appears to be the most expensive, with VFPE the cheapest,
with BPS in the middle
Actual timings in clock cycles were obtained using MIRACL’s internal AES code
(which does not use Intel’s special AES instructions). The code was run on a single
core of an Intel i5 520M, clocked at 2.4GHz. The time required for once-off AES key
scheduling is not included.
Table 2: Clock cycles vs Number of decimal digits
Somewhat to our surprise the FFX code ties for speed with BPS for smaller
input sizes. This is despite FFX using more Feistel rounds (10 as against
Overall BPS mode would be our preference. It represents a compromise between the
other two modes in terms of performance, and its unpatented status is a big
attraction. Its method of dealing with longer strings may not be ideal, but in practise
in the kind of applications where FPE is likely to be deployed, this will not be an
issue. Recall that these modes can be used with any block cipher, and a block cipher
with a larger block size is to be preferred with BPS. Here we note that the
original Rijndael (which became the AES) can be used with a block size of
256 bits, which might be an option for BPS if it must be used with longer