Add an extra reduction step to the end of RSAZ.

RSAZ has a very similar bug to mont5 from and may
return the modulus when it should return zero. As in that CL, there is
no security impact on our cryptographic primitives.

RSAZ is described in the paper "Software Implementation of Modular
Exponentiation, Using Advanced Vector Instructions Architectures".

The bug comes from RSAZ's use of "NRMM" or "Non Reduced Montgomery
Multiplication". This is like normal Montgomery multiplication, but
skips the final subtraction altogether (whereas mont5's AMM still
subtracts, but replaces MM's tigher bound with just the carry bit). This
would normally not be stable, but RSAZ picks a larger R > 4M, and
maintains looser bounds for modular arithmetic, a < 2M.

Lemma 1 from the paper proves that NRMM(a, b) preserves this 2M bound.
It also claims NRMM(a, 1) < M. That is, conversion out of Montgomery
form with NRMM is fully reduced. This second claim is wrong. The proof
shows that NRMM(a, 1) < 1/2 + M, which only implies NRMM(a, 1) <= M, not
NRMM(a, 1) < M. RSAZ relies on this to produce a reduced output (see
Figure 7 in the paper).

Thus, like mont5 with AMM, RSAZ may return the modulus when it should
return zero. Fix this by adding a bn_reduce_once_in_place call at the
end of the operation.

Change-Id: If28bc49ae8dfbfb43bea02af5ea10c4209a1c6e6
Reviewed-by: Adam Langley <>
Commit-Queue: David Benjamin <>
3 files changed
tree: 441dfa69117d7bc360f4153e20e5c2132c146e62
  1. .github/
  2. crypto/
  3. decrepit/
  4. fuzz/
  5. include/
  6. rust/
  7. ssl/
  8. third_party/
  9. tool/
  10. util/
  11. .clang-format
  12. .gitignore
  16. CMakeLists.txt
  17. codereview.settings
  20. go.mod
  21. go.sum
  24. OpenSSLConfig.cmake
  28. sources.cmake


BoringSSL is a fork of OpenSSL that is designed to meet Google's needs.

Although BoringSSL is an open source project, it is not intended for general use, as OpenSSL is. We don't recommend that third parties depend upon it. Doing so is likely to be frustrating because there are no guarantees of API or ABI stability.

Programs ship their own copies of BoringSSL when they use it and we update everything as needed when deciding to make API changes. This allows us to mostly avoid compromises in the name of compatibility. It works for us, but it may not work for you.

BoringSSL arose because Google used OpenSSL for many years in various ways and, over time, built up a large number of patches that were maintained while tracking upstream OpenSSL. As Google's product portfolio became more complex, more copies of OpenSSL sprung up and the effort involved in maintaining all these patches in multiple places was growing steadily.

Currently BoringSSL is the SSL library in Chrome/Chromium, Android (but it's not part of the NDK) and a number of other apps/programs.

Project links:

There are other files in this directory which might be helpful: