Every information I found on internet about RSA-CRT encryption/decryption uses only two primes. I'm interested in my project in doing that using multiple (up to 8) primes.

The general idea is to calculate $d_p = d\bmod(p-1)$, $d_q = d\bmod(q-1)$, and $q_\text{inv} = q^{-1}\bmod p$, where $p$ and $q$ are primes.

Encryption and decryption is based on "logical relations" between $p$ and $q$ and I'm unable to expand it to more than two primes. Can anybody explain how to use multiple primes?

  • 1
    $\begingroup$ RFC 3447, the IETF republication of PKCS1v2.1, has been on the internet since 2003 and specifies 'multi-prime' RSA (meaning more than 2 factors). Technically it was superseded by RFC8017 for v2.2 in 2016, but this content did not change. $\endgroup$ Sep 21 at 22:49

The RSA private-key operation (used for decryption and signature generation) amounts to solving for $x$ the equation $y\equiv x^e\pmod N$, knowing $y$, the factorization of the public modulus $N$ into $k\ge2$ distinct primes $N=r_1\dots r_k$, public exponent $e$ such that $\gcd(e,r_i-1)\ne1$, and that $0\le x<N$.

For an efficient implementation, we can solve this equation modulo each of the $r_i$; then use the CRT to combine solutions between products of moduli for which we already have a solution, until reaching a solution modulo $N$. The common way, implicit in PKCS#1v2 since version 2.1, is:

  • precompute the following quantities $d_i$ (the CRT exponents) and $t_i$ (the CRT inverses/coefficients), e.g. at key generation time, including the results in the private key:
    • for $i\in\{1,\dots,k\}$
      • $d_i\gets e^{-1}\bmod(r_i-1)$, or equivalently $d_i\gets d\bmod(r_i-1)$
    • $m\gets r_1$
    • for $i$ from $2$ to $k$ - $t_i\gets m^{-1}\bmod r_i$ - $m\gets m\cdot r_i$
  • when needing to use the private key and solve $y\equiv x^e\pmod N$
    • for $i\in\{1,\dots,k\}$ [note: should be parallelized if possible]
      • $x_i\gets(y\bmod r_i)^{d_i}\bmod r_i$
    • $x\gets x_1$, $m\gets r_1$
    • for $i$ from $2$ to $k$ [loop invariant: $0\le x<m$, $y\equiv x^e\pmod m$ ]
      • $x\gets x+m\cdot((x_i-x\bmod r_i)\cdot t_i\bmod r_i)$
      • $m\gets m\cdot r_i$

Correctness follows from the loop invariant. See this question for attribution. See this other one for how the bitsize of $N$ relates to a maximum reasonable number of primes. This is known as Garner’s algorithm, see the Handbook of Applied Cryptography, section 14.5.2.

Artificially small example with 3 primes:

r1=931164518537359 r2=944727352543879 r3=982273258722607
d1=558698711122415 d2=566836411526327 d3=785818606978085
                   t2=360227672914825 t3=882117903741868
x1=436496882968258 x2=903092574358267 x3=806961802724
x=710532117316769399313215266414 (when i=2)

The effort saved compared to a standard (non-CRT) implementation is by a factor at most (and near) $k^2$, if modular multiplication has cost $\mathcal O(n^2)$ for arguments of $n$ bits. The time saved can be higher, up to a factor at most (and near) $k^3$ if parallelization is used on $k$ independent modexp units.

It is critical to make a final check that $y\equiv x^e\pmod N$, and not disclose $x$ otherwise. If this precaution was not taken, the implementation would be vulnerable to the cardinal "Bellcore" fault attack: D. Boneh, R. A. DeMillo, R. Lipton; On the Importance of Eliminating Errors in Cryptographic Computations (in Journal of Cryptology 14(2), 2001).

Implementations should be adequately protected from a variety of other attacks, including timing, power analysis, and other side-channel attacks.

The question also mentions encryption, where only the public key $(N,e)$ is known, not the factorization of $N$. Hence, for that RSA public-key operation (also used for signature verification), there is no similar shortcut applying to the computation $y\gets x^e\bmod N$. However, typically, that remains of low cost compared to the RSA private-key operation, because $e$ typically is small.

Late addition: $e$ must be coprime with each of the $r_i-1$. Typically it's first chosen an odd $e$, and the $r_i$ to match this condition. The range for $e$ is a subject of debate, see discussion here and here. My opinion is that implementation attacks aside, and with padding having a security proof (RSA-KEM, RSAES-OAEP, RSASSA-PSS, ISO/IEC 9796-2 schemes 2 or 3), there's no good reason for a minimum $e$ larger than $3$; and one will not get fired for using $e=2^{16}+1$, which matches the prescription $2^{16}<e<2^{256}$ of FIPS 186-4, and is a Fermat number $F_i=2^{\left(2^i\right)}+1$ which allows the best efficiency for a given size of $e$, and prime which makes choices of $r_i$ slightly easier and in a wider set.


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