From c6278ceeab336169c72ef55b2c3399738bcc89be Mon Sep 17 00:00:00 2001 From: Christian Grothoff Date: Wed, 15 Jul 2020 22:25:06 +0200 Subject: document FDH implementation in detail, fixing #6182 --- doc/system/taler/implementation.tex | 27 ++++++++++++++++++++++++--- 1 file changed, 24 insertions(+), 3 deletions(-) (limited to 'doc/system/taler/implementation.tex') 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{http://eprint.iacr.org/2010/264.pdf}} 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 -- cgit v1.2.3