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<p>Carry bugs are fairly common, and usually too small to have big impact, or so they are considered. This one was no exception.</p>
<p><a href="https://github.com/golang/go/issues/20040">Go issue #20040</a> affected the optimized x86_64 assembly implementation of scalar multiplication on the NIST P-256 elliptic curve in the standard library.</p>
<p><code>p256SubInternal</code> computes <code>x - y mod p</code>. In order to be constant time it has to do both the math for <code>x >= y</code> and for <code>x < y</code>, it then chooses the result based on the carry bit of <code>x - y</code>. The old code chose wrong (<code>CMOVQNE</code> vs <code>CMOVQEQ</code>), but most of the times compensated by adding a carry bit that didn't belong in there (<code>ADCQ</code> vs <code>ANDQ</code>). Except when it didn't, once in a billion times (when <code>x - y < 2^256 - p</code>). <a href="https://github.com/golang/go/commit/9294fa2749ffee7edbbb817a0ef9fe633136fa9c">The whole patch is 5 lines.</a></p>
<p>The bug was found by a Cloudflare engineer because it caused ECDSA verifications to fail erroneously but the security impact was initially unclear. We devised an adaptive bug attack that can recover a scalar input to <code>ScalarMult</code> by submitting attacker-controlled points and checking if the result is correct. Elliptic Curve Diffie-Hellman involves a secret scalar, a peer-provided point, and fails to establish a key if the result is incorrect.</p>
<p>We reported this to the Go team, Go 1.7.6 and 1.8.2 were issued and the vulnerability was assigned <a href="https://cve.mitre.org/cgi-bin/cvename.cgi?name=CVE-2017-8932">CVE-2017-8932</a>.</p>
<p>At a high level, this P-256 ScalarMult implementation processes the scalar in blocks of 5 bits. We can precompute points that trigger the bug for each specific 5 bit value, and submit them. When the protocol fails, we learned 5 key bits, and we move on to the next 5, Hollywood style. In about 500 submissions on average we recover the whole key.</p>
<p>The precomputation involves a lot of unusable points and edge cases, but by modifying the optimized assembly implementation and generating points intelligently, we can produce a full round of points in seconds on 1000 machines (or spot instances). Each round depends on the previous ones, so must be computed live during each attack.</p>
<p>Normal ECDH does not offer an attacker multiple attempts against the same scalar, making the attack impossible. However, a variant of ECDH with a static scalar is used as a public key encryption scheme, for example in JSON Web Encryption. The attack can fully recover the private key in that scenario.</p>
<p>No bug is small enough.</p>