Abstract: This comprehensive research paper combines the results of three fundamental studies devoted to deep cryptanalysis of the elliptic curve secp256k1, which underlies the Bitcoin ecosystem and other cryptocurrencies. The paper thoroughly examines the deterministic nature of parameter generation (the Koblitz structure), analyzes algebraic anomalies, and examines the non-randomness of the base point selection. Particular attention is paid to the analysis of vulnerabilities arising when the curve operates on hidden prime moduli (supersingular configurations with Frobenius trace t=0 and embedding degree k=2), which open an attack vector using MOV reduction. Practical aspects of performance (GLV endomorphisms) and hardware vulnerabilities (fault injection) are considered. The paper includes historical precedents and examples of cryptanalytic computations in Python, SageMath, Magma, and PARI/GP.
1. Introduction: secp256k1 Specifics and Parameter Selection
Bitcoin’s cryptographic system uses the elliptic curve secp256k1 , specified by the SECG standards. The curve’s equation has an extremely simple Weierstrass form: y² = x³ + 7 over a finite prime field F_p . Unlike widely used NIST curves (such as P-256), whose parameters are pseudo-randomly generated (which has raised suspicions of backdoors), the parameters of secp256k1 were chosen deterministically based on the structure of Koblitz curves. This simplicity of the structure naturally arouses the interest of cryptanalysts: do the “simple rules” conceal specific structural anomalies or weaknesses?
Historical precedent: Trust in elliptic curve cryptography standards was seriously undermined by the scandal involving the Dual_EC_DRBG pseudorandom number generator, whose parameters, as proposed by the NSA, contained a proven mathematical backdoor. Satoshi Nakamoto chose secp256k1 because of the deterministic nature of its constants (the Koblitz curve), which reduced the risk of hidden vulnerabilities from the standard’s creators.
2. Non-randomness of structure, algebraic anomalies and performance
The choice of secp256k1 parameters, particularly the base point G and prime moduli, yields unique properties. On the one hand, the presence of an efficiently computable automorphism (thanks to the j-invariant being equal to 0) enables the use of Gallant-Lambert-Vanstone (GLV) optimization. An endomorphism of the form φ(x, y) = (λx, y) accelerates scalar point multiplication by an average of 30%, which has become critical for the performance of Bitcoin network nodes validating millions of ECDSA signatures.
On the other hand, this same structure opens the door to hardware attacks. As the IACR ePrint 2019/400 study shows, when introducing short-term hardware faults (fault injection), such as a microcontroller voltage reset, the processor can skip parameter checks. For curves with j=0, this means immediate degradation to trivial arithmetic in a degenerate state, leading to the complete extraction of the private key in a single operation.
3. Supersingular vulnerabilities and MOV reduction
The structure of the equation y² = x³ + 7 has a hidden but catastrophic vulnerability when considered over a Galois field F_p , where the field characteristic satisfies the condition p ≡ 2 (mod 3) . In such a finite field, the mapping x ↦ x³ is a bijection (permutation). This means that for every value of y there is exactly one root x , and the order of the group of points on the curve is strictly equal to p + 1 .
According to Hasse’s theorem, the order of a curve is given by #E(F_p) = p + 1 — t . Thus, the Frobenius trace is t = 0 , which is the definition of a supersingular elliptic curve . The embedding degree k for such curves is minimal ( k = 2 ). This condition ( p^2 ≡ 1 (mod p+1) ) makes the cryptosystem completely vulnerable to the MOV-reduction attack (Menezes-Okamoto-Vanstone) . Using Weil pairing, an attacker can polynomially reduce the elliptic curve discrete logarithm problem (ECDLP) to the discrete logarithm problem in the multiplicative field F_{p²}^* , which is solvable using the subexponential Index Calculus algorithm.
Historical precedent: In the early 1990s, supersingular curves were actively considered for standardization due to their ease of order computation and efficient arithmetic. The publication of the MOV algorithm in 1993 proved their inconsistency for classical ECDLP. In the modern context, attackers can exploit this feature to create hidden backdoors in IoT devices by replacing the modulus p with a value p ≡ 2 (mod 3) while keeping the parameters a = 0 and b = 7 constant.
4. Practical Cryptanalysis: 7 Hidden Prime Numbers
We present an analysis of the configuration where the secp256k1 parameters are considered over 7 small prime modules (11, 17, 23, 29, 41, 47, 53). These numbers satisfy the condition p ≡ 2 (mod 3) and demonstrate the supersingularity of the structure (t=0) at the software level.
4.1 Analysis using Python (SymPy)
from sympy import isprime primes = [11, 17, 23, 29, 41, 47, 53] for p in primes: if isprime(p) and p % 3 == 2: # Calculate the order of the curve y^2 = x^3 + 7 points = [(x, y) for x in range(p) for y in range(p) if (y**2) % p == (x**3 + 7) % p] order = len(points) + 1 # +1 for the point at infinity t = p + 1 — order print(f»p={p}, Order={order}, t={t} (Supersingular: {t == 0})»)
4.2 Analysis using SageMath
primes = [11, 17, 23, 29, 41, 47, 53]
for p
in primers:
if is_prime(p)
and p % 3 == 2: F = GF(p) E = EllipticCurve(F, [0, 7]) t = E.trace_of_frobenius()
print (f”p={p}, trace={t}, Supersingular={E.is_supersingular()}”)
4.3 Calculations in Magma
primes := [11, 17, 23, 29, 41, 47, 53];
for p
in primes
do if IsPrime(p)
and p
mod 3
eq 2
then F := FiniteField(p); E := EllipticCurve([F | 0, 7]); t := TraceOfFrobenius(E);
print “p =”, p, “t =”, t, “Supersingular:”, IsSupersingular(E);
end if ;
end for ;
4.4. Script for PARI/GP
primes = [11, 17, 23, 29, 41, 47, 53];
for (i=1, length(primes), p = primes[i];
if (isprime(p) && p % 3 == 2, E = ellinit([0, 7], p); t = elltrace(E); print(“p=”, p, “t=”, t, “Supersingular=”, (t==0)) )
5. Conclusion
An analysis of the secp256k1 parameters reveals the dual nature of this elliptic curve. On the one hand, the deterministic Koblitz structure (a = 0) ensures high performance and resistance to pseudorandom backdoors, which justifies Satoshi Nakamoto’s choice. On the other hand, the j-invariant equal to zero imposes strict algebraic constraints. Under hardware failures or with a deliberate modification of the modulus p to the class p ≡ 2 (mod 3) , the curve structure instantly degrades to a supersingular state. A vulnerability to MOV reduction allows the discrete logarithm problem on an elliptic curve to be reduced to a finite-field problem, nullifying the entire cryptographic strength of the system. Understanding these structural anomalies is critical for developing robust implementations and protecting hardware cryptomodules from side-channel attacks.