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Add Ed25519 point compression encoding
This implements the point compression encoding to a byte array from https://datatracker.ietf.org/doc/html/rfc8032#section-5.1.2 as function "edwards25519_encode". It assumes the input is a point (x,y) on the edwards25519 curve, with coordinates reduced mod p_25519 = 2^255 - 19, and does not check any of that.
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| // Copyright Amazon.com, Inc. or its affiliates. All Rights Reserved. | ||
| // SPDX-License-Identifier: Apache-2.0 OR ISC | ||
|
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| // ---------------------------------------------------------------------------- | ||
| // Encode edwards25519 point into compressed form as 256-bit number | ||
| // Input p[8]; output z[32] (bytes) | ||
| // | ||
| // extern void edwards25519_encode | ||
| // (uint8_t z[static 32], uint64_t p[static 8]); | ||
| // | ||
| // This assumes that the input buffer p points to a pair of 256-bit | ||
| // numbers x (at p) and y (at p+4) representing a point (x,y) on the | ||
| // edwards25519 curve. It is assumed that both x and y are < p_25519 | ||
| // but there is no checking of this, nor of the fact that (x,y) is | ||
| // in fact on the curve. | ||
| // | ||
| // The output in z is a little-endian array of bytes corresponding to | ||
| // the standard compressed encoding of a point as 2^255 * x_0 + y | ||
| // where x_0 is the least significant bit of x. | ||
| // See "https://datatracker.ietf.org/doc/html/rfc8032#section-5.1.2" | ||
| // In this implementation, y is simply truncated to 255 bits, but if | ||
| // it is reduced mod p_25519 as expected this does not affect values. | ||
| // | ||
| // Standard ARM ABI: X0 = z, X1 = p | ||
| // ---------------------------------------------------------------------------- | ||
| #include "_internal_s2n_bignum.h" | ||
|
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| S2N_BN_SYM_VISIBILITY_DIRECTIVE(edwards25519_encode) | ||
| S2N_BN_SYM_PRIVACY_DIRECTIVE(edwards25519_encode) | ||
| .text | ||
| .balign 4 | ||
|
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| #define z x0 | ||
| #define p x1 | ||
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| #define y0 x2 | ||
| #define y1 x3 | ||
| #define y2 x4 | ||
| #define y3 x5 | ||
| #define y0short w2 | ||
| #define y1short w3 | ||
| #define y2short w4 | ||
| #define y3short w5 | ||
| #define xb x6 | ||
|
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| S2N_BN_SYMBOL(edwards25519_encode): | ||
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| // Load lowest word of x coordinate in xb and full y as [y3;y2;y1;y0]. | ||
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| ldr xb, [p] | ||
| ldp y0, y1, [p, #32] | ||
| ldp y2, y3, [p, #48] | ||
|
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| // Compute the encoded form, making the LSB of x the MSB of the encoding | ||
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| and y3, y3, #0x7FFFFFFFFFFFFFFF | ||
| orr y3, y3, xb, lsl #63 | ||
|
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| // Write back in a byte-oriented fashion to be independent of endianness | ||
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| strb y0short, [z] | ||
| lsr y0, y0, #8 | ||
| strb y0short, [z, #1] | ||
| lsr y0, y0, #8 | ||
| strb y0short, [z, #2] | ||
| lsr y0, y0, #8 | ||
| strb y0short, [z, #3] | ||
| lsr y0, y0, #8 | ||
| strb y0short, [z, #4] | ||
| lsr y0, y0, #8 | ||
| strb y0short, [z, #5] | ||
| lsr y0, y0, #8 | ||
| strb y0short, [z, #6] | ||
| lsr y0, y0, #8 | ||
| strb y0short, [z, #7] | ||
|
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| strb y1short, [z, #8] | ||
| lsr y1, y1, #8 | ||
| strb y1short, [z, #9] | ||
| lsr y1, y1, #8 | ||
| strb y1short, [z, #10] | ||
| lsr y1, y1, #8 | ||
| strb y1short, [z, #11] | ||
| lsr y1, y1, #8 | ||
| strb y1short, [z, #12] | ||
| lsr y1, y1, #8 | ||
| strb y1short, [z, #13] | ||
| lsr y1, y1, #8 | ||
| strb y1short, [z, #14] | ||
| lsr y1, y1, #8 | ||
| strb y1short, [z, #15] | ||
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| strb y2short, [z, #16] | ||
| lsr y2, y2, #8 | ||
| strb y2short, [z, #17] | ||
| lsr y2, y2, #8 | ||
| strb y2short, [z, #18] | ||
| lsr y2, y2, #8 | ||
| strb y2short, [z, #19] | ||
| lsr y2, y2, #8 | ||
| strb y2short, [z, #20] | ||
| lsr y2, y2, #8 | ||
| strb y2short, [z, #21] | ||
| lsr y2, y2, #8 | ||
| strb y2short, [z, #22] | ||
| lsr y2, y2, #8 | ||
| strb y2short, [z, #23] | ||
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| strb y3short, [z, #24] | ||
| lsr y3, y3, #8 | ||
| strb y3short, [z, #25] | ||
| lsr y3, y3, #8 | ||
| strb y3short, [z, #26] | ||
| lsr y3, y3, #8 | ||
| strb y3short, [z, #27] | ||
| lsr y3, y3, #8 | ||
| strb y3short, [z, #28] | ||
| lsr y3, y3, #8 | ||
| strb y3short, [z, #29] | ||
| lsr y3, y3, #8 | ||
| strb y3short, [z, #30] | ||
| lsr y3, y3, #8 | ||
| strb y3short, [z, #31] | ||
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| // Return | ||
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| ret | ||
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| #if defined(__linux__) && defined(__ELF__) | ||
| .section .note.GNU-stack,"",%progbits | ||
| #endif |
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| (* | ||
| * Copyright Amazon.com, Inc. or its affiliates. All Rights Reserved. | ||
| * SPDX-License-Identifier: Apache-2.0 OR ISC | ||
| *) | ||
|
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| (* ========================================================================= *) | ||
| (* Encoding of point on edwards25519 in its compressed form. *) | ||
| (* ========================================================================= *) | ||
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| needs "arm/proofs/base.ml";; | ||
| needs "common/ecencoding.ml";; | ||
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| (**** print_literal_from_elf "arm/curve25519/edwards25519_encode.o";; | ||
| ****) | ||
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| let edwards25519_encode_mc = | ||
| define_assert_from_elf "edwards25519_encode_mc" "arm/curve25519/edwards25519_encode.o" | ||
| [ | ||
| 0xf9400026; (* arm_LDR X6 X1 (Immediate_Offset (word 0)) *) | ||
| 0xa9420c22; (* arm_LDP X2 X3 X1 (Immediate_Offset (iword (&32))) *) | ||
| 0xa9431424; (* arm_LDP X4 X5 X1 (Immediate_Offset (iword (&48))) *) | ||
| 0x9240f8a5; (* arm_AND X5 X5 (rvalue (word 9223372036854775807)) *) | ||
| 0xaa06fca5; (* arm_ORR X5 X5 (Shiftedreg X6 LSL 63) *) | ||
| 0x39000002; (* arm_STRB W2 X0 (Immediate_Offset (word 0)) *) | ||
| 0xd348fc42; (* arm_LSR X2 X2 8 *) | ||
| 0x39000402; (* arm_STRB W2 X0 (Immediate_Offset (word 1)) *) | ||
| 0xd348fc42; (* arm_LSR X2 X2 8 *) | ||
| 0x39000802; (* arm_STRB W2 X0 (Immediate_Offset (word 2)) *) | ||
| 0xd348fc42; (* arm_LSR X2 X2 8 *) | ||
| 0x39000c02; (* arm_STRB W2 X0 (Immediate_Offset (word 3)) *) | ||
| 0xd348fc42; (* arm_LSR X2 X2 8 *) | ||
| 0x39001002; (* arm_STRB W2 X0 (Immediate_Offset (word 4)) *) | ||
| 0xd348fc42; (* arm_LSR X2 X2 8 *) | ||
| 0x39001402; (* arm_STRB W2 X0 (Immediate_Offset (word 5)) *) | ||
| 0xd348fc42; (* arm_LSR X2 X2 8 *) | ||
| 0x39001802; (* arm_STRB W2 X0 (Immediate_Offset (word 6)) *) | ||
| 0xd348fc42; (* arm_LSR X2 X2 8 *) | ||
| 0x39001c02; (* arm_STRB W2 X0 (Immediate_Offset (word 7)) *) | ||
| 0x39002003; (* arm_STRB W3 X0 (Immediate_Offset (word 8)) *) | ||
| 0xd348fc63; (* arm_LSR X3 X3 8 *) | ||
| 0x39002403; (* arm_STRB W3 X0 (Immediate_Offset (word 9)) *) | ||
| 0xd348fc63; (* arm_LSR X3 X3 8 *) | ||
| 0x39002803; (* arm_STRB W3 X0 (Immediate_Offset (word 10)) *) | ||
| 0xd348fc63; (* arm_LSR X3 X3 8 *) | ||
| 0x39002c03; (* arm_STRB W3 X0 (Immediate_Offset (word 11)) *) | ||
| 0xd348fc63; (* arm_LSR X3 X3 8 *) | ||
| 0x39003003; (* arm_STRB W3 X0 (Immediate_Offset (word 12)) *) | ||
| 0xd348fc63; (* arm_LSR X3 X3 8 *) | ||
| 0x39003403; (* arm_STRB W3 X0 (Immediate_Offset (word 13)) *) | ||
| 0xd348fc63; (* arm_LSR X3 X3 8 *) | ||
| 0x39003803; (* arm_STRB W3 X0 (Immediate_Offset (word 14)) *) | ||
| 0xd348fc63; (* arm_LSR X3 X3 8 *) | ||
| 0x39003c03; (* arm_STRB W3 X0 (Immediate_Offset (word 15)) *) | ||
| 0x39004004; (* arm_STRB W4 X0 (Immediate_Offset (word 16)) *) | ||
| 0xd348fc84; (* arm_LSR X4 X4 8 *) | ||
| 0x39004404; (* arm_STRB W4 X0 (Immediate_Offset (word 17)) *) | ||
| 0xd348fc84; (* arm_LSR X4 X4 8 *) | ||
| 0x39004804; (* arm_STRB W4 X0 (Immediate_Offset (word 18)) *) | ||
| 0xd348fc84; (* arm_LSR X4 X4 8 *) | ||
| 0x39004c04; (* arm_STRB W4 X0 (Immediate_Offset (word 19)) *) | ||
| 0xd348fc84; (* arm_LSR X4 X4 8 *) | ||
| 0x39005004; (* arm_STRB W4 X0 (Immediate_Offset (word 20)) *) | ||
| 0xd348fc84; (* arm_LSR X4 X4 8 *) | ||
| 0x39005404; (* arm_STRB W4 X0 (Immediate_Offset (word 21)) *) | ||
| 0xd348fc84; (* arm_LSR X4 X4 8 *) | ||
| 0x39005804; (* arm_STRB W4 X0 (Immediate_Offset (word 22)) *) | ||
| 0xd348fc84; (* arm_LSR X4 X4 8 *) | ||
| 0x39005c04; (* arm_STRB W4 X0 (Immediate_Offset (word 23)) *) | ||
| 0x39006005; (* arm_STRB W5 X0 (Immediate_Offset (word 24)) *) | ||
| 0xd348fca5; (* arm_LSR X5 X5 8 *) | ||
| 0x39006405; (* arm_STRB W5 X0 (Immediate_Offset (word 25)) *) | ||
| 0xd348fca5; (* arm_LSR X5 X5 8 *) | ||
| 0x39006805; (* arm_STRB W5 X0 (Immediate_Offset (word 26)) *) | ||
| 0xd348fca5; (* arm_LSR X5 X5 8 *) | ||
| 0x39006c05; (* arm_STRB W5 X0 (Immediate_Offset (word 27)) *) | ||
| 0xd348fca5; (* arm_LSR X5 X5 8 *) | ||
| 0x39007005; (* arm_STRB W5 X0 (Immediate_Offset (word 28)) *) | ||
| 0xd348fca5; (* arm_LSR X5 X5 8 *) | ||
| 0x39007405; (* arm_STRB W5 X0 (Immediate_Offset (word 29)) *) | ||
| 0xd348fca5; (* arm_LSR X5 X5 8 *) | ||
| 0x39007805; (* arm_STRB W5 X0 (Immediate_Offset (word 30)) *) | ||
| 0xd348fca5; (* arm_LSR X5 X5 8 *) | ||
| 0x39007c05; (* arm_STRB W5 X0 (Immediate_Offset (word 31)) *) | ||
| 0xd65f03c0 (* arm_RET X30 *) | ||
| ];; | ||
|
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| let EDWARDS25519_ENCODE_EXEC = ARM_MK_EXEC_RULE edwards25519_encode_mc;; | ||
|
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| (* ------------------------------------------------------------------------- *) | ||
| (* Proof. *) | ||
| (* ------------------------------------------------------------------------- *) | ||
|
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| let p_25519 = define `p_25519 = 57896044618658097711785492504343953926634992332820282019728792003956564819949`;; | ||
|
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| let ed25519_encode = new_definition | ||
| `ed25519_encode (X,Y) = | ||
| let x = num_of_int(X rem &p_25519) | ||
| and y = num_of_int(Y rem &p_25519) in | ||
| 2 EXP 255 * x MOD 2 + y`;; | ||
|
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| let EDWARDS25519_ENCODE_CORRECT = time prove | ||
| (`!z p x y pc. | ||
| nonoverlapping (word pc,0x108) (z,32) | ||
| ==> ensures arm | ||
| (\s. aligned_bytes_loaded s (word pc) edwards25519_encode_mc /\ | ||
| read PC s = word pc /\ | ||
| C_ARGUMENTS [z; p] s /\ | ||
| bignum_pair_from_memory(p,4) s = (x,y)) | ||
| (\s. read PC s = word (pc + 0x104) /\ | ||
| (x < p_25519 /\ y < p_25519 | ||
| ==> read (memory :> bytelist(z,32)) s = | ||
| bytelist_of_num 32 (ed25519_encode (&x,&y)))) | ||
| (MAYCHANGE [PC; X2; X3; X4; X5; X6] ,, | ||
| MAYCHANGE [memory :> bytes(z,32)])`, | ||
| MAP_EVERY X_GEN_TAC [`z:int64`; `p:int64`; `x:num`; `y:num`; `pc:num`] THEN | ||
| REWRITE_TAC[C_ARGUMENTS; C_RETURN; SOME_FLAGS; NONOVERLAPPING_CLAUSES; | ||
| PAIR_EQ; bignum_pair_from_memory] THEN | ||
| STRIP_TAC THEN CONV_TAC(ONCE_DEPTH_CONV NUM_MULT_CONV) THEN | ||
| CONV_TAC(ONCE_DEPTH_CONV NORMALIZE_RELATIVE_ADDRESS_CONV) THEN | ||
| REWRITE_TAC[BIGNUM_FROM_MEMORY_BYTES] THEN ENSURES_INIT_TAC "s0" THEN | ||
| BIGNUM_LDIGITIZE_TAC "x_" `read (memory :> bytes (p,8 * 4)) s0` THEN | ||
| BIGNUM_LDIGITIZE_TAC "y_" | ||
| `read (memory :> bytes (word_add p (word 32),8 * 4)) s0` THEN | ||
| ARM_STEPS_TAC EDWARDS25519_ENCODE_EXEC (1--65) THEN | ||
| ENSURES_FINAL_STATE_TAC THEN ASM_REWRITE_TAC[] THEN | ||
| STRIP_TAC THEN REWRITE_TAC[READ_BYTELIST_EQ_BYTES] THEN | ||
| REWRITE_TAC[LENGTH_BYTELIST_OF_NUM; NUM_OF_BYTELIST_OF_NUM] THEN | ||
| MATCH_MP_TAC(MESON[MOD_LT] `x < e /\ x = y ==> x = y MOD e`) THEN | ||
| CONJ_TAC THENL | ||
| [REWRITE_TAC[ARITH_RULE `256 EXP 32 = 2 EXP (8 * 32)`] THEN | ||
| REWRITE_TAC[READ_BYTES_BOUND; READ_COMPONENT_COMPOSE]; | ||
| CONV_TAC(LAND_CONV BYTES_EXPAND_CONV) THEN ASM_REWRITE_TAC[]] THEN | ||
| DISCARD_STATE_TAC "s65" THEN | ||
| ASM_SIMP_TAC[ed25519_encode; INT_OF_NUM_REM; MOD_LT; NUM_OF_INT_OF_NUM] THEN | ||
| CONV_TAC(TOP_DEPTH_CONV let_CONV) THEN | ||
| SUBGOAL_THEN `y = y MOD 2 EXP 255` SUBST1_TAC THENL | ||
| [CONV_TAC SYM_CONV THEN MATCH_MP_TAC MOD_LT THEN | ||
| UNDISCH_TAC `y < p_25519` THEN REWRITE_TAC[p_25519] THEN ARITH_TAC; | ||
| ALL_TAC] THEN | ||
| MAP_EVERY EXPAND_TAC ["x"; "y"] THEN | ||
| REWRITE_TAC[ARITH_RULE | ||
| `z = x + y MOD 2 EXP 255 <=> | ||
| 2 EXP 255 * y DIV 2 EXP 255 + z = x + y`] THEN | ||
| CONV_TAC(ONCE_DEPTH_CONV BIGNUM_OF_WORDLIST_DIV_CONV) THEN | ||
| REWRITE_TAC[BIGNUM_OF_WORDLIST_SPLIT_RULE(1,3)] THEN | ||
| REWRITE_TAC[ARITH_RULE `2 EXP 64 * x = 2 * 2 EXP 63 * x`; MOD_MULT_ADD] THEN | ||
| REWRITE_TAC[BIGNUM_OF_WORDLIST_SING] THEN | ||
| REWRITE_TAC[bignum_of_wordlist] THEN | ||
| CONV_TAC WORD_BLAST);; | ||
|
|
||
| let EDWARDS25519_ENCODE_SUBROUTINE_CORRECT = time prove | ||
| (`!z p x y pc returnaddress. | ||
| nonoverlapping (word pc,0x108) (z,32) | ||
| ==> ensures arm | ||
| (\s. aligned_bytes_loaded s (word pc) edwards25519_encode_mc /\ | ||
| read PC s = word pc /\ | ||
| read X30 s = returnaddress /\ | ||
| C_ARGUMENTS [z; p] s /\ | ||
| bignum_pair_from_memory(p,4) s = (x,y)) | ||
| (\s. read PC s = returnaddress /\ | ||
| (x < p_25519 /\ y < p_25519 | ||
| ==> read (memory :> bytelist(z,32)) s = | ||
| bytelist_of_num 32 (ed25519_encode (&x,&y)))) | ||
| (MAYCHANGE_REGS_AND_FLAGS_PERMITTED_BY_ABI ,, | ||
| MAYCHANGE [memory :> bytes(z,32)])`, | ||
| ARM_ADD_RETURN_NOSTACK_TAC EDWARDS25519_ENCODE_EXEC | ||
| EDWARDS25519_ENCODE_CORRECT);; |
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