Compendium of pairing-based cryptography optimizations
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# Optimizations
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This document lists the optimizations relevant to an elliptic curve or pairing-based cryptography library and whether Constantine has them implemented.
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The optimizations can be of algebraic, algorithmic or "implementation details" nature. Using non-constant time code is always possible, it is listed if the speedup is significant.
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## Big Integers
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- Conditional copy
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- [x] Loop unrolling
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- [x] x86: Conditional Mov
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- [x] x86: Full Assembly implementation
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- [ ] SIMD instructions
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- Add/Sub
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- [x] int128
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- [x] add-with-carry, sub-with-borrow intrinsics
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- [x] loop unrolling
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- [x] x86: Full Assembly implementation
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- Multiplication
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- [x] int128
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- [x] loop unrolling
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- [x] Comba multiplication / product Scanning
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- [ ] Karatsuba
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- [ ] Karatsuba + Comba
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- [x] x86: Full Assembly implementation
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- [x] x86: MULX, ADCX, ADOX instructions
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- [x] Fused Multiply + Shift-right by word (for Barrett Reduction and approximating multiplication by fractional constant)
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- Squaring
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- [ ] Dedicated squaring functions
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- [ ] int128
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- [ ] loop unrolling
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- [ ] x86: Full Assembly implementation
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- [ ] x86: MULX, ADCX, ADOX instructions
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## Finite Fields & Modular Arithmetic
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- Representation
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- [x] Montgomery Representation
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- [ ] Barret Reduction
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- [ ] Unsaturated Representation
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- [ ] Mersenne Prime (2^k - 1),
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- [ ] Generalized Mersenne Prime (NIST Prime P256: 2^256 - 2^224 + 2^192 + 2^96 - 1)
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- [ ] Pseudo-Mersenne Prime (2^m - k for example Curve25519: 2^255 - 19)
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- [ ] Golden Primes (φ^2 - φ - 1 with φ = 2^k for example Ed448-Goldilocks: 2^448 - 2^224 - 1)
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- [ ] any prime modulus (lazy carry)
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- Montgomery Reduction
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- [x] int128
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- [x] loop unrolling
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- [x] x86: Full Assembly implementation
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- [x] x86: MULX, ADCX, ADOX instructions
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- Addition/substraction
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- [x] int128
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- [x] add-with-carry, sub-with-borrow intrinsics
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- [x] loop unrolling
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- [x] x86: Full Assembly implementation
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- [x] Addition-chain for small constants
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- Montgomery Multiplication
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- [x] Fused multiply + reduce
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- [x] int128
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- [x] loop unrolling
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- [x] x86: Full Assembly implementation
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- [x] x86: MULX, ADCX, ADOX instructions
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- [x] no-carry optimization for CIOS (Coarsely Integrated Operand Scanning)
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- [x] FIPS (Finely Integrated Operand Scanning)
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- Montgomery Squaring
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- [ ] Dedicated squaring functions
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- [ ] Fused multiply + reduce
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- [ ] int128
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- [ ] loop unrolling
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- [ ] x86: Full Assembly implementation
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- [ ] x86: MULX, ADCX, ADOX instructions
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- [ ] no-carry optimization for CIOS (Coarsely Integrated Operand Scanning)
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- Exponentiation
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- [x] variable-time exponentiation
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- [x] fixed window optimization _(sliding windows are not constant-time)_
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- [ ] NAF recoding
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- [ ] windowed-NAF recoding
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- [ ] SIMD vectorized select in window algorithm
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- [ ] Almost Montgomery Multiplication, https://eprint.iacr.org/2011/239.pdf
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- [ ] Pippenger multi-exponentiation (variable-time)
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- [ ] parallelized Pippenger
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- Inversion (constant-time baseline, Little-Fermat inversion via a^(p-2))
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- [x] Constant-time binary GCD algorithm by Möller, algorithm 5 in https://link.springer.com/content/pdf/10.1007%2F978-3-642-40588-4_10.pdf
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- [x] Addition-chain for a^(p-2)
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- [ ] Constant-time binary GCD algorithm by Bernstein-Young, https://eprint.iacr.org/2019/266
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- [ ] Constant-time binary GCD algorithm by Pornin, https://eprint.iacr.org/2020/972
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- [ ] Simultaneous inversion
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- Square Root (constant-time)
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- [x] baseline sqrt via Little-Fermat for `p ≡ 3 (mod 4)`
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- [ ] baseline sqrt via Little-Fermat for `p ≡ 5 (mod 8)`
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- [ ] baseline sqrt via Little-Fermat for `p ≡ 9 (mod 16)`
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- [x] baseline sqrt via Tonelli-Shanks for any prime.
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- [x] sqrt via addition-chain
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- [x] Fused sqrt + testIfSquare (Euler Criterion or Legendre symbol or Kronecker symbol)
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- [x] Fused sqrt + 1/sqrt
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- [x] Fused sqrt + 1/sqrt + testIfSquare
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## Extension Fields
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- [ ] Lazy reduction via double-width base fields
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- [x] Sparse multiplication
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- Fp2
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- [x] complex multiplication
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- [x] complex squaring
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- [x] sqrt via the constant-time complex method (Adj et al)
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- [ ] sqrt using addition chain
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- [x] fused complex method sqrt by rotating in complex plane
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- Cubic extension fields
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- [x] Toom-Cook polynomial multiplication (Chung-Hasan)
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## Elliptic curve
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- Weierstrass curves:
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- [x] Affine coordinates
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- [x] Homogeneous projective coordinates
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- [x] Projective complete formulae
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- [x] Mixed addition
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- [x] Jacobian projective coordinates
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- [x] Jacobian complete formulae
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- [x] Mixed addition
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- [ ] Conjugate Mixed Addition
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- [ ] Composites Double-Add 2P+Q, tripling, quadrupling, quintupling, octupling
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- [x] scalar multiplication
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- [x] fixed window optimization
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- [ ] constant-time NAF recoding
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- [ ] constant-time windowed-NAF recoding
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- [ ] SIMD vectorized select in window algorithm
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- [x] constant-time endomorphism acceleration
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- [ ] using NAF recoding
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- [x] using GLV-SAC recoding
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- [x] constant-time windowed-endomorphism acceleration
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- [ ] using wNAF recoding
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- [x] using windowed GLV-SAC recoding
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- [ ] SIMD vectorized select in window algorithm
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- [ ] Fixed-base scalar mul
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- [ ] Multi-scalar-mul
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- [ ] Strauss multi-scalar-mul
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- [ ] Bos-Coster multi-scalar-mul
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- [ ] Pippenger multi-scalar-mul (variable-time)
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- [ ] parallelized Pippenger
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## Pairings
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- Frobenius maps
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- [x] Sparse Frobenius coefficients
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- [x] Coalesced Frobenius in towered Fields
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- [x] Coalesced Frobenius powers
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- Line functions
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- [x] Homogeneous projective coordinates
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- [x] D-Twist
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- [x] M-Twist
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- [x] Fused line add + elliptic curve add
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- [x] Fused line double + elliptic curve double
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- [ ] Jacobian projective coordinates
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- [ ] D-Twist
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- [ ] M-Twist
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- [ ] Fused line add + elliptic curve add
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- [ ] Fused line double + elliptic curve double
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- [x] Sparse multiplication line * Gₜ element
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- [x] 6-way sparse
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- [ ] Pseudo 8-sparse
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- [x] D-Twist
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- [x] M-Twist
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- Miller Loop
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- [x] NAF recoding
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- [ ] Quadruple-and-add and Octuple-and-add
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- [ ] addition chain
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- Final exponentiation
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- [x] Cyclotomic squaring
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- [ ] Karabina's compressed cyclotomic squarings
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- [x] Addition-chain for exponentiation by curve parameter
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- [x] BN curves: Fuentes-Castañeda
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- [ ] BN curves: Duquesne, Ghammam
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- [ ] BLS curves: Ghamman, Fouotsa
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- [x] BLS curves: Hayashida, Hayasaka, Teruya
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- [ ] Multi-pairing
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- [ ] Line accumulation
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- [ ] Parallel Multi-Pairing
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## Hash-to-curve
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- Clear cofactor
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- [x] BLS G1: Wahby-Boneh
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- [ ] BLS G2: Scott et al
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- [ ] BLS G2: Fuentes-Castañeda
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- [x] BLS G2: Budroni et al, endomorphism accelerated
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- [ ] BN G2
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- [ ] BW6-761 G1
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- [ ] BW6-761 G2
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- Subgroup check
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- [ ] BLS G1: Bowe, endomorphism accelerated
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- [ ] BLS G2: Bowe, endomorphism accelerated
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