diff options
authorChristian Grothoff <>2020-07-15 22:25:06 +0200
committerChristian Grothoff <>2020-07-15 22:25:06 +0200
commitc6278ceeab336169c72ef55b2c3399738bcc89be (patch)
parent9e92cc608932abc6caf53dce7cd96984f793f4e3 (diff)
document FDH implementation in detail, fixing #6182
1 files changed, 24 insertions, 3 deletions
diff --git a/doc/system/taler/implementation.tex b/doc/system/taler/implementation.tex
index 973e9789..4b095b77 100644
--- a/doc/system/taler/implementation.tex
+++ b/doc/system/taler/implementation.tex
@@ -1536,9 +1536,30 @@ We write $\mathbb{Z}^*_N$ for the multiplicative group of integers modulo $N$.
Given an $r \in \mathbb{Z}^*_N$, we write $r^{-1}$ for the multiplicative
inverse modulo $N$ of $r$.
-We write $H(m)$ for the SHA-512 hash of a bit string,
-and $\FDH(N,m)$ for the full domain hash that maps the bit string $m$ to an element
-of $\mathbb{Z}^*_N$.
+We write $H(m)$ for the SHA-512 hash of a bit string.
+We write $\FDH(N,m)$ for the full domain hash that maps the bit string $m$ to
+an element of $\mathbb{Z}^*_N$. Specifically, $\FDH(N,m)$ is computed by
+first computing $H(m)$. Let $b := \lceil \log_2 N\rceil$. The full domain
+hash is then computed by iteratively computing a HKDF to obtain $b$ bits of
+output until the $b$-bit value is below $N$. The inputs to the HKDF are a
+``secret key'', a fixed context plus a 16-bit counter (in big endian) as a
+context chunk that is incremented until the computation succeeds. For the
+source key material, we use a binary encoding of the public RSA key with
+modulus $N$.\footnote{So technically, it is $\FDH(N,e,m)$, but we use the
+ simplified notation $\FDH(N,m)$.} Here, the public RSA key is encoded by
+first expressing the number of bits of the modulus and the public exponent as
+16-bit numbers in big endian, followed by the two numbers (again in unsigned
+big endian encoding).\footnote{See
+ \texttt{GNUNET\_CRYPTO\_rsa\_public\_key\_encode()}.} For the context, the
+C-string ``RSA-FDA FTpsW!'' (without 0-termination) is used. For the KDF, we
+instantiate the HKDF described in RFC 5869~\cite{rfc5869} using HMAC-SHA512 as
+XTR and HMAC-SHA256 as PRF*.\footnote{As suggested in
+ \url{}} Let the result of the first
+successful iteration of the HKDF function be $r$ with $0 \le r < N$. Then, to
+protect against a malicious exchange when blinding values, the $FDH(N,m)$
+function checks that $\gcd(r,n) = 1$. If not, the $\FDH(n,m)$ calculation
+fails because $n$ is determined to be malicious.
The expression $x \randsel X$ denotes uniform random selection of an element
$x$ from set $X$. We use $\algo{SelectSeeded}(s, X) \mapsto x$ for pseudo-random uniform