Abstract:  This paper presents a deep cryptanalytic analysis of the secp256k1 elliptic curve, which underlies the Bitcoin ecosystem. We examine the paradox of non-random parameter generation, where a special structure (the Koblitz curve) provides computational gain without theoretically losing cryptographic strength under standard conditions. Particular attention is given to the vulnerability to MOV reduction when extending the curve equation over hidden prime modules, where the curve becomes supersingular with an embedding degree of 2.

Architecture and non-randomness of secp256k1 parameters

The standard curve secp256k1 is described by an equation  y² = x³ + 7 over a large prime field. For a long time, the method for choosing the base point generator  G remained opaque. Research in 2025–2026 (in particular, the work of John Zweng) mathematically proved the non-randomness of these parameters. It was discovered that when multiplying the generator by the inverse of 2 (the operation of dividing a point by 2,  H = G·2⁻¹) for the family of SEC 2 curves (secp160k1, secp192k1, secp224k1, secp256k1), the X coordinate contains a common 152-bit constant.

This statistically rules out random selection of the generator and indicates a deterministic generation algorithm that violates the NUMS (Nothing Up My Sleeve) paradigm. However, this structure alone does not weaken ECDSA, provided a cryptographically secure random number generator is used to generate nonces (k).

A real-life example:  The opacity of Certicom’s SEC 2 standards is often compared to the Dual_EC_DRBG pseudorandom number generator, promoted by the US NSA in NIST standards. In Dual_EC_DRBG, the P and Q points were generated in an unknown way, allowing for a backdoor. In the case of secp256k1, the generator’s hidden structure didn’t lead to a direct hack, but it did raise reasonable suspicions in the cryptographic community regarding the use of unknown seed values.

Performance without losing cryptographic strength

The special structure of secp256k1 (where the parameter is  a = 0) places it in the class of Koblitz curves with complex multiplication constant  (-1+√-3)/2. This property allows for the use of efficiently computable endomorphisms (the Galois-Laval-Vencuresan (GLV) method).

Thanks to GLV endomorphism, the scalar multiplication operation  k·P is broken down into a sum  k₁P + k₂Φ(P), where  Φ is a fast coordinate transformation. This provides a performance gain of up to 30% for signing and verifying transactions. Importantly, this mathematical structure does not reduce the complexity of the discrete logarithm problem (ECDLP) for Pollard’s Rho algorithms, maintaining the stated level of 128-bit cryptographic security.

A real-life example:  Satoshi Nakamoto chose secp256k1 (over the then-popular NIST P-256) precisely because of concerns about hidden vulnerabilities in NIST’s pseudorandom parameters and to optimize Bitcoin node performance. GLV optimization was subsequently implemented in the libsecp256k1 library (Bitcoin Core), dramatically speeding up block validation.

Supersingular Variations over Hidden Primes: The MOV Vulnerability

The equation secp256k1  y² = x³ + 7 is catastrophically vulnerable when considered over a Galois field  F_q, where the characteristic is  p ≡ 2 (mod 3). In such a field, the mapping  x ↦ x³ is a permutation, meaning that for every value of ,  y there is exactly one  x. As a result, the order of the curve is strictly equal to  p + 1.

By Hasse’s theorem  #E(F_p) = p + 1 - t. Therefore, the Frobenius trace  t = 0. Curves with  t = 0 are  supersingular . For such curves, the embedding degree  k satisfies the condition  p^k ≡ 1 (mod (p+1)), which yields  k = 2. This makes them vulnerable to MOV-reduction (Menezes-Okamoto-Vanstone), which, using the Weyl pairing, reduces the ECDLP on the curve to the discrete logarithm problem in the multiplicative group of the field  F_{p²}*, solved by the subexponential algorithm Index Calculus.

A real-life example:  In the early 1990s, supersingular curves were actively proposed for cryptography due to their ease of point counting and efficient arithmetic. However, the publication of the MOV attack in 1993 demonstrated that they are completely insecure against ECDLP. Hidden parameter modifications, where an attacker replaces the modulus  p with a number  p ≡ 2 (mod 3)while leaving the parameters  a=0, b=7 unchanged, are a classic attack vector for implanting bugs in IoT devices.

Practical implementation: 7 supersingular curves

Below are examples for 4 cryptographic systems demonstrating how 7 primes (11, 17, 23, 29, 41, 47, 53, where  p ≡ 2 mod 3) make the secp256k1 structure supersingular with  t=0.

1. Python (using SymPy)

from sympy import isprime

def check_supersingular():
    # Ищем скрытые простые p = 2 mod 3
    primes = [p for p in range(10, 60) if isprime(p) and p % 3 == 2][:7]
    for p in primes:
        # Для y^2 = x^3 + 7 mod p (где p = 2 mod 3), t всегда 0
        order = p + 1
        t = p + 1 - order
        print(f"Prime: {p}, Order: {order}, Trace of Frobenius (t): {t}, Embedding Degree: 2")

check_supersingular()

2. SageMath

p_list = [11, 17, 23, 29, 41, 47, 53]
for p in p_list:
    E = EllipticCurve(GF(p), [0, 7])
    is_super = E.is_supersingular()
    t = E.trace_of_frobenius()
    print(f"p={p} | Supersingular: {is_super} | t={t} | Order={E.order()}")

3. Magma

p_list := [11, 17, 23, 29, 41, 47, 53];
for p in p_list do
    F := FiniteField(p);
    E := EllipticCurve([F!0, F!7]);
    t := TraceOfFrobenius(E);
    printf "p=%o, Order=%o, t=%o, IsSupersingular=%o\n", p, #E, t, IsSupersingular(E);
end for;

4. PARI/GP

p_list = [11, 17, 23, 29, 41, 47, 53];
for(i=1, #p_list, \
    p=p_list[i]; \
    E=ellinit([0,7], p); \
    t=elltrace(E); \
    print("p=", p, " | t=", t, " | order=", ellcard(E)) \
)