# Decimal encryption

Greg Rose ggr at qualcomm.com
Wed Aug 27 12:34:15 EDT 2008

```Philipp Gühring wrote:
> Hi,

G'day Philipp,

> I am searching for symmetric encryption algorithms for decimal strings.
>
> Let's say we have various 40-digit decimal numbers:
> 2349823966232362361233845734628834823823
> 3250920019325023523623692235235728239462
> 0198230198519248209721383748374928601923
>
> As far as I calculated, a decimal has the equivalent of about 3,3219
> bits, so with 40 digits, we have about 132,877 bits.
>
> Now I would like to encrypt those numbers in a way that the result is a
> decimal number again (that's one of the basic rules of symmetric
> encryption algorithms as far as I remember).
>
> Since the 132,877 bits is similar to 128 bit encryption (like eg. AES),
> I would like to use an algorithm with a somewhat comparable strength to AES.
> But the problem is that I have 132,877 bits, not 128 bits. And I can't
> cut it off or enhance it, since the result has to be a 40 digit decimal
> number again.
>
> Does anyone know a an algorithm that has reasonable strength and is able
> to operate on non-binary data? Preferrably on any chosen number-base?

There is a fairly standard technique for handling things like this.

1. encode your number N into a 133-bit string S
2. encrypt S with your favourite 133-bit block cipher (see below)
3. decode S to a number N'
4. if N' >= 10^40, goto 2 (that is, re-encrypt until it is in range)

This is uniquely invertible, although a little slow (since on average it
will take 8.9% or so more encryptions because of the inner loop, and
some side-channel information leaks when it does the extra encryptions.
Decryption is exactly the same operation except step 2 uses decryption
mode of the block cipher.

So, you don't have a 133-bit block cipher lying around? No worries, I'll
sell you one ;-). Actually that is easy too. Take a trustworthy 128-bit
block cipher like AES. To encrypt, do:

1. Encrypt the first 128 bits (ECB mode)
2. Encrypt the last 128 bits (also ECB mode).

To decrypt, do decryptions in the reverse order, obviously. It's easy to
see that this is a secure permutation if AES itself is, depending on
your definition of secure; if you add a third step, to re-encrypt the
first 128 bits, it is truly secure. (Without the third step, tweaking a
bit in the first 5 bits will often leave the last 5 unchanged on
decryption, which is clearly a distinguishing attack; the third
encryption makes it an all-or-nothing transform.)

I believe the above gives you a secure enough block cipher on 40 digit
strings, and if you only ever encrypt single chunks, ECB mode should be
fine... of course that depends on the real threat analysis of your
system. It does about 2.19 AES encryptions per 40 digits, should be fast
enough.

hope that helps,
Greg.

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