From 3226a5e2032e93421c432781dc5054f211717dc2 Mon Sep 17 00:00:00 2001 From: Youssef El Housni Date: Mon, 1 Jun 2026 17:29:45 -0400 Subject: [PATCH 1/4] Fix silent (0, 0) return in Fp2 sqrt for purely-real non-residue inputs --- .../math/extension_fields/square_root_fp2.nim | 51 +++++++++++++++++++ tests/math_extension_fields/t_fp2_sqrt.nim | 45 ++++++++++++++++ 2 files changed, 96 insertions(+) diff --git a/constantine/math/extension_fields/square_root_fp2.nim b/constantine/math/extension_fields/square_root_fp2.nim index 00ebc1d92..94981b67c 100644 --- a/constantine/math/extension_fields/square_root_fp2.nim +++ b/constantine/math/extension_fields/square_root_fp2.nim @@ -157,8 +157,22 @@ func sqrt_if_square_generic(a: var Fp2): SecretBool = # Gora Adj, Francisco Rodríguez-Henríquez, 2012, # https://eprint.iacr.org/2012/685 # Made constant-time and optimized to fuse sqrt and inverse sqrt + # + # Note on the purely-real-input edge case: + # Adj-Rodríguez Algorithm 8 (and Scott §6.3) has an unstated precondition + # that a.c1 ≠ 0. For input (a0, 0) with a0 a non-residue in Fp, the + # algorithm silently picks t2 = 0 below, sqrt_invsqrt(0) = (0, 0), and the + # function reports `result = true` with `a` rewritten to (0, 0). But + # (a0, 0) is *always* a square in Fp² (a0^((p²−1)/2) = (a0^(p−1))^((p+1)/2) + # = 1), and the correct sqrt is (0, sqrt(a0 / β)) where β = u² is the + # quadratic non-residue defining Fp². We compute that fallback + # unconditionally in constant time and ccopy it in iff a.c1 was zero. var t1{.noInit.}, t2{.noInit.}, t3{.noInit.}: typeof(a.c0) + # Save what we need for the purely-real-input fallback before mutating a. + let a0_orig = a.c0 + let a1_isZero = a.c1.isZero() + t1.square(a.c0) # a0² t2.square(a.c1) # - β a1² with β = 𝑖² in a complex extension field when a.fromComplexExtension(): @@ -185,6 +199,43 @@ func sqrt_if_square_generic(a: var Fp2): SecretBool = t3 *= a.c1 a.c1.ccopy(t3, result) + # Purely-real-input fallback (see the comment at the top of this function). + # Compute the two candidate purely-real roots of (a0_orig, 0): + # - candidateA = (sqrt_fp(a0), 0) valid iff a0 is a QR in Fp + # - candidateB = (0, sqrt_fp(a0 / β)) valid iff a0/β is a QR in Fp + # For any nonzero a0 in Fp, at least one of {a0, a0/β} is a QR (since β + # itself is a non-residue: QR × non-QR = non-QR, non-QR × non-QR = QR). + # We always run both candidate computations for constant time, then + # constant-time-select. + var fbC0{.noInit.}, fbC1{.noInit.}: typeof(a.c0) + fbC0.setZero() + fbC1.setZero() + + var candA = a0_orig + let candAOk = candA.sqrt_if_square() # candA = sqrt(a0) iff a0 ∈ QR(Fp) + fbC0.ccopy(candA, candAOk) + + var candB = a0_orig + when a.fromComplexExtension(): + # β = −1, so a0/β = −a0. Compute candidateB = (0, sqrt(−a0)). + candB.neg() + else: + # General case: compute 1/β via the NonResidue marker. + var betaInv{.noInit.}: typeof(a.c0) + betaInv.setOne() + betaInv *= NonResidue # = β + betaInv.inv() # = 1/β + candB *= betaInv # = a0/β + let candBOk = candB.sqrt_if_square() # candB = sqrt(a0/β) iff QR + # Use candidateB iff candidateA was not a QR but candidateB is. + fbC1.ccopy(candB, (not candAOk) and candBOk) + + # Override the output with the fallback iff the input was purely-real. + a.c0.ccopy(fbC0, a1_isZero) + a.c1.ccopy(fbC1, a1_isZero) + # The result flag is already correct: for (a0, 0) with a0 ≠ 0 the norm + # a0² is a square in Fp, so `result = true` was already set above. + func sqrt_if_square*(a: var Fp2): SecretBool = ## If ``a`` is a square, compute the square root of ``a`` ## if not, ``a`` is unmodified. diff --git a/tests/math_extension_fields/t_fp2_sqrt.nim b/tests/math_extension_fields/t_fp2_sqrt.nim index 427826539..c934987c3 100644 --- a/tests/math_extension_fields/t_fp2_sqrt.nim +++ b/tests/math_extension_fields/t_fp2_sqrt.nim @@ -69,6 +69,44 @@ proc randomSqrtCheck(Name: static Algebra, gen: RandomGen) = # bool(r == s) bool(s == a or s == na) +proc purelyRealNonResidueSqrtCheck(Name: static Algebra) = + ## Regression test for the silent-failure bug in sqrt_if_square on + ## purely-real inputs whose Fp coordinate is a non-residue. + ## + ## For any nonzero a0 ∈ Fp, the element (a0, 0) is always a square in Fp² + ## (Euler's criterion: a0^((p²−1)/2) = (a0^(p−1))^((p+1)/2) = 1). When a0 is + ## a non-residue in Fp, the actual square root lies in Fp² \ Fp and has the + ## shape (0, sqrt_fp(a0/β)) where β = u² is the QNR defining Fp². + ## + ## Adj-Rodríguez Algorithm 8 ("complex method", ePrint 2012/685) and Scott + ## §6.3 (ePrint 2020/1497) implicitly assume a.c1 ≠ 0 and silently return + ## (0, 0) on this stratum. The optimized path used by other curves + ## (sqrt_if_square_opt with rotation extension) handles it correctly via + ## the alignment step, but BLS12_377 cannot use it (sqrt of QNR is not in + ## Fp²) and was therefore vulnerable on this input shape. + for v in 2'u32 ..< 20'u32: + var x: Fp2[Name] + x.fromUint(v) # (v, 0) + if bool(x.c0.isSquare()): + continue # only test non-QR a0 values + + # (non-QR, 0) is *always* a square in Fp² + check: bool x.isSquare() + + var root = x + let ok = root.sqrt_if_square() + check: bool ok + + var sq = root + sq.square() + check: bool(sq == x) + check: not bool(root.isZero()) + return + + # If we reach here, the search range exhausted without finding a non-QR, + # which is statistically impossible for any reasonable p. + doAssert false, "no small non-residue found in Fp[" & $Name & "]" + proc main() = suite "Modular square root" & " [" & $WordBitWidth & "-bit words]": staticFor(curve, TestCurves): @@ -77,6 +115,13 @@ proc main() = randomSqrtCheck(curve, gen = HighHammingWeight) randomSqrtCheck(curve, gen = Long01Sequence) + test "[𝔽p2] (non-QR, 0) sqrt regression for " & $curve: + # Exercises the purely-real-input edge case that the Adj-Rodríguez + # Algorithm 8 complex method (used by BLS12_377) silently mishandled + # prior to the purely-real fallback added to sqrt_if_square_generic. + # Pre-fix behaviour on BLS12_377: returns (0, 0) with `ok = true`. + purelyRealNonResidueSqrtCheck(curve) + suite "Modular square root - 32-bit bugs highlighted by property-based testing " & " [" & $WordBitWidth & "-bit words]": test "sqrt_if_square invalid square BLS12_381 - #64": var a: Fp2[BLS12_381] From 29657490295b34eb768594820b978f6e2f0496d8 Mon Sep 17 00:00:00 2001 From: Youssef El Housni Date: Mon, 1 Jun 2026 17:53:57 -0400 Subject: [PATCH 2/4] regression loop covering every NQR in [2,20) per curve --- tests/math_extension_fields/t_fp2_sqrt.nim | 7 +++---- 1 file changed, 3 insertions(+), 4 deletions(-) diff --git a/tests/math_extension_fields/t_fp2_sqrt.nim b/tests/math_extension_fields/t_fp2_sqrt.nim index c934987c3..d61e4b108 100644 --- a/tests/math_extension_fields/t_fp2_sqrt.nim +++ b/tests/math_extension_fields/t_fp2_sqrt.nim @@ -84,6 +84,7 @@ proc purelyRealNonResidueSqrtCheck(Name: static Algebra) = ## (sqrt_if_square_opt with rotation extension) handles it correctly via ## the alignment step, but BLS12_377 cannot use it (sqrt of QNR is not in ## Fp²) and was therefore vulnerable on this input shape. + var tested = 0 for v in 2'u32 ..< 20'u32: var x: Fp2[Name] x.fromUint(v) # (v, 0) @@ -101,11 +102,9 @@ proc purelyRealNonResidueSqrtCheck(Name: static Algebra) = sq.square() check: bool(sq == x) check: not bool(root.isZero()) - return + inc tested - # If we reach here, the search range exhausted without finding a non-QR, - # which is statistically impossible for any reasonable p. - doAssert false, "no small non-residue found in Fp[" & $Name & "]" + doAssert tested > 0, "no small non-residue found in Fp[" & $Name & "]" proc main() = suite "Modular square root" & " [" & $WordBitWidth & "-bit words]": From 6e021f4ec1a4a9037e28e856e04e2bf6db9424c8 Mon Sep 17 00:00:00 2001 From: Youssef El Housni Date: Mon, 1 Jun 2026 17:56:23 -0400 Subject: [PATCH 3/4] use sqrt_ratio_if_square --- .../math/extension_fields/square_root_fp2.nim | 19 ++++++++++-------- tests/math_extension_fields/t_fp2_sqrt | Bin 0 -> 266232 bytes 2 files changed, 11 insertions(+), 8 deletions(-) create mode 100755 tests/math_extension_fields/t_fp2_sqrt diff --git a/constantine/math/extension_fields/square_root_fp2.nim b/constantine/math/extension_fields/square_root_fp2.nim index 94981b67c..656c8b73c 100644 --- a/constantine/math/extension_fields/square_root_fp2.nim +++ b/constantine/math/extension_fields/square_root_fp2.nim @@ -215,18 +215,21 @@ func sqrt_if_square_generic(a: var Fp2): SecretBool = let candAOk = candA.sqrt_if_square() # candA = sqrt(a0) iff a0 ∈ QR(Fp) fbC0.ccopy(candA, candAOk) - var candB = a0_orig + var candB{.noInit.}: typeof(a.c0) + var candBOk{.noInit.}: SecretBool when a.fromComplexExtension(): # β = −1, so a0/β = −a0. Compute candidateB = (0, sqrt(−a0)). + candB = a0_orig candB.neg() + candBOk = candB.sqrt_if_square() else: - # General case: compute 1/β via the NonResidue marker. - var betaInv{.noInit.}: typeof(a.c0) - betaInv.setOne() - betaInv *= NonResidue # = β - betaInv.inv() # = 1/β - candB *= betaInv # = a0/β - let candBOk = candB.sqrt_if_square() # candB = sqrt(a0/β) iff QR + # General case: use sqrt_ratio_if_square to compute √(a0/β) without an + # explicit field inversion of β (the fused routine handles the ratio + # via a single invsqrt, saving ~70-100 Fp muls per call). + var beta{.noInit.}: typeof(a.c0) + beta.setOne() + beta *= NonResidue # beta = β + candBOk = candB.sqrt_ratio_if_square(a0_orig, beta) # Use candidateB iff candidateA was not a QR but candidateB is. fbC1.ccopy(candB, (not candAOk) and candBOk) diff --git a/tests/math_extension_fields/t_fp2_sqrt b/tests/math_extension_fields/t_fp2_sqrt new file mode 100755 index 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Mon, 1 Jun 2026 18:00:03 -0400 Subject: [PATCH 4/4] guard both ccopies --- .../math/extension_fields/square_root_fp2.nim | 12 +++++++----- tests/math_extension_fields/t_fp2_sqrt | Bin 266232 -> 0 bytes 2 files changed, 7 insertions(+), 5 deletions(-) delete mode 100755 tests/math_extension_fields/t_fp2_sqrt diff --git a/constantine/math/extension_fields/square_root_fp2.nim b/constantine/math/extension_fields/square_root_fp2.nim index 656c8b73c..444e2a922 100644 --- a/constantine/math/extension_fields/square_root_fp2.nim +++ b/constantine/math/extension_fields/square_root_fp2.nim @@ -233,11 +233,13 @@ func sqrt_if_square_generic(a: var Fp2): SecretBool = # Use candidateB iff candidateA was not a QR but candidateB is. fbC1.ccopy(candB, (not candAOk) and candBOk) - # Override the output with the fallback iff the input was purely-real. - a.c0.ccopy(fbC0, a1_isZero) - a.c1.ccopy(fbC1, a1_isZero) - # The result flag is already correct: for (a0, 0) with a0 ≠ 0 the norm - # a0² is a square in Fp, so `result = true` was already set above. + # Override the output with the fallback iff the input was purely-real and + # is a square. The `and result` guard upholds the "a unmodified on failure" + # contract; for (a0, 0) with a0 ≠ 0 the norm a0² is a square in Fp so + # `result` is always true here, but the guard makes that explicit. + let useFallback = a1_isZero and result + a.c0.ccopy(fbC0, useFallback) + a.c1.ccopy(fbC1, useFallback) func sqrt_if_square*(a: var Fp2): SecretBool = ## If ``a`` is a square, compute the square root of ``a`` diff --git a/tests/math_extension_fields/t_fp2_sqrt b/tests/math_extension_fields/t_fp2_sqrt deleted file mode 100755 index d045c47b1945115fac69fed86fc1a2d611f5ef6b..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 266232 zcmc${3wTu3x$wXCOhWcd5^kB05U5GO%S^xvhf5L*W)e_Cz)NF#^Rx|ksguOZhoV9O zlSr*01L^>3oAd}?+LIZiM=hw((@)~%G$e{3*4m!))qwVENW29RGAQQzy?f72Cc^}= z_WA$wJd>H(d+oK>d%vuAt$F?U2VbNpr78YwyvFh>9ih}cYBQgd8qcegS7qgu#g|@u z^~L2^$km~L#=}D|o%Bumd7Flnl@%9%uVUzJUw+&7(w!a7NHT&B&*=#T06C%&Xd!xVGmfl)*^9_-kvHe}S#q95b$fV?Z^xx2L 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